Designing and Analyzing
Cost-Sharing Mechanisms
under fundamental performance objectives
Dissertation
von Yvonne Bleischwitz
Schriftliche Arbeit zur Erlangung des Grades
Doktor der Naturwissenschaften
an der Fakultät für
Elektrotechnik, Informatik und Mathematik
der Universität Paderborn
Paderborn, 14. April 2008
Alles wird gut.
Preface
A problem is a chance for you to do your best.
Edward Kennedy "Duke" Ellington (1899–1974)
Acknowledgements
I am indepted to many people who have accompanied me on the long and winding
road to finally writing up my thesis. First and foremost, I would like to thank my
advisor Prof. Dr. Burkhard Monien for his scientific guidance and patience. I have
sincerely appreciated his continuous support. Through his own valuable research,
comments, and questions, he has encouraged and enlightened me. He taught me to
be persistent – and passionate – to accomplish a goal. I am grateful for this and
many other lessons learned.
I have especially enjoyed the relaxed and amicable atmosphere within the research
group of Prof. Dr. Monien in which I always felt very comfortable. For providing this
convenient background for my research, I would like to express thanks to all my
present and former colleagues. Special thanks go to:
- Florian Schoppmann, for the fruitful collaboration over the last two years. It was
a pleasure to work with you!
- Dr. Rainer Feldmann, for encouraging me to aim for a doctorate degree and
reassuring me when I was on the verge of giving up.
- Dr. Ulf-Peter Schröder, for giving valuable advice on scientific and non-scientific
issues.
- Bernard Bauer, Marion Rohloff, Sigrid Gundelach, Uli Ahlers, and Thomas
Thissen for assisting me in many ways.
The first three years of my research were financed by the ‘International Graduate
School of Dynamic Intelligent Systems’, which I very much appreciate.
Life would not be enjoyable without my dear friends. Thank you for sharing the
bright side of life in particular and standing by me through hard times as well!
Most importantly, I thank my family on whose constant encouragement and un-
conditional love I can always rely. Thank you for enduring my moods and keeping
me sane!
I cannot end without thanking Alberto for his love, support, and patience over
the last two years, and for his courage to be willing to break new grounds together.
Contents
1 Introduction .................................................... 1
1.1 Motivation and Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 The Model ...................................................... 13
2.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Binary Demand Cost Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Cost-Sharing Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Incentive-Compatibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 The Service Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.4 Budget-Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.5 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.6 Cost-Sharing Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.7 Cost-Sharing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 General Demand Cost Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Marginal Cost-Sharing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 The Cost-Sharing Problems ..................................... 21
3.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 The Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Scheduling on Related Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Bin Packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.3 Facility Location and Fault Tolerant Facility Location . . . . . . . . 22
3.2.4 Steiner Forest and Generalized Steiner Forest . . . . . . . . . . . . . . . 23
3.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Solution Properties for Makespan Scheduling . . . . . . . . . . . . . . . . . . . . . 26
4 Moulin Mechanisms and Cost-Sharing Methods in the
Approximate Core .............................................. 31
4.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Moulin Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
XII Contents
4.4 Lower Bounds on Budget-Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Bounds on Social Cost Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 Applications To Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.6.1 Lower Bounds on Budget-Balance and Efficiency. . . . . . . . . . . . . 35
4.6.2 Moulin Mechanisms for Identical Jobs . . . . . . . . . . . . . . . . . . . . . . 36
4.6.3 Moulin Mechanisms for the General Setting . . . . . . . . . . . . . . . . . 38
4.6.4 Efficiency Considerations for Identical Jobs . . . . . . . . . . . . . . . . . 39
4.6.5 Cost-Sharing Methods in the Approximate Core . . . . . . . . . . . . . 42
4.7 Conclusion and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Group-Strategyproof Non-Moulin Mechanisms . . . . . . . . . . . . . . . . . . 47
5.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Preference-Ordered Cost-Sharing Methods . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 Symmetric and Subadditive Costs and One Price . . . . . . . . . . . . . . . . . . 48
5.5 Symmetric Costs and Two Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5.1 Two-Price Cost-Sharing Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5.2 Validity of Two-Price Cost-Sharing Forms. . . . . . . . . . . . . . . . . . . 50
5.5.3 GSP Mechanisms for Two-Price Cost-Sharing Forms . . . . . . . . . 52
5.5.4 √17+1
4-BB Two-Price Cost-Sharing Forms for Subadditive Costs 55
5.5.5 Applications To Scheduling Identical Jobs . . . . . . . . . . . . . . . . . . 59
5.6 Non-Symmetric Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.6.1 Applications To The General Scheduling Setting . . . . . . . . . . . . . 60
5.6.2 Applications To Scheduling On Identical Machines . . . . . . . . . . . 62
5.7 Conclusion and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 Egalitarian Mechanisms ......................................... 67
6.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Collectors’ Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3.1 New Behavioral Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3.2 Sufficient Conditions for Unique Cost Shares . . . . . . . . . . . . . . . . 70
6.4 Egalitarian Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.4.1 Set Selection and Price Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.4.2 Computing Egalitarian Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 73
6.4.3 Validity of Set Selection and Price Functions . . . . . . . . . . . . . . . . 74
6.5 Egalitarian Mechanisms are CGSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.5.1 Acyclic Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.5.2 Acyclic Mechanisms are CGSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5.3 Egalitarian Mechanisms are Acyclic . . . . . . . . . . . . . . . . . . . . . . . . 77
6.6 Efficiency of Egalitarian Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.7 Computational Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.8 Applications to Scheduling and Bin Packing . . . . . . . . . . . . . . . . . . . . . . 84
6.9 Conclusion and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 Group-Strategyproof Mechanisms for General Demand . . . . . . . . . . 89
7.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.3 Generalized Moulin Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.3.1 Validity of Marginal Cost-Sharing Methods . . . . . . . . . . . . . . . . . 90
Contents XIII
7.3.2 Level Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.4 Applications to Fault Tolerant Facility Location . . . . . . . . . . . . . . . . . . . 95
7.4.1 The Marginal Cost-Sharing Method . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4.2 The Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4.3 Budget-Balance and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.4.4 Comparison to the Method of Mehta et al. . . . . . . . . . . . . . . . . . . 103
7.5 Applications to Generalized Steiner Forest . . . . . . . . . . . . . . . . . . . . . . . . 104
7.5.1 The Marginal Cost-Sharing Method . . . . . . . . . . . . . . . . . . . . . . . . 105
7.5.2 The Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.5.3 Budget-Balance and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.6 Conclusion and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A Budget-Balance and Efficiency Bounds for Moulin Mechanisms
and Acyclic Mechanisms ........................................ 109
B Incremental and Groves Mechanisms ............................ 111
B.1 Incremental (Sequential Stand Alone) Mechanisms. . . . . . . . . . . . . . . . . 111
B.2 Groves Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C Characterization of GSP Mechanisms for Submodular Costs . . . . 119
D Further Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
References .......................................................... 127
1
Introduction
1.1 Motivation and Framework
Do you always tell the truth when participating in an auction? Probably not. You
would take the opportunity to reduce your price by cheating. Not that we wish to
suggest that you have dishonest motives; we rather agree with the common eco-
nomic assumption that people act selfishly in order to maximize their profit. On the
other hand, if you sell something by auction, what would you do to prevent such
manipulation? What incentives would you offer for truth-telling?
This thesis considers truth elicitation in the context of cost sharing, where a
certain service is auctioned off by a service provider who selects which players to
serve and at what price. Next to designing the rules for this selection process, the
service provider faces another difficulty: depending on the kind of service, he has
to solve an underlying optimization problem in order to actually make the service
accessible to the selected players. In addition, all of the provider’s tasks have to be
accomplishable in a reasonable amount of time.
We examine cost-sharing problems under both players’ and provider’s objectives.
While on the application level we mainly focus on scheduling scenarios, we also con-
sider the problem of satisfying players’ connectivity requirements within a network.
In particular, our work is motivated by the following three examples:
Example 1.1. A computing center uses an auction-based approach in order to process
customers’ jobs. A customer request involves stating the required processing time of
his job and a price limit. On the basis of these requests, the computing center accepts
a selection of requests and determines corresponding prices. Minimizing the actual
cost of scheduling the chosen jobs requires the application of adequate scheduling
algorithms. A potential underlying optimization problem is the problem of schedul-
ing on related parallel machines, where machines are simply specified by speeds. A
possible cost measure is given by the maximum completion time over all jobs (the
makespan). The problem of minimizing the makespan is denoted by (Q||Cmax)(see
Section 3.2.1 and, e.g., [20, 26, 43–46, 51, 52, 58, 68, 69, 138]).
Example 1.2. A county plans to establish a huge industrial area for industries with
high energy demand (e.g., iron and steel works, chemical plants, or large cement
works). Assume that each company has already been assigned a potential site. To
provide access to the electricity network, the county needs to build several trans-
former stations that reduce voltage for distribution. As the cost for building the
transformer stations and establishing the power supply lines is to be shared among
the companies, an auction is conducted. Hence, companies with insufficient budget
2 1 Introduction
risk to be excluded. Having determined the successful companies, the county has
to decide on the number and locations of stations to erect and on the power supply
lines between stations and companies. To prevent power shortage due to malfunction-
ing stations, the county may allow companies to bid on any amount of connections
to distinct stations and then realize the number of connections purchased by the
auction. The underlying optimization problem of minimizing the total cost of a real-
ized solution is the fault-tolerant facility location problem (FaultTolerantFL).
If at most one connection to a station is allowed, this restricted problem be-
comes the facility location problem (FacilityLocation) (see Section 3.2.3 and,
e.g., [18, 23, 64, 65, 75, 77, 78, 89, 90, 119, 120, 123, 133]).
Example 1.3. Cooperating companies aim to establish a train network for exchanging
their goods. Required connections between pairs of companies have already been
identified. In order to guarantee punctual delivery of time-critical goods even in case
of track failure, the number of disjoint connections between a pair of companies
should be proportional to some priority measure. Given each company’s willingness
to contribute to the train project, it has to be decided on the exclusion or connection
reduction of financially weak companies, the payments of the remaining companies,
and on a solution that meets the requirements for the remaining companies. The
underlying optimization problem with respect to minimizing the cost of the solution
can be modeled as a generalized Steiner forest problem (GenSteinerForest), or as
the restricted Steiner forest problem (SteinerForest) if each company has at most
one connection requirement(see Section 3.2.4 and, e.g., [1, 9, 56, 74, 80, 135, 136]).
Examples 1.1 to 1.3 address providing players with a service. In the context of cost
sharing, the notion of service is a broad term. Obviously, ‘job-processing’ is a service.
On a more abstract level, ‘power supply’ or ‘being connected within a transport
network’ are services as well. Three basic tasks common to all examples are:
1. Collect bids from players indicating the amount of money they are willing to pay.
2. On the basis of these bids, determine the set of served players and their payments
(usually referred to as cost shares).
3. Compute a solution in order to establish the service for the selected players.
We assume that this whole process is conducted by a central authority, referred to
as the service provider. In order to decide about which players to serve and at what
price, the provider applies a cost-sharing mechanism, which is simply a protocol
defining the rules for this decision process. This mechanism is public knowledge.
There are many conflicting interests from several perspectives. Players are as-
sumed to be rational and to act selfishly in order to maximize their own benefit.
Thus, they may communicate untruthful bids to the service provider that differ from
their true valuations for receiving the service. Players might even collude and agree
on collective deceit if this is beneficial for all of them. The service provider’s pri-
mary goal is therefore to create incentives for telling the truth. He achieves this by
employing incentive-compatible cost-sharing mechanisms. The most common notions
of incentive compatibility are strategyproofness (SP), where no single player can
unilaterally improve his outcome by submitting an untruthful bid, or even group-
strategyproofness (GSP), where no coalition can jointly submit false bids such that
the resulting outcome is at least as good for all coalition members and strictly better
for at least one member.
In addition, the service provider wants to recover the cost incurred by serving the
selected players, while at the same time he needs to be able to offer competitive prices.
1.2 Classification 3
In the best case, leaving aside polynomial-time computability, the provider computes
a minimum cost solution for serving the selected players and applies a cost-sharing
mechanism whose cost shares sum up to the corresponding cost. A mechanism that
always achieves this is called budget-balanced (1-BB).
An economic objective is to design efficient (1-EFF) mechanisms that optimally
trade off the service cost and the valuations of the players. The most established
notion of efficiency is maximizing the social welfare, which is the sum of the served
players’ valuations minus the minimum cost of serving them. Equivalent to maxi-
mizing the social welfare is minimizing the social cost, which is the minimum cost of
serving the selected players plus the sum of the valuations of the excluded players.
In the majority of cases it will not be possible to achieve incentive-compatibility,
1-BB, and 1-EFF at the same time, since already SP places high restrictions upon the
cost shares. Furthermore, the underlying optimization problems may be NP-hard1
and the provider has to resort to approximate solutions for the sake of computabil-
ity. We follow the line of previous work (see, e.g., [16, 19, 66, 73, 76, 81, 82, 86, 97,
106, 114, 115]) and approach this problem by designing incentive-compatible mech-
anisms with approximate budget-balance and approximate efficiency guarantees. We
especially consider β-budget-balance (β-BB, β≥1), demanding that the sum of the
cost shares of the served players is not larger than βtimes the optimal cost and not
smaller than the actual cost of serving them. We focus on approximating social cost
efficiency, where a mechanism is called γ-efficient (γ-EFF, γ≥1), if the actual social
cost of the computed solution is not larger than γtimes the optimal social cost.
To summarize, given an underlying optimization problem Π, the problem is to
determine a mechanism and an approximation algorithm (both polynomial-time com-
putable), such that the mechanism is incentive-compatible, β-BB, and γ-EFF in the
strongest possible sense (i.e., preferably GSP, and β,γas small as possible), where
the actual cost is induced by the approximate solution. We term these problems
Π-cost-sharing problems. We remark that economics mainly considers cost-sharing
problems that ignore how the service is provided to the selected players. These prob-
lems are specified by a single cost function, and actual and optimal costs coincide.
Most research on cost-sharing assumes binary demand, where players are ‘served’
or ‘not served’. We also consider the general demand setting, providing service levels
ranging from 0 to some maximum number. This is of particular interest when play-
ers require different qualities of service. For connectivity problems within a network,
the service level of a player is the number of his distinct connections. More connec-
tions correspond to a higher quality of service, for reasons including throughput and
resistance to link failure.
1.2 Classification
Problem settings involving self-interested individuals who pursue competing goals are
vital research topics. Many of these settings involve making decisions while individ-
uals’ actual preferences are not publicly observable. Important applications include
auctions, pricing, deciding about public projects, and voting. Naturally, economics
and game theory provide tools for modeling and solving such problems.
Economics studies the production, distribution, and consumption of goods and
services. Its origin in its modern sense is conventionally accredited to Adam Smith [121].
1For the complexity theory basics, we refer the reader to [51, 107, 111, 134]
4 1 Introduction
Political economy, considered as a branch of the science of a statesman or
legislator, proposes two distinct objects: first, to supply a plentiful revenue or
product for the people, or, more properly, to enable them to provide such a
revenue or subsistence for themselves; and secondly, to supply the state or
commonwealth with a revenue sufficient for the public services. It proposes to
enrich both the people and the sovereign [121].
Game theory attempts to mathematically capture behavior in strategic situations,
where an individual’s success in making choices according to pre-defined strategies
depends on the choices of others. An outstanding goal is finding equilibria of strategy
choices where individuals are unlikely to change their behavior. The work of John
von Neumann and Oskar Morgenstern [132] is widely considered to be path breaking
for present-day game theory as it established the basis for the economic application
of classical game theory.
We shall attempt to utilize only some commonplace experience concerning
human behavior which lends itself to mathematical treatment and which is
of economic importance. [...] We shall find it necessary to draw upon tech-
niques of mathematics which have not been used heretofore in mathematical
economics, and it is quite possible that further study may result in the future
in the creation of new mathematical disciplines. [132].
Microeconomics is a branch of economics which examines the economic behavior
of players (including individuals and firms) and their interactions through markets,
given scarcity of resources and government regulation. Mechanism design, as a sub-
field of microeconomics, especially studies how private information can be elicited
and how this affects making a choice that responds to individual preferences. The
seminal work of Leonid Hurwicz [71, 72] marks the birth of mechanism design. It
especially introduced the key notion of incentive-compatibility. Erik Maskin [91] pio-
neered work on implementation theory which deals with the problem of the potential
co-existence of inferior equilibria along with the desired ones. Roger Myerson [102]
introduced incomplete information to mechanism design theory.2
While for a long time economic literature (and mechanism design in particular)
concerned itself mainly with incentives and not with the algorithmic efficiency of
its acquired methods, computer science traditionally did the opposite. Obviously
however, there are many applications that need to consider both incentives and com-
putational complexity. An outstanding example is the design of network protocols for
the Internet; they need to enable efficient data interchange between a large number
of autonomous systems. Most notably, the pioneering work of Nisan and Ronen [104]
initiated the study of mechanisms under computational aspects.
1.3 Related Work
We first consider binary demand cost-sharing and review the few results known for
general-demand cost-sharing at the end of this section. The optimization problems
mentioned below are standard computer science problems. Consult Chapter 3 and
Appendix D for details. When we say in the following that a mechanism is β-BB
or γ-EFF for a Π-cost-sharing problem, we indirectly imply the existence of an
approximation algorithm for Πthat induces the actual cost.
2We find it noteworthy that the most recent Nobel price for economics in 2007 has been awarded
to Hurwicz, Maskin, and Myerson for ‘laying the foundations of mechanism design theory’.
1.3 Related Work 5
General Impossibility Results
For the objectives incentive-compatibility, 1-BB, and 1-EFF, a significant negative
result has been known for over thirty years. Already in the 70’s, Green and Laf-
font [60, 61] ruled out the existence of SP mechanisms that simultaneously guarantee
1-BB and 1-EFF, even for a simple cost function. As a consequence, at least one ob-
jective has to be (at least partially) sacrificed. Moreover, Feigenbaum et al. [36]
showed that the request for SP and a constant-factor approximation of budget-
balance generally precludes a simultaneous constant-factor approximation of social
welfare efficiency. With respect to computational tractability, Feigenbaum et al. [37]
proved that there are cost functions derived from optimal solutions to SteinerTree
problems, for which the social welfare cannot be approximated to within a constant
factor in polynomial time, assuming P 6=NP.
Moulin Mechanisms
Primary works on cost sharing (see, e.g., [76, 81, 97, 106]) completely ignored the ef-
ficiency objective and focused on achieving GSP and β-BB for βas small as possible.
In this context, Moulin mechanisms [97] have attracted the greatest deal of atten-
tion. Essential ingredients of Moulin mechanisms are cross-monotonic cost-sharing
methods. A cost-sharing method is simply a function that maps each set of players
to a vector of cost shares; it is cross-monotonic, if the cost-share of a player in a spe-
cific set never increases as the set expands. Thomson [125, 126] was one of the first
to introduce cross-monotonicity. Naturally, β-BB of a cost-sharing method (defined
analogously to β-BB of a mechanism) implies β-BB of the corresponding Moulin
mechanism [76]. To summarize, β-BB and cross-monotonic cost-sharing methods
imply the existence of β-BB and GSP mechanisms realized by Moulin mechanism.
In fact, almost all known GSP mechanisms are Moulin mechanisms. Moreover,
Moulin mechanisms even characterize all 1-BB and GSP mechanisms in case that
costs are submodular, meaning that the marginal cost of adding a player set Sto a
player set Tis never more than the marginal cost of adding Sto S∩T(confer [97] and
Theorem C.3, p. 120). While there are always 1-BB and GSP Moulin mechanisms
for submodular cost (we discuss some of them in the next but one paragraph), this is
not always true for non-submodular costs. We summarize all known (im)possibility
results on the performance of Moulin mechanisms for diverse cost-sharing problems
with underlying optimization problems that induce non-submodular costs in Ta-
bles A.1 and A.2 on pages 109 and 110, and resort to discussing some of the primary
results and those directly related to this thesis.
Among the first 1-BB and cross-monotonic cost-sharing methods for non-sub-
modular costs were the methods by Kent and Skorin-Kapov [81] for optimal Span-
ningTree costs. Jain and Vazirani [76] presented a whole family of 1-BB and cross-
monotonic cost-sharing methods for these costs. As there are approximation algo-
rithms for SteinerTree and TravelingSalesman that compute costs that are
within twice the cost of an optimum spanning tree (see [93, 124, 129]), the cost-
sharing methods of [76, 81] yield 2-BB cost-sharing methods for the induced cost-
sharing problems. In addition, Könemann et al. [82] gave SteinerTree cost-sharing
problems for which no cross-monotonic cost-sharing method can be β-BB for β < 2.
Cross-monotonic cost-sharing methods that are of importance within this thesis
are the 3-BB methods proposed by Pál and Tardos [106] for FacilityLocation cost-
sharing problems, and the 2-BB methods by Könemann et al. [82] for SteinerFor-
6 1 Introduction
est cost-sharing problems. These guarantees are tight under the cross-monotonicity
requirement as shown by Immorlica et al. [73] and Könemann et al. [83].
The point of departure and motivation for this thesis is the work of Bleischwitz
and Monien [10]. For (Q||Cmax)cost-sharing problems for mmachines and njobs
with ddifferent processing times, we showed that in general, no cross-monotonic
cost-sharing methods that are β-BB for β < d can exist. Furthermore, we gave
cross-monotonic cost-sharing methods that are 2d-BB. In case that either jobs or
machines are identical (have the same processing times or speeds), we even presented
2m
m+1 -BB cross-monotonic cost-sharing methods. For cost-sharing problems that aim
to minimize the sum of completion times (denoted by (Q||PCi)), Brenner and
Schäfer[16] showed that already for the setting with one machine, cross-monotonic
cost-sharing methods can generally not be β-BB for β < n+1
2.
Motivated by the impossibility result of Feigenbaum et al. [36], Roughgarden
and Sundararajan [114] measured efficiency by social cost instead of social welfare.
They identified a criterion of cost-sharing methods termed summability, allowing for
simultaneous budget-balance and social cost efficiency approximations by Moulin
mechanisms within certain bounds. On the other hand, they as well showed that
constant-factor approximations of social welfare efficiency are already impossible for
constant costs; here, no Moulin mechanism can be better than Hn-EFF, where nis
the number of players and Hnis the n-th harmonic number3. Hence, in case that
constant costs are induced by some underlying optimization problem, O(log n)-EFF
is generally the best that can be hoped for. We remark that this is especially the
case for (Q||Cmax),FacilityLocation, and SteinerForest.
The work of Roughgarden and Sundararajan [114] allowed to show that the meth-
ods of Pál and Tardos [106] for FacilityLocation cost-sharing problems actu-
ally yield O(log n)-EFF Moulin mechanism, and that the Moulin mechanisms based
on the methods of Könemann et al. [82] for SteinerForest cost-sharing prob-
lems are O(log2n)-EFF [19], which is also asymptotically tight [115]3. Brenner and
Schäfer [16] identified a shortcoming of the approximate efficiency of the Moulin
mechanisms in [10] for (Q||Cmax)cost-sharing problems for identical machines by
showing that they are generally no better than O(n)-EFF. They presented a new
cross-monotonic cost-sharing method yielding O(log n)-EFF while slightly impair-
ing the budget-balance approximation. For a summary of the multitude of work on
both upper and lower bounds on the approximate social cost efficiency of Moulin
mechanisms we refer to Tables A.1 and A.2 on pages 109 and 110.
The Core
Closely related to finding 1-BB cross-monotonic cost-sharing methods is finding so-
lutions for cooperative games with transferable payoffs (see, e.g., [8, 96, 98, 103, 105,
137]) that lie in the core. Adapted to our scenario, the task is to determine cost
shares for a fixed set of players that exactly split their optimal service cost among
them and are stable in the sense that no coalition pays more than the optimal cost
of serving the coalition alone. The core was proposed in 1881 by Edgeworth [35] who
defined the set of coalitionally stable states of an economy as ‘final settlements’. It
was rediscovered and introduced to game theory by Gillies [53].
The β-core is a relaxation of the core, demanding that the actual service cost
is recovered while every coalition does not pay more than βtimes the optimal cost
3The original bounds from [114, 115] assumed 1-BB, due to a different notion of approximate
budget-balance. We elaborate on these two models in Section 4.5.
1.3 Related Work 7
for serving only the coalition (again, in the classical sense, actual and optimal costs
coincide and are given by a cost function).
A major drawback of this concept is that the set of players for which the cost
shares are determined is fixed. Hence, if the players fail to fully cooperate, it is not
specified how to adjust cost-shares for a coalition. Generalized concepts were de-
manded and investigated by, among others, Dutta and Ray [34], Thomson [125–127],
Sprumont [122], Chun [24], and Moulin [95]. Consequently, cost-sharing methods
were considered. Cost-sharing methods are in the β-core if for each set of play-
ers, the computed cost shares are in the β-core. Natural limitations on the β-BB
of cross-monotonic cost-sharing methods can be derived from the observation that
every β-BB cross-monotonic cost-sharing method specifies cost shares in the β-core
(confer Section 4.4, p. 33).
From the multitude of economic literature that studies the core we only mention
some that we think are of interest in this thesis. Traditionally, the non-emptiness
of the core is established by the Bondareva-Shapley theorem [15, 117]. The theorem
identifies a certain balance condition for costs to be necessary and sufficient for
the non-emptiness of the core. A way to provide lower bounds for the β-core was
proposed by Jain and Vazirani [76] for costs that are implicitly defined by minimum
solutions to covering integer programs, examples of which are SteinerTree and
FacilityLocation. For these costs, Jain and Vazirani [76] showed that βis bounded
by the integrality gap of the linear program relaxation of the problem by utilizing that
these relaxation costs are balanced in the sense of Bondareva and Shapley [15, 117].
Immorlica et al. [73] noted that by using the approach of Jain and Vazirani [76], a
lower bound of 1.463 for FacilityLocation cost-sharing problems can be obtained.
Furthermore, 2is a lower bound for SteinerTree cost-sharing problems due to the
integrality gap of to 2(see, e.g., [128]). Meggido [92] had previously shown that the
core is empty for SteinerTree cost-sharing problems. Goemans and Skutella [54]
and Chudak [23] give cost shares for FacilityLocation cost-sharing problems in
the e+2
e-core, where e+2
e≈1.736.
Cost shares based on some considerations of equity between players are for ex-
ample obtained by the Shapley value [116] and the egalitarian solution proposed by
Dutta and Ray [34]. For submodular costs, Shapley [118] showed that the Shap-
ley value is in the core, and Dutta and Ray [34] observed that this is true for the
egalitarian solution as well.
Moulin Mechanisms for Submodular Costs
In case that costs are submodular, the Shapley value and the egalitarian solution play
a major role as they are not only in the core, but give 1-BB cross-monotonic cost-
sharing methods [33, 70, 112, 122]. Moreover, cross-monotonic cost shares are also
computed by the 1-BB sequential stand alone mechanisms that consider players in a
predefined order and let them pay the marginal cost of being added to the already
selected set (confer Appendix B.1, p. 111). However, the Shapley value and the
egalitarian solution are more equitable. Furthermore, already for constant costs, the
approximation guarantee of sequential stand alone mechanisms with respect to social
cost efficiency is no better than n(Appendix B.1, p. 111). Contrary, Roughgarden
and Sundararajan [114] showed that Moulin mechanisms applying the 1-BB cross-
monotonic cost-sharing methods computed by the Shapley value are Hn-EFF.
Moreover, Moulin and Shenker [99] showed that Moulin mechanisms comput-
ing cost shares via the Shapley value have the smallest social welfare efficiency loss
8 1 Introduction
among all Moulin mechanisms. Mutuswami [101] proved that, under some technical
restrictions, Moulin mechanisms with cost shares computed by the egalitarian solu-
tion maximize the expected size of the set of service-receiving players over all Moulin
mechanisms, assuming that valuations are independently and identically distributed.
The problem of using the Shapley value or the egalitarian solution is that they
are in general not polynomial-time computable. However, for Multicast problems
with fixed transmission trees which induce submodular costs, Feigenbaum et al. [37]
observed that the Shapley value is polynomial-time computable. We remark that
(Q||Cmax)problems do not yield submodular cost (confer Appendix B.1, p. 111).
Group-Strategyproof Non-Moulin Mechanisms
What are the chances to improve on the performance of Moulin mechanisms? More
generally, do GSP non-Moulin mechanisms with good performance exist at all?
The first reasonable approach to design such mechanisms was made by Immorlica
et al. [73], who gave a GSP and 1-BB non-Moulin mechanism for three players and a
simple cost function. To the best of our knowledge, the only other approach was made
by Bleischwitz et al. [12], where we gave GSP and 1-BB non-Moulin mechanisms for
every symmetric cost function (costs only depend on the cardinality of the set) for
three players. We further observed in [12] that for 4players, symmetry is not sufficient
for the existence of GSP and 1-BB mechanisms. We do not elaborate on these results,
as this goes beyond the scope of this thesis.
In the case that only non-negative bids are allowed, Immorlica et al. [73] were the
first to note that there are trivial mechanisms that are GSP and 1-BB for every cost
function. According to an arbitrary order, the mechanisms find the first player that
can pay for himself and all the players behind him in that order. The whole burden of
paying the cost is put on this player while the subsequent players receive the service
for free. Next to their inequity, these mechanisms also yield the unrealistic property
that players have in general no choice to refuse the service. If we allow negative bids
to enable that choice, these mechanisms can only be guaranteed to be SP. Penna
and Ventre [108] pursue a similar approach for SteinerTree cost-sharing problems
which exhibits the same unsatisfactory properties.
These sparse results suggest that our initial questions seem exceedingly difficult
to answer. To tackle these questions, we summarize what can be concluded from the
rare characterizations of GSP mechanisms.
Moulin [97] showed that cost shares computed by a GSP mechanism are solely
dependent on the set of served players (confer Theorem C.1, p. 119). Thus, the ‘new’
mechanisms have to induce cost-sharing methods. However, these cost-sharing meth-
ods must not be cross-monotonic, as Moulin [97] established that GSP mechanisms
employing cross-monotonic cost shares are equivalent to Moulin mechanisms (confer
Corollary C.4, p. 123).
A result by Immorlica et al. [73] in particular implies that a GSP mechanism
yields cross-monotonic cost shares if and only if it is upper-continuous, i.e., if for
every player iit holds that if igets the service for every bid value greater than
xholding other bids fixed, then igets the service if he bids x. They also showed
that a 1-BB and GSP mechanism yields cross-monotonic cost shares if and only if
the computed cost shares lie in the core. Hence, the ‘new’ mechanisms must violate
upper-continuity and, if 1-BB is aspired, must not lie in the core.
Penna and Ventre [109] characterize upper-continuous and GSP mechanisms that
offer no opportunity for players to refuse the service.
1.3 Related Work 9
The characterization by Moulin [97] tells us that for submodular costs, one has to
resort to mechanisms that are not 1-BB. In case that costs are supermodular, meaning
that the marginal cost of adding a player set Sto a player set Tis never less than
the marginal cost of adding Sto S∩T, the sequential stand alone mechanisms can
be adapted to be 1-BB and GSP. Due to their unsatisfactory efficiency and the fact
that (Q||Cmax)cost-sharing problems do not induce supermodular costs in general
(see Appendix B.1, p. 111), we do not further follow this approach.
Relaxing GSP
The scarcity of contributions to finding GSP non-Moulin mechanisms with reason-
able performance or characterizing them, respectively, suggests that these tasks are
intrinsically complicated. An alternative is to consider relaxations of GSP.
Mehta et. al [94] proposed acyclic mechanisms that strictly generalize Moulin
mechanisms and outperform Moulin mechanisms in many cases (confer Table A.3,
p. 110). The price they pay is the deterioration of performance with respect to
incentive-compatibility as acyclic mechanisms can only be guaranteed to be weakly
GSP (WGSP), meaning that only those coalitions are prevented that make all of its
members better off. Specific acyclic mechanisms were given by Devanur et al. [31]
for SetCover and FacilityLocation cost-sharing problems. At the time when
they were introduced, only SP was established. Mehta et al. [94] reconsidered these
mechanisms and obtained the efficiency approximations in Table A.3 on page 110.
Further restricting incentive-compatibility to SP, 1-BB and SP can always be
achieved by the sequential stand alone mechanisms (lacking good efficiency approx-
imations). On the other hand, 1-EFF and SP cost-sharing mechanisms can be de-
rived from the seminal Groves mechanisms [25, 63, 131] which are essentially the
only 1-EFF mechanisms among all SP mechanisms [60]. The main two drawbacks
of Groves mechanisms are that they cannot guarantee a reasonable budget-balance
approximation [61], and that they are generally not GSP [62] (confer Appendix B.2,
p. 114, Example B.7, p. 116, Example B.8, p. 117, and Example B.9, p. 117).
General Demand Cost-Sharing
For general-demand cost-sharing, incremental and acyclic mechanisms have been
generalized [94, 97]. However, incremental mechanisms are only known to be GSP
for costs that are either sub- or supermodular (in a generalized way). Specifically,
this questions their applicability to general demand cost-sharing when already the
restricted binary demand problems do not induce sub- or supermodular costs. This
is in particular the case for FaultTolerantFL and GenSteinerForest cost-
sharing problems (confer Appendix B.1, p. 111).
For each FaultTolerantFL cost-sharing problem, Mehta et al. [94] gave O(L2)-
BB and O(L2·(1 + log n))-EFF generalized acyclic mechanisms, where Lis the
maximum connectivity requirement. In the full version of their paper, they further
gave Hn-BB and (Hn·(Hn+HL+ 1))-EFF generalized acyclic mechanisms for the
non-metric case4.
4These results are adjusted to our notion of β-BB, confer Section 4.5.
10 1 Introduction
1.4 Summary of Results
Next to developing and generalizing Moulin mechanisms, we tackle one of the most
important tasks for present cost-sharing: designing non-Moulin mechanisms with
comparable or even better performance than Moulin mechanisms.
•We investigate what can be achieved for symmetric costs if we hold up the demand
for GSP (published in [12]). Already for 4players, 1-BB is generally infeasible.
We omit this result and rather show that √17+1
4-BB can be obtained if costs
are subadditive as well, where √17+1
4≈1.28 (Theorem 5.14, p. 55). Notably, we
identify symmetric and subadditive costs for which no Moulin mechanism can be
better than 2-BB (Theorem 4.7, p. 35). For any set of served players, our √17+1
4-
BB mechanisms charge at most two different cost shares and are polynomial-time
computable if the cost function can be evaluated in polynomial time. We further
show that √17+1
4-BB is tight when using our technique (Theorem 5.17, p. 58).
Generally, we give mechanisms for symmetric costs that use so-called two-price
cost-sharing forms (2P-CSFs) and show that our mechanisms are GSP if the
employed 2P-CSFs meet a certain validity requirement (Theorem 5.12, p. 54).
Moreover, polynomial-time computability of a 2P-CSF implies polynomial-time
computability of the corresponding mechanism (Lemma 5.13, p. 55).
The drawback of our framework is that it is generally only applicable if efficiency
is only a secondary goal, as we identify symmetric and subadditive costs for which
the corresponding mechanism is no better than Ω(n)-EFF (Lemma 5.20, p. 60).
•We break new grounds on assumptions for coalition forming by proposing that
coalitions are unlikely to form if some member would lose service and are likely
to form if at least one member wins service, even when paying his true valuation
(published in [11]). We term mechanisms that prevent coalitions with respect to
this behavior GSP against collectors (CGSP) (confer Definition 6.1, p. 68).
We show that in particular, CGSP is incomparable to GSP, and that CGSP
is stronger than all relaxations of GSP considered so far (Lemma 6.3, p. 68).
Surprisingly, we show that already a relaxation of both CGSP and GSP induces
a unique cost-sharing method (Theorem 6.4, p. 70). This strictly improves on the
characterization result by Moulin [97] (confer Theorem C.1, p. 119).
Utilizing the idea of [34] for computing the egalitarian solution for submodular
costs, we give egalitarian mechanisms that are CGSP and 1-BB for arbitrary costs
(Lemma 6.9, p. 74 and Theorem 6.10, p. 75). We establish their 2Hn-EFF in case
that costs are subadditive (thus in particular for submodular costs) and show that
this is tight up to a factor of 2(Theorems 6.18, p. 79 and 6.23, p. 80).
On the other hand, approximability of social cost efficiency for arbitrary costs
is infeasible (Lemma 6.24, p. 80). However, the cost function that provides this
impossibility result is rather unnatural as costs ‘explode’. We extract a condition
on the cost shares that quantifies efficiency loss and thus may be helpful for
investigating the approximate social cost efficiency of egalitarian mechanisms for
reasonable cost functions in the future (confer Theorem 6.21, p. 79).
We further provide a framework for polynomial-time computability of egalitarian
mechanisms in case that each player is endowed with some size. The clue here are
monotonic cost functions that do not increase when one player is substituted by
another player with smaller size (see Section 6.7, p. 80).
1.4 Summary of Results 11
•We apply Moulin mechanisms and our new mechanisms to (Q||Cmax)cost-sharing
problems and their subproblems5. For their β-BB and γ-EFF, see Table 1.1.
Table 1.1. BB and EFF guarantees of best known polynomial-time mechanisms for (Q||Cmax),
EFF and BB entries in one ‘box’ are simultaneously achieved by one mechanism.
GSP GSP CGSP
Moulin Mech. 2P-CSF Mech. Egalitarian Mech.
Problem BB EFF BB BB EFF
(Q||Cmax)2d 2d ·(1+Hn)√17+1
4·d 2 4Hn
(Thm. 4.14, p. 38), [14] (Lem. 5.21, p. 60), [12] (Thm. 6.34, p. 86), [11]
(P||Cmax)12m
m+1 Ω(n)√17+1
4·d 1 +ε2(1+ε)Hn
[10] [16] (Lem. 5.21, p. 60), [12] (Thm. 6.35, p. 86), [11]
2m−1
m
2m−1
m·(1 + Hn)4
3−1
3m 2(4
3−1
3m )Hn
[16] [16] (Thm. 6.32, p. 84), [11]
(Q|pi=1|Cmax)2m
m+1
2m
m+1·(1+Hn)√17+1
41 2Hn
(Thm. 4.12, p. 36), [14] (Thm. 5.18, p. 59), [12] (Lem. 6.36, p. 86), [11]
(P|pi=1|Cmax)2m
m+1
2m
m+1·(1+Hn)1 1 2Hn
(Thm. 4.12, p. 36), [14] (Lem. 5.19, p. 59), [12] (Lem. 6.36, p. 86), [11]
n, m, d, Hn: number of jobs, machines, different processing times, n-th harmonic number
1CGSP mechanisms: Upper result based on PTAS with running time exponential in 1
ε
– A fine grained analysis for our Moulin mechanisms for (Q|pi=1|Cmax)estab-
lishes O(1)-EFF for many cases (Theorem 4.16, p. 39). We further show in
Section 4.6.1 that no Moulin mechanism for (Q||Cmax)cost-sharing problems
can be better than (max{d, 2m
m+1 })-BB and (max{d, Hn})-EFF in general.
– We generalize our 2P-CSFs and obtain GSP and 1-BB for a scheduling problem
with non-symmetric costs (see Section 5.6.2).
– The β-BB with β > 1for egalitarian mechanisms is due to polynomial-time
computability. In addition, the egalitarian mechanisms for (Q||Cmax)stem
from 2-BB and 4Hn-EFF egalitarian mechanisms for the related BinPacking
cost-sharing problems (Theorem 6.33, p. 85). We further obtain polynomial-
time 1-BB and 2Hn-EFF egalitarian mechanisms for other scheduling models.
– Despite the devastating bound of dfor the β-BB of cross-monotonic cost-
sharing methods for (Q||Cmax)cost-sharing problems, we give cost-sharing
methods in the 2m
m+1 -core (Theorem 4.17, p. 42), which is the best possible
(Theorem 4.8, p. 36).
•For general demand, we generalize Moulin mechanisms and provide a general
framework for designing GSP mechanisms (published in [13], see Section 7.3). For
FaultTolerantFL cost-sharing problems, these mechanisms strictly improve
on the results by Mehta et al. [94] by tightening weak GSP to GSP and reducing
approximate budget-balance and efficiency each by a factor of L(the maximum
connectivity requirement) (Theorem 7.13, p. 96). We are the first to consider
GenSteinerForest cost-sharing problems, and give GSP mechanisms that are
O(log L)-BB and O(log2n·log L)-EFF (Theorem 7.23, p. 104). For both problems,
our mechanisms are polynomial-time computable.
5(P||Cmax): identical speeds of machines, (Q|pi=1|Cmax): identical job-processing times,
(P|pi=1|Cmax): both identical speeds and processing times (confer Section 3.2.1)
12 1 Introduction
1.5 Publications
The results described in this thesis are published in parts as joint work in the Pro-
ceedings of the 6th Italian Conference on Algorithms and Complexity (CIAC’06) [10],
the Proceedings of the 32th International Symposium on Mathematical Foundations
of Computer Science (MFCS’07) [12], the Proceedings of the 3rd International Work-
shop on Internet and Network Economics (WINE’07) [11], the Elsevier Information
Processing Letters (IPL’08) [14], and the Proceedings of the 1st International Sym-
posium on Algorithmic Game Theory (SAGT’08) [13].
The work from [10] has also been accepted for publication in the Elsevier Journal
of Discrete Algorithms.
1.6 Organization
Chapter 2 explains the general cost-sharing model and gives the necessary defini-
tions. The cost-sharing problems with their underlying optimization problems that
we consider in this thesis are presented in Chapter 3 together with related work
besides cost sharing, as well as properties that we exploit within the proofs.
Our results are arranged in Chapters 4 to 7. Chapter 4 covers Moulin mechanisms,
Chapter 5 is concerned with mechanisms on two-price cost-sharing forms, Chapter 6
examines egalitarian mechanisms, and Chapter 7 deals with generalizing Moulin
mechanisms for general demand cost-sharing.
We provide further information in the appendix. Particularly, Appendix A pro-
vides a complete overview of existing Moulin and acyclic mechanisms. With respect
to other mechanisms, we introduce incremental mechanisms in Appendix B.1, and
give consideration to the significance of Groves mechanisms for mechanism design
in Appendix B.2, where we stress their applicability to cost-sharing. Appendix C
provides a revised proof of Moulin’s theorem in [97] adapted to our notation, sta-
ting that for submodular costs, Moulin mechanisms are essentially the only 1-BB
and GSP mechanisms, and particularly implying that GSP mechanisms with cross-
monotonic cost shares are equivalent to Moulin mechanisms. Finally, Appendix D
defines further optimization problems that we discuss in the related work section and
that are stated in the tables in Appendix A.
2
The Model
2.1 Organization
Section 2.2 states the notational conventions used throughout this thesis. Most of the
results presented in this thesis are for binary demand cost sharing where players are
‘served’ or ‘not served’. Therefore, we first introduce the restricted model for binary
demand in Section 2.3. We give the necessary adaptations to general demand cost
sharing where players receive service levels in Section 2.4.
2.2 Notation
Let n∈N. We define [n] := {1, . . . , n},[n]0:= {0, . . . , n}, and denote the n-th
harmonic number by Hn:= Pn
i=1 1
i∈(log n, 1 + log n).
•Sets:
– The function in : 2[n]→ {0,1}nindicates membership in a set, where for all
S⊆[n]and all i∈[n],ini(S) = 1 ⇔i∈S.
– The rank of an element i∈S⊆[n]is given by rank(i, S):=|{j∈S|j≤i}|.
– For S⊆[n]and k∈[|S|],MINkSis the set of the ksmallest elements in S.
•Vectors:
– The vectors 0,1, and eidenote the zero, one, and i-th standard basis vector
(dimension will be clear from the context).
– For x∈Rnand S⊆[n], we define x(S) := Pi∈Sxi.
– For vectors x,y∈Rn, we write x≥yif for all i∈[n],xi≥yi.
– Given x,y∈Rnand S⊆[n], we define xS:= (xi)i∈S∈R|S|and
x−S:= x[n]\S. We define (xS,y−S)∈Rnto denote the vector containing
the components of xfor Sand the components of yfor [n]\S.
– We restrict a vector x∈Nn
0to values of at most j∈Nby defining x≤j∈[j]n
0,
where x≤j
i:= min{xi, j}for all i∈[n].
– We define xj:= in({i∈[n]|xi≥j}), indicating all entries of xwith value at
least j.
2.3 Binary Demand Cost Sharing
There are nplayers, numbered from 1to n, i.e., the set of players is [n]. We denote the
true valuation of player i∈[n]by vi∈R, and let bi∈Rbe the actual bid submitted
by player i. Players’ true valuations are private information. We call v∈Rnthe
valuation vector and b∈Rnthe bid vector.
14 2 The Model
2.3.1 Cost-Sharing Mechanisms
A bid vector b∈Rnconstitutes the input for a cost-sharing mechanism:
Definition 2.1. Acost-sharing mechanism
M= (Q, x)
is a pair of functions Q:Rn→2[n]and x:Rn→Rn
≥0, where
•Q(b)∈2[n]is the set of players to be served and
•x(b)∈Rn
≥0is the vector of cost shares.
There are three standard properties of cost-sharing mechanisms:
Definition 2.2. A cost-sharing mechanism M= (Q, x)meets
•no positive transfers (NPT), if players never get paid:
for all b∈Rn:x(b)≥0
•voluntary participation (VP), if players never pay more than they bid and are
only charged when served:
for all b∈Rn, i ∈[n] : i∈Q(b)⇒xi(b)≤biand i /∈Q(b)⇒xi(b) = 0
•consumer sovereignty (CS), if for every player i∈[n]there is a threshold bid
b+
i∈R≥0such that iis served if bidding at least b+
i, regardless of the other bids:
for all i∈[n]there is a b+
i∈R≥0such that for all b∈Rn:i∈Q(b+
i,b−i)
Note that in our model, VP and NPT imply that players may opt to not participate
(by submitting a negative bid). This property together with CS is referred to as
strict CS. An alternative approach to ensure strict CS when negative bids are not
allowed is to require that payments are always positive. The economic term for this
requirement is ‘no free riders’.
Assumption 2.1. We assume that all cost-sharing mechanisms referred to within
this thesis meet VP, NPT and strict CS, unless stated otherwise.
We consider quasi-linear utilities derived from the output of a mechanism:
Definition 2.3. The utility function ui:Rn×R→Rof player i∈[n]is given by
ui(b, vi):=vi·ini(Q(b)) −xi(b).
The second argument of uidenotes the true valuation of player i. For bi6=vi,
ui((vi,b−i), vi)and ui(b, vi) = ui((bi,b−i), vi)correspond to i’s utility for truth-
telling and for submitting the untruthful bid bi, respectively. We remark that our
definition of utilities assumes a given mechanism M= (Q, x). More generally, we
could define utilities independent of mechanisms by u0
i: 2n×Rn×R→Rwhere
u0
i(Q, x, vi) = vi·ini(Q)−xi. Then, ui(b, vi)corresponds to u0
i(Q(b), x(b), vi). As we
mostly consider utilities in conjunction with mechanisms, we use Definition 2.3 for
notational convenience.
Definition 2.4 introduces welfare equivalence of two mechanisms. In particular, if
two mechanisms are welfare-equivalent and are assumed to receive true valuations,
each player derives the same utility from both mechanisms.
Definition 2.4. Two cost-sharing mechanisms M= (Q, x)and M0= (Q0, x0)are
welfare equivalent if for all bid vectors b∈Rnand all players i∈[n]it holds that
bi·ini(Q(b)) −xi(b) = bi·ini(Q0(b)) −x0
i(b).
2.3 Binary Demand Cost Sharing 15
2.3.2 Incentive-Compatibility
Besides fulfilling the mandatory NPT, VP, and strict CS, cost-sharing mecha-
nisms should create incentives for all players to tell the truth out of self-interest.
The most common notions of incentive-compatibility are strategyproofness, group-
strategyproofness, and weak group-strategyproofness. A cost-sharing mechanism is
strategyproof if no player can strictly increase his utility by misreporting his val-
uation, independent of the bids submitted by the other players:
Definition 2.5. A mechanism is strategyproof (SP) if for every player iand every
true valuation vi∈Rthere is no bid vector b∈Rnsuch that ui(b, vi)> ui((vi,b−i), vi).
A stronger notion of incentive compatibility is group-strategyproofness that even
prevents manipulation by coalitions. A mechanism is group-strategyproof if no coali-
tion of players can jointly misreport (some or all of) their valuations such that this
strictly increases the utility of at least one of its members and does not strictly de-
crease the utility of any other member, independent of the bids from non-coalitional
players:
Definition 2.6. A mechanism is group-strategyproof (GSP) if for every coalition
K⊆[n]and every true valuation vector vK= (vi)i∈K∈R|K|, there is no bid vector
b∈Rnsuch that
•ui(b, vi)≥ui((vK,b−K), vi)for all i∈Kand
•ui(b, vi)> ui((vK,b−K), vi)for at least one i∈K.
A relaxation of GSP which is still stronger than SP is the notion of weak GSP. A
mechanism is weakly GSP if no coalition of players can jointly misreport (some or all
of) their valuations such that this strictly increases the utility of all of its members,
independent of the bids from non-coalitional players:
Definition 2.7. A mechanism is weakly GSP (WGSP) if for every coalition K⊆[n]
and every true valuation vector vK= (vi)i∈K∈R|K|, there is no bid vector b∈Rn
such that
•ui(b, vi)> ui((vK,b−K), vi)for all i∈K.
2.3.3 The Service Cost
Incentive compatibility on its own is not sufficient for a cost-sharing mechanism,
since it does not provide performance guarantees with respect to the cost of serving
the selected players. We specify this cost by the cost function C: 2[n]→R≥0and
require that C(∅) = 0 and that Cis non-decreasing, i.e., for all S, T ⊆[n]with
T⊆Sit holds that C(T)≤C(S). We define some properties of cost functions that
are relevant within this thesis:
Definition 2.8. A cost function Cis submodular if for any two sets S, T ⊆[n]it
holds that the marginal cost of adding a set of players Sto a set of players Tis never
more than the marginal cost of adding Sto the set S∩T:
C(S∪T)−C(T)≤C(S)−C(S∩T)
Definition 2.9. A cost function Cis supermodular if for any two sets S, T ⊆[n]
it holds that the marginal cost of adding a set of players Sto a set of players Tis
never less than the marginal cost of adding Sto the set S∩T:
C(S∪T)−C(T)≥C(S)−C(S∩T)
16 2 The Model
Definition 2.10. A cost function Cis subadditive if for any two sets S, T ⊆[n]it
holds that the marginal cost of adding a set of players Sto a set of players Tis never
more than the stand-alone cost of the set S:
C(S∪T)−C(T)≤C(S)
Definition 2.11. A cost function Cis symmetric if for any two sets S, T ⊆[n]it
holds that the cost of a set Sonly depends on its cardinality |S|:
|S|=|T|=⇒C(S) = C(T)
For a symmetric cost function Cwe use the respective lower case letter to define
c: [n]→R≥0by c(i):=C(S)for every i∈[n]and an arbitrary S⊆[n]with |S|=i.
For applications, costs typically stem from solutions to a combinatorial minimization
problem and are defined only implicitly (confer Examples 1.1– 1.3). In the follow-
ing, C(S)denotes the value of a minimum-cost solution for the instance induced
by the player set S. Since computing optimal costs may take exponential time, the
service provider therefore resorts to approximate solutions with actual costs C0(S).
Section 2.3.6 explains in detail how these costs are derived from optimization prob-
lems.
2.3.4 Budget-Balance
The budget-balance performance measure relates the overall cost share to the cost
functions Cand C0. We want a mechanism M= (Q, x)to recover the cost C0(Q(b))
incurred by serving the selected players Q(b), while at the same time the overall
cost-share should be reasonably bounded with respect to the optimal cost C(Q(b))
of serving Q(b):
Definition 2.12. A mechanism M= (Q, x)is β-budget-balanced (β-BB) for β∈
R≥1and cost functions Cand C0if for all b∈Rnit holds that
C0(Q(b)) ≤X
i∈Q(b)
xi(b)≤β·C(Q(b)) .
We say that a mechanism is budget-balanced if it is 1-BB. In that case, C0=C.
We remark that an alternative definition of β-BB has been used in some works
(e.g. [14, 19, 94, 113–115]), where 1
β·C0(Q(b)) ≤Pi∈Q(b)xi(b)≤C(Q(b)) is required.
2.3.5 Efficiency
With respect to the efficiency performance measure, we introduce two established
ways to quantify efficiency loss: the social welfare objective and the social cost objec-
tive. As both objectives are defined on the true valuations of the players, a mechanism
has to rely on receiving truthful bids in order to meet these objectives. Naturally,
the question if all players can be assumed to bid truthfully within a specific scenario
depends on the underlying interpretation of incentive-compatibility.
A mechanism is social welfare efficient for a cost function Cif, assuming truthful
bids, it always selects a set Sof players that maximizes Pi∈Smax{vi,0}−C(S). It is
social cost efficient for Cif, assuming truthful bids, it always selects a set of players
2.3 Binary Demand Cost Sharing 17
that minimizes C(S) + Pi/∈Smax{vi,0}. Formally, for a set S⊆[n], true valuations
v, and a cost function C, define the social welfare SW C(S, v)by
SW C(S, v):=X
i∈S
max{vi,0}−C(S).
Correspondingly, we define the social cost SC C(S, v)to be
SC C(S, v):=C(S) + X
i/∈S
max{vi,0}.
Definition 2.13. A mechanism M= (Q, x)is social welfare efficient for cost func-
tion Cif for all true valuations v∈Rnand all sets S⊆[n]it holds that
SW C(Q(v),v)≥SW C(S, v).
Definition 2.14. A mechanism M= (Q, x)is social cost efficient for cost function
Cif for all true valuations v∈Rnand all sets S⊆[n]it holds that
SC C(Q(v),v)≤SC C(S, v).
Since SC C(S, v) = Pi∈[n]max{vi,0} − SW C(S, v)for every cost function C,
every v∈Rn, and every S⊆[n], a subset maximizes the social welfare if and only if
it minimizes the social cost. We refer to social welfare efficient (and thus social cost
efficient) mechanisms simply as efficient mechanisms.
In this thesis, we consider approximating the social cost objective:
Definition 2.15. A mechanism M= (Q, x)is γ-social cost efficient (γ-EFF) for
γ∈R≥1and cost functions Cand C0if for all true valuations v∈Rnand all
subsets S⊆[n]it holds that
SC C0(Q(v),v)≤γ·SC C(S, v).
2.3.6 Cost-Sharing Problems
A cost-sharing problem in the classical economic sense is simply specified by a cost
function C. The aim is to define a mechanism with strong incentive-compatibility and
good performance with respect to β-BB and γ-EFF, where C0=Cin Definitions 2.12
and 2.15. In a sense, a cost-sharing problem is a multiobjective optimization problem,
and an optimal solution is a mechanism that is GSP, 1-BB, and 1-EFF (which we
have discussed to be generally infeasible in Section 1.3).
Definition 2.16. A binary demand cost-sharing problem is specified by a cost func-
tion C: 2[n]→R≥0. The aim is to give a cost-sharing mechanism Msuch that
1. Mis incentive-compatible in a sense as strong as possible.
2. Mis β-BB for C,C0=C, and β∈R≥1as small as possible.
3. Mis γ-EFF for C,C0=C, and γ∈R≥1as small a possible.
In this thesis we especially consider cost-sharing problems for which Cis only
implicitly defined by an underlying optimization problem. Furthermore, we seek for
polynomial-time computability of cost-sharing mechanisms as well as solutions to
provide the service to players. The latter requires the application of approximation
algorithms where the actual costs C0come into play.
18 2 The Model
In order to clarify how the cost functions Cand C0are actually defined, consider
an optimization problem Πwhose instances are partially defined by a set of players.
Let (IS)S⊆[n]be a tuple of 2ninstances of Πthat only differ from each other by
the set of players. All other parameters, especially player-specific data, are fixed. For
every S⊆[n], we define C(S)to be the value of a minimum cost solution for instance
IS. Given an approximation algorithm ALG for Π, we further define ALG(IS)to be
the solution computed by ALG for instance ISand CALG(S)to be the value of this
solution. Then, C0(S):=CALG(S)for all S⊆[n].
Definition 2.17. A binary demand Π-cost-sharing problem for an optimization
problem Πis specified by a tuple of instances (IS)S⊆[n]of Π. The aim is to give
a cost-sharing mechanism Mand an approximation algorithm ALG such that for C
and CALG defined by (IS)S⊆[n],
1. Mis incentive-compatible in a sense as strong as possible.
2. Mis β-BB for C,CALG, and β∈R≥1as small as possible.
3. Mis γ-EFF for C,CALG, and γ∈R≥1as small a possible.
4. Mand ALG are polynomial-time computable in the size of the succinct represen-
tation of (IS)S⊆[n].
We conclude this subsection with a note on the terminology. When considering a
cost-sharing problem specified by cost function C, we say that a mechanism Mis
β-BB for Cmeaning that Mis β-BB in the sense of Definition 2.12 with C0=C.
When considering a Π-cost-sharing problem specified by (IS)S⊆[n], the cost function
Cimmediately results from (IS)S⊆[n]. Thus, given an algorithm ALG for Π, we simply
say that a mechanism Mis β-BB for CALG meaning that Mis β-BB in the sense
of Definition 2.12 for Cand C0=CALG. Furthermore, when we do not want to
specify the approximation algorithm, we only say that Mis β-BB. We adapt this
terminology to γ-EFF as well. Furthermore, we use it for β-BB and the β-core for
cost-sharing methods introduced in Section 2.3.7.
2.3.7 Cost-Sharing Methods
For the decision process of selecting players and determining their payments in par-
ticular, many mechanisms employ cost-sharing methods:
Definition 2.18. Acost-sharing method is a function ξ: 2[n]→Rn
≥0that maps each
set of players S⊆[n]to a vector of cost shares ξ(S), where for all S⊆[n]and all
i /∈Sit holds that ξi(S) = 0.
Notably, for any GSP mechanism M= (Q, x)we can define a unique cost-sharing
method ξby setting ξ(S):=x(b)for b∈Rwith bi=b+
iif i∈Sand bi<0otherwise.
This can be deduced from a result by Moulin [97] (and a more general result in this
thesis), stating that for all b,b0with Q(b) = Q(b0)it holds that x(b) = x(b0)(confer
Theorems 6.4 and C.1).
β-BB of cost-sharing mechanisms directly translates to cost-sharing methods:
Definition 2.19. A cost-sharing method ξis β-budget-balanced (β-BB) for
β∈R≥1and cost functions Cand C0if for all S⊆[n]it holds that
C0(S)≤X
i∈S
ξi(S)≤β·C(S).
2.4 General Demand Cost Sharing 19
In terms of stability and fairness there are two main attributes of cost-sharing meth-
ods, the β-core property and cross-monotonicity. The β-core requires that the actual
cost is recovered while no coalition jointly pays more than βtimes the cost of serving
only the coalition:
Definition 2.20. A cost-sharing method ξis in the β-core for β∈R≥1and cost
functions Cand C0if for all S, T ⊆[n]with T⊆Sit holds that
•Pi∈Sξi(S)≥C0(S)and
•Pi∈Tξi(S)≤β·C(T).
Cross-monotonicity requires that the cost-share charged to any player in a group
does not increase as the group expands:
Definition 2.21. A cost-sharing method ξ: 2[n]→Rn
≥0is cross-monotonic if for all
S, T ⊆[n]and all i∈[S]it holds that ξi(S∪T)≤ξi(S).
The rather abstract α-summability property was introduced as an important tool
that along with the approximate budget-balance of a cost-sharing method character-
izes the approximate social cost efficiency of the corresponding Moulin mechanisms
(confer Section 4.5).
Definition 2.22. A cost-sharing method ξ: 2[n]→Rn
≥0is α-summable (α-SUM) for
α∈R≥0and cost function Cif for all S⊆[n]and every order i1, . . . , i|S|of Swith
Sj:= {i1, . . . , ij}it holds that P|S|
j=1 ξij(Sj)≤α·C(S).
2.4 General Demand Cost Sharing
For general demand, each player i∈[n]has a maximum level of service Li∈Nhe
can receive. We define L:= maxi∈[n]{Li}to be the maximum service level. We call
a representation of service levels for each player allocation.
Definition 2.23. We define A:= [L1]0×. . . ×[Ln]0to denote the allocation space.
An allocation is a vector a∈ A.
In our model, each player i∈[n]submit his bids for the different levels as a bid
vector bi∈RLiconsisting of the marginal bids bi,` of receiving level `additionally to
level `−1. The vector viof true valuations (composed of the marginal valuations) is
represented analogously. The overall true valuation of ifor level kis then Pk
`=1 vi,`.
Our work is based on the common assumption of diminishing marginal returns, i.e.,
marginal valuations are non-increasing in the service level:
Assumption 2.2. For all i∈[n]and all v∈RLiit holds that vi,1≥. . . ≥vi,Li.
Clearly, marginal bids are subject to Assumption 2.2 as well, as otherwise lying would
be immediately discovered.
Definition 2.24. We denote the bid space by R:= RL1×. . . ×RLn.
We call V= (v1, . . . , vn)∈ R the valuation matrix and B= (b1, . . . , bn)∈ R abid
matrix.
Obviously, cost-sharing mechanisms for general demand have to compute an al-
location instead of only a set of served players:
20 2 The Model
Definition 2.25. Ageneral demand cost-sharing mechanism
M= (q, x)
is a pair of functions q:R → A and x:R → Rn
≥0, where
•q(B)∈ A is the allocation according to which players are served, and
•x(B)∈Rn
≥0is the vector of cost shares.
NPT, VP and strict CS are generalized as follows:
Definition 2.26. A general demand cost-sharing mechanism M= (q, x)meets
•NPT, if for all B∈ R :x(B)≥0.
•VP, if for all B∈ R and all i∈[n] : qi(B)>0 =⇒xi(B)≤Pqi(B)
`=1 bi,`and
qi(B) = 0=⇒xi(B) = 0.
•strict CS, if for all i∈[n]and all `∈[Li]0, there is a bid vector b+`
i∈RLisuch
that for all B∈ R :qi(b+`
i,B−i) = `.
Just like for binary demand, we consider quasi-linear utilities:
Definition 2.27. The utility function ui:R×RLi→Rof player i∈[n]is of the
form
ui(B,vi) :=
qi(B)
X
`=1
vi,` −xi(B).
The definitions of SP, GSP, WGSP, β-BB and γ-EFF naturally carry over to general
demand, where cost functions are now given by C:A → R≥0, and the social cost is
now defined by
SC C(a,V):=C(a) +
n
X
i=1
Li
X
`=ai+1
max{0, vi,`}.
We look at optimization problems Πwhose instances are partially defined by an
allocation a∈ A. Then, a general demand Π-cost-sharing problem is specified by a
tuple of instances (Ia)a∈A that only differ from each other by the allocation.
2.4.1 Marginal Cost-Sharing Methods
Our general demand mechanisms use marginal cost-sharing methods that specify
separate cost shares for each service level.
Definition 2.28. We define R≥0:= RL1
≥0×. . .×RLn
≥0to denote the space of marginal
cost shares. A marginal cost-sharing method is a function χ:A → R≥0, where for
all a∈ A, all i∈[n], and all `∈[Li]with ` > aiit holds that χi,`(a) = 0.
We also define β-BB for marginal cost-sharing methods:
Definition 2.29. A marginal cost-sharing method χis β-budget-balanced (β-BB) for
β∈R≥1and cost functions Cand C0if for all a∈ A it holds that
C0(a)≥
n
X
i=1
ai
X
`=1
χi,`(a)≤β·C(a).
3
The Cost-Sharing Problems
3.1 Organization
We give the Π-cost-sharing problems and the underlying optimization problems Π
examined in this thesis in Section 3.2. Section 3.3 provides related work on optimiza-
tion and game-theoretic settings other than cost-sharing. Section 3.4 gives relevant
properties of solutions to specific scheduling optimization problems.
3.2 The Problems
3.2.1 Scheduling on Related Machines
Given njobs, mmachines, a vector p∈Nnof processing times pifor each job i∈[n],
and a vector s∈Nmof speeds sgfor each machine g∈[m], a scheduling instance
for S⊆[n]is defined by IS:= (pS,s).
Jobs are called identical if p=1. Machines are called identical if s=1. For
arbitrary speeds, machines are termed related.
Assumption 3.1. It holds that p1≥. . . ≥pnand s1≥. . . ≥sm.
The aim is to assign the jobs in Sto the machines such that a given objective function
is minimized. Formally, an assignment for S⊆[n]is a function φ:S→[m], where
φ(i) = gindicates that job iis assigned to machine g. For T⊆S, we let mφ(T)be
the set of machines that φuses to assign T:
mφ(T):={φ(i)}i∈T.
We mainly focus on minimizing the maximum completion time over all machines,
termed makespan. Given an assignment φ:S→[m], the completion time of g∈[m]
is simply p({i∈S|φ(i)=g})
sg. We define the makespan of an optimal assignment for Sby
MSP(S) := min
φmax
g∈mφ(S)p({i∈S|φ(i) = g})
sg .
In the context of cost-sharing, we assume that the service provider acts as a
machine administrator and that each player iowns exactly one job. Thus, a player
receives the service if and only if his job is scheduled. Accordingly, we will use S⊆[n]
to denote players and jobs interchangeably. The succinct representation of (IS)S⊆[n]is
(p,s). The function MSP : 2[n]→R≥0defines the optimal service cost. For identical
22 3 The Cost-Sharing Problems
jobs, MSP is symmetric. In that case we define msp : [n]→R≥0by msp(i):=MSP(S)
for every i∈[n]and an arbitrary S⊆[n]with |S|=i(confer Definition 2.11, p. 16).
We remark that we do not further specify how jobs on a specific machine are
scheduled. However, in this thesis we marginally consider minimizing the sum of
completion times of jobs, where such a specification is necessary. The completion
time of a job iassigned to machine gdepends on the order in which all jobs assigned
to gare processed. Let Tbe the set of jobs processed on gprevious to i. Then
the completion time of iis p(T∪{i})
sg. The makespan then also corresponds to the
maximum completion time over all jobs.
Scheduling Notation
We adopt the (α|β|γ)notation introduced by Graham et al. [59] to classify scheduling
problems, where αdescribes the machine environment, βprovides job characteristics
and scheduling constraints, and γstates the objective function to minimize.
In this thesis, we consider α∈ {1,Pm, Qm}, where 1stands for the setting of one
single machine, Pmfor midentical machines, and Qmfor mrelated machines. When
mis clear from the context, we simply write Por Q. The βfield takes the values
β∈ {pi=p, pi∈N, rj,pmtn}, where pi=pmeans that all jobs have processing
time p,pi∈Nrestricts processing time to those specified by N⊆N,ridenotes the
case in which all jobs have release dates ri, and pmtn states that preemption is al-
lowed. Note that for β∈ {ri,pmtn}, schedules have to ensure that no job is executed
previous to its release date or account for preemption, respectively. Furthermore,
γ∈ {Cmax,PCi,PwiCi}, for the objectives to minimize the makespan, the sum of
all job completion times, or the weighted sum of all job completion times for a given
w∈Nn. Our core results on scheduling within this thesis are for (Q||Cmax)problems
and their subproblems (Q|pi=1|Cmax),(P||Cmax), and (P|pi=1|Cmax). However, we
touch on the other problems as well.
3.2.2 Bin Packing
Given nobjects and a vector ς∈(0,1]nof object sizes ςifor each object i∈[n], a
BinPacking instance for S⊆[n]is defined by IS:= ςS.
Assumption 3.2. Objects are ordered such that ς1≥. . . ≥ςn.
The aim is to find an assignment of the objects in Sto bins that uses a minimum
amount of bins and does not exceed the unit capacity of any bin. We let BP(S)be
the minimum number of bins needed to assign object set S.
In the context of cost-sharing, we assume that the service provider owns the bins
and that each player iowns exactly one object. Thus, a player receives the service
if and only if his object is assigned to some bin. We will use S⊆[n]to denote
players and objects interchangeably. The succinct representation of (IS)S⊆[n]is ς.
The function BP : 2[n]→R≥0defines the optimal service cost.
3.2.3 Facility Location and Fault Tolerant Facility Location
Given nplayers, a set of facilities F, a vector o∈N|F|of opening costs offor each
facility f∈F, and a function d: ([n]∪F)×([n]∪F)→N0defining the distances
between all pairs of players and facilities, a FaultTolerantFL instance for an
allocation a∈[|F|]n
0is defined by Ia:= (F, o, d, a).
3.2 The Problems 23
Assumption 3.3. We assume that dis a metric, i.e., for all i, j, k ∈[n]∪F,dsatisfies
(d(i, j) = 0 ⇔i=j),d(i, j) = d(j, i), and d(i, j)≤d(i, k) + d(k, j).
The aim is to open a set of facilities and connect each player i∈[n]to aidistinct
open facilities, such that the total opening and connection cost is minimized.
For k∈N, let Fk:= {F0⊆F| |F0| ≥ k}be all sets of facilities with cardinality
at least k. For i∈[n]and F0∈ Fmaxj{aj}, let F0
idenote a set of aiclosest distinct
facilities in F0to i. Then the cost of an optimal solution is
FTFL(a) := min
F0∈Fmaxj{aj}
X
f∈F0
of+X
i∈[n]X
f∈F0
i
d(i, f)
.
In the context of general demand cost-sharing, given maximum service levels Li
for all i∈[n]that define the allocation space A= [L1]0×. . . ×[Ln]0, we only
consider FaultTolerantFL problems with |F| ≥ maxiLiand a∈ A. A succinct
representation of (Ia)a∈A is (o, D), where D∈Nn+|F|
0×Nn+|F|
0is a triangular matrix
containing the respective values computed by function d. For clarity, we use the
representation (F, o, d). The function FTFL :A → R≥0defines the optimal service
cost.
Restricting ato a∈ {0,1}n, we obtain FacilityLocation optimization and bi-
nary demand cost-sharing problems. For a FacilityLocation cost-sharing problem,
the optimal cost function is denoted by FL : 2[n]→R≥0, where FL(S) := FTFL(a)
for all S⊆[n]and a:= in(S).
3.2.4 Steiner Forest and Generalized Steiner Forest
Given nplayers, an undirected graph G= (V, E), vector w∈N|E|of edge weights
wefor all e∈E, and vectors s∈Vnand t∈Vnrepresenting pairs (si, ti)of nodes
for each player i∈[n], a GenSteinerForest instance for an allocation a∈Nn
0is
defined by Ia:= (G, w,s,t,a).
The aim is to determine a subgraph with minimum overall edge weight that has
aiedge-disjoint paths between siand tifor all i∈[n]. The cost of an optimal solution
for allocation ais denoted by GSF(a).
Assumption 3.4. We assume a simplification allowing to use an arbitrary amount
of edge copies, where the cost of each edge copy is equal to the cost of the edge, i.e.,
we seek for a multiset of edges that contain aiedge disjoint path between siand ti
for all i∈[n].
In the context of general demand cost-sharing, given maximum service levels Li
for all i∈[n]that define the allocation space A= [L1]0×. . . ×[Ln]0, we only
consider GenSteinerForest problems with a∈ A. A succinct representation of
(Ia)a∈A is (G,s,t), where G ∈ N|V|,|V|represents a triangular adjacency matrix
with weight entries. For clarity, we use the representation (G, w,s,t). The function
GSF :A → R≥0defines the optimal service cost.
Restricting ato a∈ {0,1}n, we obtain SteinerForest optimization and binary
demand cost-sharing problems. For a SteinerForest cost-sharing problem, the
optimal cost function is denoted by SF : 2[n]→R≥0, where SF(S) := GSF(a)for all
S⊆[n]and a:= in(S).
24 3 The Cost-Sharing Problems
3.3 Related Work
Garey and Johnson showed that already (P2||Cmax)and (P||Cmax)problems are
NP-hard [51] and even strongly NP-hard [52], respectively. On the positive side,
Hochbaum and Shmoys [68] developed a PTAS for (P||Cmax)and subsequently also
a PTAS for (Q||Cmax)[69]. Their algorithms are based on a decision procedure which
tests if there exists a schedule for a given problem instance where all jobs are com-
pleted by time t. Naturally, this decision problem can be viewed as a BinPacking
problem (with variable bin sizes for (Q||Cmax)). The minimum t is computed by
a simple binary search procedure. Similar approaches have been used to develop
other approximation algorithms for (P||Cmax)and (Q||Cmax)based on the Multifit
algorithm proposed by Coffman et al. [26] (see, e.g., [20, 43, 45, 46, 138]).
A very simple approach for solving (Q||Cmax)problems is to consider the jobs in
an arbitrary order and assign each job to a machine for which the completion time
is minimized, taking into account the jobs that have been assigned already. This
approximation algorithm is known as List-Scheduling. Graham [58] showed that it
yields an approximation ratio of 2−1
mfor (P||Cmax). Since the worst case arises
when the last job has the largest processing time, Graham [58] suggested the LPT
algorithm (longest processing time first) that considers the jobs by non-increasing
processing times. LPT is optimal for (Q|pi=1|Cmax), achieves an approximation ratio
of 4
3−1
3mfor (P||Cmax)[58], and an approximation ratio of 5
3for (Q||Cmax)[44]. Its
running time is O(nlog n+nm)in general and O(nlog n+nlog m)for (P||Cmax)
when using a priority queue for job assignment. If jobs are identical, we do not have to
sort the jobs and can run LPT for (Q|pi=1|Cmax)problems in time O(nlog m), again
using a priority queue. For (P|pi=1|Cmax)problems, clearly O(n)time is sufficient.
Recently, makespan scheduling problems have been considered in a game-theoretic
setting where there is no central authority that assigns the jobs. Instead, each player
assigns his job to a machine himself. This scenario is motivated by interpreting
machines as links (e.g., different kinds of streets like highways, freeways, or lanes) on
which a player routes his traffic. The latency of his traffic then corresponds to the
completion time of the respective machine. For this scenario, one aims at computing
a Nash equilibrium assignment, meaning that no player can improve the latency
of his traffic by unilaterally deviating from the current assignment (i.e., choosing
another link). Notably, the LPT algorithm has the property that in every iteration,
the current assignment is a Nash equilibrium [42].
It is well known that Nash equilibria do not always correspond to an optimal
assignment. To better understand the performance of Nash equilibria, Papadimitriou
and Koutsoupias [85] proposed to consider the ratio between the optimum and the
worst Nash equilibrium with special regard to (P||Cmax)problems. This gave rise to
many other works on this topic, see, e.g., [30, 40, 41, 47–49, 84] and the references
therein. We also refer the interested reader to the surveys of Gairing et al. [50] and
Czumaj [29].
Makespan scheduling has also been considered for mechanism design problems in
which the private data is part of the input of the optimization problem. Nisan and
Ronen [104] introduced a model in which players own machines instead of jobs. They
considered unrelated machines, meaning that each machine ghas data tg,1, . . . , tg,n
denoting the times that gneeds to process jobs 1, . . . , n. This data is the private
information of each player. Given the players’ bids on the execution times of their
machines, a mechanism computes an allocation and payments given as a benefit to
the players. The utility of a player is the negative time his machine needs to process
3.3 Related Work 25
its assigned jobs plus the payment he receives. The goal is to design an SP mech-
anism according to these utilities while computing an allocation that approximates
the optimal makespan as good as possible. Note that in this model, approximating
budget-balance is not a primary goal; in general, the mechanism does not care how
much he pays the players.
Nisan and Ronen [104] gave a polynomial-time computable SP mechanism whose
allocation approximates the optimal makespan within a factor of n. In addition, they
showed that no SP mechanism (polynomial-time or not) can achieve a better approx-
imation factor than 2. This lower bound was recently improved by Christodoulou et
al. [22] to 1 + √2for 3or more machines. On the other hand, Nisan and Ronen [104]
gave a randomized SP mechanism that yields a 7
4-approximation with respect to the
expected makespan. This approximation was recently improved by Lu and Yu [88]
to 1.6737. Archer and Tardos [5] considered the restricted (Q||Cmax)model and de-
signed an exponential time SP mechanism that computes an optimal assignment.
Moreover, they gave a randomized SP mechanism yielding an approximation factor
of 3, which was later improved to 2by Archer [3]. In the same framework, Andelmann
et al. [2] presented a deterministic SP mechanism computing a 5-approximation.
Nisan and Ronen [104] also considered a model in which payments are given to
the players only after the execution of the jobs. In case that the mechanism knows
the times that the machines spend on computing each job afterwards, it may punish
those players who pretended to be capable of processing jobs more quickly. These
mechanisms are termed mechanisms with verification. For mechanisms with verifica-
tion for scheduling problems, we refer to the work of Nisan and Ronen [104], Auletta
et al. [6, 7], Ventre [130], and Ferrante et al. [39]. For further reading on scheduling
problems, we suggest the books [87, 110, 129] and [17].
For BinPacking, there exists a negative result stating that for any ε > 0,Bin-
Packing is NP-hard to approximate within a factor of 3
2−ε(see, e.g. [129]). Thus,
there is no PTAS for BinPacking, assuming that P 6=NP. On the other hand,
there are asymptotic PTASs for BinPacking proposed by Fernandez de la Vega
and Lueker [38] and Karmarkar and Karp [79].
We now review two simple approximation algorithms for BinPacking prob-
lems relevant to this thesis, the FFD (first fit decreasing) and the NFD (next fit
decreasing) heuristic. Here, objects are first ordered by non-increasing sizes. Accord-
ing to this order, FFD assigns an object to a new bin, if there is no existing bin that
has sufficient free capacity to hold the object. Otherwise, it is assigned to the first
such bin (where bins are ordered by the times they were first used). For NFD, an
object is assigned to a new bin, if it is the first object in the order or does not fit
into the same bin as the previous object. Otherwise, it is assigned to this bin. The
running time of NFD and FFD is O(nlog n).
Very recently, Dósa [32] showed that the tight bound of FFD is 11
9·opt+6
9. Further-
more, FFD is known to produce an optimal packing if all object sizes are powers of
two, as shown by Coffman et al. [27]. NFD is a 2-approximation algorithm (e.g. [28])
and, as shown by Murgolo [100], monotonic (contrary to FFD). Monotonicity is of
vital importance with respect to the applicability of specific results presented in this
thesis. It requires that decreasing the size of a single object does not increase the so-
lution value. For more information on BinPacking problems, we refer to the survey
of Coffman et al. [28].
26 3 The Cost-Sharing Problems
For FacilityLocation, Guha and Kuller [64] showed a lower bound of 1.463 on
polynomial-time approximability, assuming NP /∈DTIME(nO(log log n)). The first con-
stant factor approximation for FacilityLocation was given by Shmoys et al. [120],
whose 3.16-approximation algorithm is based on LP rounding. Similar techniques
were used to improve their result, e.g., by Chudak [23] and Guha and Khuller [64],
finally leading to 1.582-approximations presented by Sviridenko [123]. Jain and Vazi-
rani [77] gave a combinatorial primal-dual 3-approximation algorithm. A subsequent
primal-dual algorithm for FacilityLocation by Mahdian et al. [89] provided an
1.861-approximation. This result was further improved by Mahdian et al. [90] and
Jain et al. [75] to an 1.52-approximation. Recently, Byrka [18] presented the cur-
rently best known 1.5-approximation algorithm based on LP rounding. For a survey
on FacilityLocation problems, we confer to the lecture notes of Vygen [133].
FaultTolerantFL problems were first considered by Jain and Vazirani [78]
who gave a 3HL-approximation algorithm (where Lis the maximum requirement)
by iteratively applying their 3-approximation algorithm [77]. Algorithms based on
LP rounding improved on their result: Guha et al. [65] obtained factor 2.47 and,
currently best, Shmoys et al. [119] provided a factor of 2.076.
SteinerForest problems are NP-hard as particularly the subclass of Stein-
erTree problems is NP-hard due to Karp [80], and even APX-hard due to Bern and
Plassmann [9]. On the positive side, Agrawal et al. [1] provided a first approximation
algorithm for SteinerForest that computes 2−1
napproximations in polynomial
time. Previously, only exact solutions or approximations for special classes of graphs
had been considered (see, e.g., [135, 136]).
Agrawal et al. [1] also presented the first polynomial-time algorithm for Gen-
SteinerForest with a guarantee of (2 −1
n)·dlog(L+ 1)e, where Lis the largest
requirement. Goemans and Williamson [56] simplified and generalized the algorithm
from [1]. Jain [74] significantly improved on the results in [1, 56] by presenting a
2-approximation algorithm for GenSteinerForest. His algorithm also works for
the setting where edge copies are not allowed. For both settings, these approximation
guarantees are the best known.
We note that Agrawal et al. [1] term SteinerForest and GenSteinerForest
problems ‘minimum cost R-join’ and ‘minimum cost R-multijoin’ problems, respec-
tively. GenSteinerForest problems are also called ‘survivable network problems’
by, e.g., Chien [21] or Gomory and Hu [57].
3.4 Solution Properties for Makespan Scheduling
Properties of Optimal Solutions
Fix p∈Nnand s∈Nm. First observe that obviously, the optimal makespan is non-
decreasing, i.e., for all S, T ⊆[n]with S⊆Tit holds that MSP(S)≤MSP(T)and
for all i, j ∈[n]with i≤jit holds that msp(i)≤msp(j), respectively. Moreover, if for
a set S⊆[n]there is a t∈Nsuch that pi=tfor all i∈S, then MSP(S) = t·msp(|S|).
Further properties are summarized in Lemma 3.1 and Lemma 3.2.
Lemma 3.1. For given p∈Nnand s∈Nm, the function MSP : 2[n]→Rn
≥0as
defined in Section 3.2.1 satisfies the properties below:
3.4 Solution Properties for Makespan Scheduling 27
•For all S, T ⊆[n]it holds that :
MSP(S∪T)≤MSP(S) + MSP(T)(3.1)
•For a set S⊆[n]and each optimal assignment φ:S→[m]for S,
p(S)
s(mφ(S)) ≤MSP(S).(3.2)
Proof. Property (3.1) states that optimal makespan costs are subadditive. Con-
sider S, T ⊆[n]and two corresponding optimal assignments with cost MSP(S)and
MSP(T). Define a feasible assignment for S∪Tby simply combining the two assign-
ments and breaking ties for i∈S∩Tarbitrarily. This gives a new feasible assignment
with makespan cost not larger than MSP(S) + MSP(T).
To see (3.2), observe that by definition, p({i∈S|φ(i) = g})≤sg·MSP(S)for
all g∈mφ(S). Summing up over all g∈mφ(S)yields the desired result.
Lemma 3.2. For a given s∈Nm, the function msp : [n]→R≥0as defined in
Section 3.2.1 satisfies the properties below, with σ:= s([m]) denoting the sum of
machine speeds:
For all i, j ∈[n] : msp(i+j)≤msp(i) + msp(j),(3.3)
especially msp(2i)
2i≤msp(i)
i
For all i∈[n] : msp(i)≥i
σ(3.4)
For all i∈[n] : msp(i·σ) = i(3.5)
For all i∈ {m+ 1, . . . , n}:msp(i)≤i
σ·2m
m+ 1 (3.6)
For all i∈[n] : min
j∈[i]
msp(j)
j≥1
σ(3.7)
For all i∈[n] : If k∈[i]is maximum with (3.8)
msp(k)
k= min
j∈[i]
msp(j)
j,
then msp(i)≤2·msp(k).
Proof. Properties (3.3) and (3.4) follow from (3.1) and (3.2), respectively.
Property (3.5) is due to the fact that for each optimal assignment for i·σidentical
jobs, all machines have the same completion time, since exactly i·sgidentical jobs
are placed on machine gfor all g∈[m]. Any other assignment has to strictly increase
the completion time of at least one machine and thus the makespan.
To prove (3.6), consider an optimal assignment φfor i≥m+ 1 identical jobs.
Without loss of generality assume that the set of jobs is [i]. Let Tbe the set of
machines whose completion times are equal to msp(i). We change φto φ0by moving
a job kwith φ(k)∈Tfrom machine φ(k)to machine hif the new completion time
of his strictly smaller than msp(i). This can be done at most |T|−1times as φis
optimal. Now consider a machine g∈Twhose completion time has not changed. It
holds that
28 3 The Cost-Sharing Problems
p({j∈[i]|φ0(j) = g}) = msp(i)·sg,
and for all h∈[m]\{g}we have that
p({j∈[i]|φ0(j) = h})+1≥msp(i)·sh.
Summing up over all machines yields i+m−1≥msp(i)·σ. Consequently, with
i+ (m−1) ·i
m+1 ≥msp(i)·σwe get the bound.
In order to show Property (3.7), let minj∈[i]
msp(j)
j=msp(k)
kfor a k∈[i]. By (3.4)
it holds that msp(k)
k≥1
σ.
Finally, for (3.8), observe that msp(2k)
2k≤msp(k)
kby (3.3). If now i≥2k, obviously
kcannot be maximum. Thus, i < 2k. Since msp is non-decreasing and by (3.3),
msp(i)≤msp(2k)≤2·msp(k).
The LPT Algorithm and Properties of LPT Solutions
Given a (Q||Cmax)problem instance (pS,s)for S⊆[n], Algorithm 3.1 formally
computes the LPT assignment for S:
Algorithm 3.1 (computing LPT assignments).
Input: set S⊆[n], processing times pS∈N|S|, speeds s∈Nm
Output: assignment φ:S→[m]
1: T:= ∅.set of assigned jobs
2: while S6=∅do
3: h:= arg ming∈[m]
pmin S+p({j∈T|φ(j) = g})
sg
4: φ(min S):=h
5: T:= T∪{min S};S:= S\{min S}
6: return φ
We now formalize that LPT computes a Nash Equilibrium (in the sense of [42], see
page 24). Assume that exactly job set Tis already assigned by LPT’s assignment φ.
Then, for each job i∈Twith φ(i) = hit holds that
for all g∈[m]\h:p({j∈T|φ(j) = g}) + pi
sg≥p({j∈T|φ(j) = h})
sh
.
We utilize the Nash equilibrium property in Lemma 3.3.
Lemma 3.3. Consider a (Q||Cmax)problem instance (pS,s)for S⊆[n]. Let ˆ
S⊆S
be the jobs that LPT assigns until there is exactly one machine that has a completion
time equal to CLPT(S). Furthermore, let φ0:S→[m]be the assignment that LPT
computes for S.
•There is an optimal assignment φ:T→[m]for every T⊆ˆ
Ssuch that
mφ(T)⊆mφ0(ˆ
S).(3.9)
•If at least two jobs of ˆ
Sare assigned to the same machine then
CLPT(S)≤2·|mφ0(ˆ
S)|
|mφ0(ˆ
S)|+ 1 ·p(ˆ
S)
s(mφ0(ˆ
S)) .(3.10)
3.4 Solution Properties for Makespan Scheduling 29
Proof. First we prove Property (3.9). Fix T⊆ˆ
S. If CLPT(ˆ
S) = MSP(T), we define
φby adopting the assignment for Tgiven by φ0. If CLPT(ˆ
S)>MSP(T), consider
a machine g /∈mφ0(ˆ
S). It is sufficient to show that g /∈mφ(T)for any optimal
assignment φfor T. Among all jobs in ˆ
S, let ibe the job last assigned by LPT.
It holds that pi= min{pj|j∈ˆ
S}. Since iwas not assigned to g, it holds that
pi
sg≥CLPT(S) = CLPT(ˆ
S)>MSP(T). Consequently, no job with size larger or
equal to piis assigned to gin an optimal assignment for T. Since piis the smallest
processing time in ˆ
S, no job in Tis assigned to gin an optimal assignment. Hence,
g /∈mφ(T).
We continue to prove Inequality (3.10). For the sake of readability, we write
τ:= |mφ0(ˆ
S)|. Among all jobs in ˆ
S, let ibe the job last assigned by LPT, and define
g:= φ0(i). Especially, pi= min{pj|j∈ˆ
S}. Since the assignment computed by LPT
for ˆ
Sis a Nash equilibrium,
p({i∈ˆ
S|φ0(i) = g}) = CLPT(S)·sg
and for all h∈mφ0(ˆ
S)\g,
p({i∈ˆ
S|φ0(i) = h}) + pi≥CLPT(S)·sh.
Summation yields
p(ˆ
S)+(τ−1) ·pi≥CLPT(S)·s(mφ0(ˆ
S)) .
Since piis the smallest processing time of a job in ˆ
S, and the set ˆ
Sconsists of at
least τ+ 1 jobs, it holds that p(ˆ
S)≥(τ+ 1) ·pi. This leads to
1 + τ−1
τ+ 1·p(ˆ
S)≥CLPT(S)·s(m(ˆ
S)) ,
yielding Inequality (3.10).
Finally, we prove a property of CLPT in Lemma 3.4 that reminds of subadditivity
but is significantly weaker. It requires that all players in the one set have the same
processing time which is not larger than any processing time in the other set:
Lemma 3.4. For p∈Nnand s∈Nm, consider two (Q||Cmax)instances (pS,s)and
(pT,s)for S, T ⊆[n]with S∩T=∅and pi=tfor all i∈T, where t∈Nsuch that
t≤mini∈S{pi}. Then CLPT(S) + CLPT(T)≥CLPT(S∪T).
Proof. Let φS,φTand φS∪Tbe the assignments that LPT computes for S,Tand
S∪T, respectively. Without loss of generality we may assume that LPT for S∪T
assigns elements in Swith size tprior to elements in T, implicating φS(i) = φS∪T(i)
for all i∈S. We additionally assume without loss of generality that the runs for LPT
on input Tand S∪Tprocess the jobs in Tin the same order. Let T:= {i1, . . . , i|T|}
according to this order and define Tj:= {i1, . . . , ij}for all j∈[|T|].
Assume now that CLPT(S)+CLPT(T)< CLPT(S∪T)and consider the first element
ijin Tsuch that CLPT(S)+CLPT(Tj)< CLPT(S∪Tj). In particular, CLPT(S∪Tj−1)<
CLPT(S∪Tj). We show that
for all g∈[m] : |{i∈Tj−1|φS∪T(i) = g}| ≥ |{i∈Tj−1|φT(i) = g}| (3.11)
there exists g0∈[m] : |{i∈Tj−1|φS∪T(i) = g0}| >|{i∈Tj−1|φT(i) = g0}| (3.12)
30 3 The Cost-Sharing Problems
Assume that (3.11) does not hold for a machine h∈[m]. We get a contradiction by
CLPT(S) + CLPT(Tj)≥p({i∈S|φS∪T(i) = h}) + p({i∈Tj|φT(i) = h})
sh
≥p({i∈S∪Tj−1|φS∪T(i) = h}) + t
sh
≥p({i∈S∪Tj|φS∪T(i) = φS∪T(ij)})
sφS∪T(ij)
=CLPT(S∪Tj).
If (3.11) holds with equality for all g∈[m], we get
CLPT(S∪Tj)≤p({i∈S∪Tj−1|φS∪T(i) = φT(ij)}) + t
sφT(ij)
=p({i∈S|φS(i) = φT(ij)}) + p({i∈Tj|φT(i) = φT(ij)})
sφT(ij)
≤CLPT(S) + CLPT(Tj)
Now (3.11) and (3.12) clearly yield a contradiction.
4
Moulin Mechanisms and Cost-Sharing Methods in the
Approximate Core
4.1 Contribution
•For every (Q||Cmax)cost-sharing problem with ddifferent processing times, we
give Moulin mechanisms that are 2d-BB and 2d·(1 + Hn)-EFF for CLPT and
computable in polynomial time. Up to a factor of 2, this is in general the best
budget-balance approximation possible for cross-monotonic cost shares [10]. We
further derive that no Moulin mechanism for (Q||Cmax)cost-sharing problems
can generally be better than max{d, Hn}-EFF.
•Restricting attention to (Q|pi=1|Cmax)cost-sharing problems, we show that for
every such problem, there are Moulin mechanisms that are even 2m
m+1 -BB and
2m
m+1 ·(1 + Hn)-EFF for CLPT and computable in polynomial time. By identifying
a(P|pi=1|Cmax)cost-sharing problem for which there is no cost-sharing method
in the β-core for β < 2m
m+1 , we show that 2m
m+1 -BB is generally the best that can be
achieved by Moulin mechanisms. Since the efficiency bound of Hnis also obtained
for a (P|pi=1|Cmax)cost-sharing problem, the efficiency guarantee of our Moulin
mechanisms for (Q|pi=1|Cmax)cost-sharing problems is asymptotically tight.
•The upper and lower bounds on the social cost efficiency of our presented Moulin
mechanisms are obtained by investigating the summability of our cost-sharing
methods and applying a result by Roughgarden and Sundararajan [114]. We con-
duct a fine-grained analysis of the approximate social cost efficiency of our Moulin
mechanisms for (Q|pi=1|Cmax)cost-sharing problems that shows that the upper
bounds are overly pessimistic. It turns out that for many cases, we can guarantee
2m
m+1 -EFF instead of O(log n)-EFF.
•Despite the devastating bound of dfor the approximate budget-balance for
(Q||Cmax)cost-sharing problems, we introduce cost sharing methods for every
such problem that are in the 2m
m+1 -core for CLPT. Though they do not lead to
GSP mechanisms, they nevertheless distribute the cost in a stable way. The cost-
shares for a fixed set can be computed in polynomial time.
The results we introduce in this chapter are published in [10] and [14]. Although 2d-
BB and 2m
m+1 -BB Moulin mechanisms for (Q||Cmax)and (Q|pi=1|Cmax)were already
introduced in [10], we only present the mechanisms from [14] as the underlying
cross-monotonic cost-sharing methods are much simpler and thus allow for the first
summability and efficiency results.
32 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
4.2 Organization
Section 4.3 introduces Moulin mechanisms. Subsequently, Section 4.4 details the in-
terconnection between cost sharing methods in the β-core and β-BB cross-monotonic
cost-sharing methods. The bounds proposed by Roughgarden and Sundararajan [114]
on the approximate social-cost-efficiency of Moulin mechanisms are provided in Sec-
tion 4.5. Section 4.6 presents our results on applications to (Q||Cmax)cost-sharing
problems and its subproblems. Section 4.7 concludes.
4.3 Moulin Mechanisms
Given a bid vector b∈Rn, the Moulin mechanism Moulinξ= (Q, x)for cost-sharing
method ξcan be computed by a very simple algorithm: Initially, consider serving
all players. Then repeatedly eliminate players whose bids are below their current
cost shares (as implied by ξ) until all remaining players can afford their cost shares.
Formally, Moulinξis computed by Algorithm 4.1.
Algorithm 4.1 (computing Moulinξ(b) = (Q(b), x(b))).
1: Q:= [n]
2: while there exists i∈Qwith bi< ξi(Q)do
3: Q:= Q\{j}for an arbitrary j∈Qwith bj< ξj(Q)
4: return (Q, ξ(Q))
It is easy to verify that for every β-BB cost-sharing method ξ,Moulinξis β-BB and
meets NPT, VP, and strict CS. Since Theorem 4.1 is a seminal result that has lead
to almost all known GSP mechanisms so far, we provide its rather straightforward
proof (see [76, 99] for similar proofs).
Theorem 4.1 ([97]). For any cross-monotonic cost-sharing method ξ,Moulinξis
GSP.
Proof. Assume that Moulinξ= (Q, x)is not GSP. Then, there exists a coalition
K⊆[n]with true valuations vK= (vi)i∈K∈R|K|and a bid vector b∈Rnsuch
that ui(b, vi)≥ui((vK,b−K), vi)for all i∈K, with at least one strict inequality.
Without loss of generality, let v:= (vK,b−K).
•We first show that Q(b)⊆Q(v):
We show that every player who receives the service for input balso receives
the service for input v. Assume the opposite and consider the first time that
Moulinξwith input vrejects a player j∈Q(b)by deleting jfrom set Qin line
3 of Algorithm 4.1, in particular vj< ξj(Q). For this set Q, it is Q⊇Q(b)
and by cross-monotonicity, ξj(Q)≤ξj(Q(b)). From jin Q(b), it follows that
ξj(Q(b)) ≤bj. However, vj< ξj(Q)≤ξj(Q(b)) ≤bjimplies that j∈Kwhich
contradicts uj(v, vj) = 0 > vj−ξj(Q(b)) = uj(b, vj). Therefore, Q(b)⊆Q(v).
•We now show that there is no j∈Kwith uj(b, vj)> uj(v, vj):
For each player j∈Kwith uj(b, vj)> uj(v, vj)it has to hold that j∈Q(b)and
thus j∈Q(v)as well. Exploiting cross-monotonicity again, we get the contradic-
tion that uj(b, vj) = vj−ξj(Q(b)) ≤vj−ξj(Q(v)) = uj(v, vj).
4.5 Bounds on Social Cost Efficiency 33
In order to determine the runtime of Moulinξfor a specific ξ, we introduce
Lemma 4.2:
Lemma 4.2. For each cost-sharing method ξsuch that for every S⊆[n],ξ(S)is
computable in time O(t),Moulinξis computable in time O(n·t).
Proof. In each iteration with current set Q,Moulinξhas to compute ξi(Q)for all
i∈Qin the worst case. Furthermore, there are at most niterations.
4.4 Lower Bounds on Budget-Balance
It is easy to see that each cross-monotonic cost-sharing method ξthat is β-BB for
Cand C0is in the β-core for Cand C0:
Lemma 4.3. Let ξbe a cross-monotonic cost-sharing method that is β-BB for Cand
C0. Then ξis in the β-core for Cand C0.
Proof. The fact that for all S⊆[n]it holds that Pi∈Sξi(S)≥C0(S)trivially follows
from the β-BB of ξ. Additionally utilizing cross-monotonicity of ξ, we see that for
all T, S ⊆[n]with T⊆Sit holds that
X
i∈T
ξi(S)≤X
i∈T
ξi(T)≤β·C(T).
From Lemma 4.3 we can conclude that if there is no cost-sharing method in the
β-core for Cand C0, no β-BB cross-monotonic cost-sharing method can exist. On
the other hand, there can be cost-sharing methods that are in the β-core despite
not being cross-monotonic. As an example, our results give cost-sharing methods for
(Q||Cmax)cost-sharing problems in the 2m
m+1 -core for CLPT, while no cross-monotonic
cost-sharing method for (Q||Cmax)can in general be better than d-BB, where dis
the number of different processing times [10].
4.5 Bounds on Social Cost Efficiency
Roughgarden and Sundararajan [114] provide Theorem 4.4 in order to approximate
social cost efficiency of Moulin mechanisms:
Theorem 4.4 ([114]). Let C0and Cbe cost functions, and ξbe a cross-monotonic
cost-sharing method. Let α≥0be the smallest number such that ξis α-SUM for
C, and let β≥1be the smallest number such that ξis β-BB for C0and C. Then
Moulinξis (α+β)-EFF and no better than max{α, β}-EFF for C0and C.
We remark that in the original work of Roughgarden and Sundararajan [114],
β-BB is defined as 1
β·C0(S)≤Pi∈Sξi(S)≤C(S). However, adapting the proof to
our definition of budget-balance (confer Definition 2.19) yields the same bounds. We
give the modified proof below for reasons of clearness. In general, when adapting a
Moulin mechanism Moulinξthat is β-BB in the sense of [114] to our definition of
β-BB, we have to multiply the cost-shares given by ξby β. However, this means
34 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
that if ξwas α-SUM before, it is now (β·α)-SUM and the guaranteed efficiency
approximation by Theorem 4.4 has impaired.
The advantage to define β-BB in the sense of [114] is that a natural trade-off
between budget-balance and efficiency can be observed: dividing the cost shares of
aβ-BB (in the sense of [114]) and α-SUM cost-sharing method by γ > 1yields a
(γ·β)-BB (in the sense of [114]) and (α
γ)-SUM cost-sharing method. Thus, sacrificing
budget-balance to gain social cost efficiency is possible within the bounds of The-
orem 4.4. However, sacrificing budget-balance according to our definition of β-BB
means to multiply the cost shares by a factor γ > 1, and automatically leads to
sacrificing summability and efficiency, accordingly.
We nevertheless think that attaching more importance to cost-recovery is far more
reasonable, as the service provider incurs a loss otherwise.
Proof (of Theorem 4.4). Fix cost functions Cand C0, a cross-monotonic cost-sharing
method ξ, and α, β that meet the requirements in Theorem 4.4.
We first look at the lower bounds. The proof from [114] showing that Moulinξis
no better than α-EFF does not involve any budget-balance considerations and still
applies. Now let S⊆[n]with Pi∈Sξi(S) = β·C(S)and define vby vi:= ξi(S)−ε
for all i∈Sand vi:= −εotherwise. Then, by cross-monotonicity of ξ,Moulinξ
with input vserves no player and induces a social cost of β·C(S)−n·ε. On the
other hand, the optimal social cost is not larger than C(S). Therefore, Moulinξis
no better than β-EFF.
We continue to show the upper bound. Fix v∈Rnand let Qbe the set that is
output by Moulinξwith input v. Let S∗be the set that minimizes social cost. By
definition of social cost,
SC C0(Q, v)−SC C0(Q∩S∗,v) = C0(Q)−C0(Q∩S∗)−X
i∈Q\S∗
vi.(4.1)
Furthermore, by β-BB and cross-monotonicity,
X
i∈Q\S∗
ξi(Q) = X
i∈Q
ξi(Q)−X
i∈Q∩S∗
ξi(Q)
≥C0(Q)−X
i∈Q∩S∗
ξi(Q∩S∗)
≥C0(Q)−β·C(Q∩S∗).(4.2)
Using Inequality (4.2) within Equation (4.1) and utilizing that ξi(Q)≤vifor all
i∈Q, we get
SC C0(Q, v)≤SC C0(Q∩S∗,v) + β·C(Q∩S∗)−C0(Q∩S∗)
=X
i/∈Q∩S∗
vi+β·C(Q∩S∗)
=X
i/∈S∗
vi+X
i∈S∗\Q
vi+β·C(Q∩S∗)
Let (S∗\Q) = {i1, . . . , i|S∗\Q|}ordered reverse to the order in which Moulinξdeletes
these players, and for each j∈[|S∗\Q|], define (S∗\Q)j:= {i1, . . . , ij}to be the
first jplayers in S∗\Q. Furthermore, let Qjbe the set from which ijis deleted by
4.6 Applications To Scheduling 35
Moulinξ. In particular, Qj⊇(S∗\Q)j. By cross-monotonicity and the fact that ξis
α-SUM, we conclude
SC C0(Q, v)<X
i/∈S∗
vi+|S∗\Q|
X
j=1
ξij(Qj) + β·C(Q∩S∗)
≤X
i/∈S∗
vi+|S∗\Q|
X
j=1
ξij((S∗\Q)j) + β·C(Q∩S∗)
≤X
i/∈S∗
vi+α·C(S∗\Q) + β·C(S∗)
≤(α+β)·SC C(S∗,v).
We adopt an idea of Roughgarden and Sundararajan [114] to show a lower
bound of Hnon the approximate social cost efficiency for constant costs. The bound
from [114] holds under 1-BB and improves for relaxations (its value is Hn
β); in our
model, Hnis a lower bound under any budget-balance approximation. Lemma 4.5 is
a direct corollary of Lemma 4.6 and Theorem 4.4.
Lemma 4.5. For a cost function Cwith C(S) = afor an a∈R>0and all S([n],
there is no Moulin mechanism Moulinξwith Pi∈Sξi(S)≥afor all S⊆[n]that is
γ-EFF for γ < Hn.
Lemma 4.6. For a cost function Cwith C(S) = afor an a∈R>0and all S([n],
there is no cost-sharing method ξwith Pi∈Sξi(S)≥afor all S⊆[n]that is α-SUM
for α < Hn.
Proof. Consider an arbitrary cost-sharing method ξwith Pi∈Sξi(S)≥afor all
S([n]. We define an order i1, . . . , inof [n]with Sj:= {i1, . . . , ij}for all j∈[n]and
show that Pn
j=1 ξij(Sj)≥Hn·C([n]).
Note that due to Pi∈[n]ξi([n]) ≥athere is a player k∈[n]with ξk([n]) ≥a
n. We
let in:= k. Similarly, from Pi∈[n]\{k}ξi([n]\{k})≥a, we know that there is a player
k0∈[n]\{k}with ξk0([n]\{k})≥a
n−1, and we let in−1:= k0. Iteratively, we define
in−2, . . . , i1by the same procedure. Then, Pn
j=1 ξij(Sj)≥Pn
j=1 a
j=Hn·C([n]).
4.6 Applications To Scheduling
4.6.1 Lower Bounds on Budget-Balance and Efficiency
In order to obtain lower bounds on β-BB for cross-monotonic cost-sharing methods
for scheduling problems, we use the interrelation to the β-core and show that al-
ready for identical jobs and machines, βcannot be smaller than 2m
m+1 in general. The
situation gets worse when allowing for different processing times and speeds [10].
Theorem 4.7. There is a (P|pi=1|Cmax)cost-sharing problem for which there is no
cross-monotonic cost-sharing method that is β-BB for β < 2m
m+1 .
Theorem 4.7 is a simple corollary of Lemma 4.3 and Theorem 4.8:
36 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
Theorem 4.8. There is a (P|pi=1|Cmax)cost-sharing problem for which there is no
cost-sharing method in the β-core for β < 2m
m+1 .
Proof. Consider a (P|pi=1|Cmax)cost-sharing problem with m=n−1machines and
a cost-sharing method ξin the β-core. For every S⊆[n]with |S|=mit holds that
Pi∈Sξi([n]) ≤β·MSP(S) = β. Thus, there exists a player j∈[n]with ξj([n]) ≤β
m.
It follows that
X
i∈[n]
ξi([n]) ≤β
m+X
i∈[n]\{j}
ξi([n])
≤β
m+β
=m+ 1
m·β .
From 2 = MSP([n]) ≤Pi∈[n]ξi([n]) ≤m+1
m·β, we conclude that β≥2m
m+1 .
Theorem 4.9 ([10]). There is a (Q||Cmax)cost-sharing problem with ddifferent
processing times for which there is no cross-monotonic cost-sharing method that is
β-BB for β < d.
For approximating social cost efficiency we can conclude that in general, we cannot
be better than Hn-EFF even in the case of identical jobs and identical machines. For
the general problem, the lower bound dfor approximate budget-balance even implies
that in general, no Moulin mechanism can be better than max{d, Hn}-EFF.
Lemma 4.10. There is a (P|pi=1|Cmax)cost-sharing problem for which there is no
Moulin mechanism that is γ-EFF for γ < Hn.
Proof. Consider a (P|pi=1|Cmax)cost-sharing problem with m=nmachines. With
MSP(S) = 1 for all S⊆[n]and Lemma 4.5, the claim follows.
Lemma 4.11. There is a (Q||Cmax)cost-sharing problem with ddifferent processing
times for which there is no Moulin mechanisms that is γ-EFF for γ < d.
Proof. The lemma simply follows from Theorem 4.4 and Theorem 4.9.
4.6.2 Moulin Mechanisms for Identical Jobs
Theorem 4.12 is the main theorem of this section and follows directly from Theo-
rem 4.13, Theorem 4.1, Theorem 4.4, and Lemma 4.2.
Theorem 4.12. For each (Q|pi=1|Cmax)cost-sharing problem there is a cost-sharing
method ξsuch that Moulinξis GSP, 2m
m+1 -BB and 2m
m+1 ·(1 + Hn)-EFF for CLPT.
Furthermore, Moulinξis computable in time O(n2·log m).
Theorem 4.13. For each (Q|pi=1|Cmax)cost-sharing problem there is a cost-sharing
method ξthat is cross-monotonic, 2m
m+1 -BB for CLPT and 2m
m+1 ·Hn-SUM. For each
S⊆[n], the vector of cost shares ξ(S)is computable in time O(n·log m).
4.6 Applications To Scheduling 37
Proof. Fix s∈Nmand consider the cost-sharing problem (1,s).
For each S⊆[n], define method ξ: 2[n]→Rn
≥0by
ξi(S):=(2m
m+1 ·minj∈[|S|]
msp(j)
jif i∈S
0otherwise.
Cross-monotonicity is obvious. Fix S⊆[n]. We start proving 2m
m+1 -BB. Directly from
the definition, it follows that Pi∈Sξi(S)≤2m
m+1 ·msp(|S|) = 2m
m+1 ·MSP(S).
•If |S| ≤ m, let k≤ |S|be maximum with msp(k)
k=minj∈[|S|]
msp(j)
j. If k=|S|
then Pi∈Sξi(S) = 2m
m+1 ·MSP(|S|). If k < |S|we get by (3.8) and k+1 ≤ |S| ≤ m
that
X
i∈S
ξi(S) = 2m
m+ 1 ·|S|
k·msp(k)
≥2m
m+ 1 ·k+ 1
2k·msp(|S|)
≥2m
m+ 1 ·m
2m−2·msp(|S|)
≥msp(|S|)
=MSP(S)
=CLPT(S).
•If |S| ≥ m+ 1, we know by (3.6) that msp(|S|)≤|S|
s([m]) ·2m
m+1 . With (3.7), we get
X
i∈S
ξi(S)≥2m
m+ 1 ·|S|
s([m])
≥msp(|S|)
=MSP(S)
=CLPT(S).
For summability, let i1, . . . , i|S|be an arbitrary order of S, and let Sj:= {i1, . . . , ij}
denote the set of the first jelements. Then,
|S|
X
j=1
ξij(Sj)≤2m
m+ 1 ·|S|
X
i=1
msp(i)
i
≤2m
m+ 1 ·Hn·msp(|S|)
=2m
m+ 1 ·Hn·MSP(S).
Since LPT on |S|identical jobs simultaneously computes all values for msp(i)for
all i∈[|S|], the time to compute minj∈[|S|]
msp(j)
j=ξi(S)for all i∈Sis O(n·log m).
38 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
4.6.3 Moulin Mechanisms for the General Setting
We generalize the idea for identical jobs from Section 4.6.2 to the general scheduling
setting to obtain Theorem 4.14 which follows from Theorem 4.15, Theorem 4.1,
Theorem 4.4, and Lemma 4.2.
Theorem 4.14. For each (Q||Cmax)cost-sharing problem with ddifferent process-
ing times there is a cost-sharing method ξsuch that Moulinξis GSP, 2d-BB, and
2d·(1 + Hn)-EFF for CLPT. Furthermore, Moulinξis computable in time O(n2·log m).
Theorem 4.15. For each (Q||Cmax)cost-sharing problem with ddifferent process-
ing times there is a cost-sharing method ξthat is cross-monotonic, 2d-BB, and
(2d·Hn)-SUM. For each S⊆[n], the vector of cost shares ξ(S)is computable in
time O(n·log m).
Proof. Fix p∈Nn,s∈Nmand consider the cost-sharing problem (p,s).
For S⊆[n], we use the subsequent notations within this proof:
• P(S) := {pi|i∈S}is the set of different processing times of the jobs in S. We
let P(S):={t1, . . . , t|P(S)|}.
•S(t):={i∈S|pi=t}is the set of players in S with processing time t.
For each S⊆[n]define the method ξ: 2[n]→Rn
≥0by
ξi(S) := (2·pi·minj∈[|S(pi)|]
msp(j)
jif i∈S
0otherwise.
Cross-monotonicity is obvious. Fix S⊆[n]. To show 2d-BB, we first bound the sum
of the cost shares from above by 2·|P(S)|·MSP(S). We have that
X
i∈S
ξi(S)=2·X
t∈P(S)
t·|S(t)|· min
j∈[|S(t)|]
msp(j)
j
≤2X
t∈P(S)
t·msp(|S(t)|)
= 2 X
t∈P(S)
MSP(S(t))
≤2·|P(S)|·MSP(S).
For the lower bound, let k(t)∈[|S(t)|]such that msp(k(t))
k(t)= minj∈[S(t)]
msp(j)
jfor all
t∈ P(S). Applying 2·msp(k(t)) ≥msp(|S(t)|)by (3.8) we get
X
i∈S
ξi(S)=2 X
t∈P(S)
t·|S(t)|· min
j∈[S(t)]
msp(j)
j
= 2 X
t∈P(S)
t·|S(t)|· msp(k(t))
k(t)
≥2X
t∈P(S)
t·msp(k(t))
≥X
t∈P(S)
t·msp(|S(t)|),
4.6 Applications To Scheduling 39
and thus Pi∈Sξi(S)≥Pt∈P(S)MSP(S(t)) = Pt∈P(S)CLPT(S(t)) ≥CLPT(S). The
last inequality is due to Lemma 3.4.
For summability, let i1, . . . , i|S|be an arbitrary order of S, and let Sj:= {i1, . . . , ij}
denote the first jelements. Then,
|S|
X
j=1
ξij(Sj)=2·|S|
X
j=1 pij·min
j∈[|Sj(pij)|]
msp(j)
j!
= 2 ·|P(S)|
X
k=1
tk
|S(tk)|
X
l=1
min
j∈[l]
msp(j)
j
≤2·|P(S)|
X
k=1
|S(tk)|
X
l=1
tk·msp(l)
l
≤2·|P(S)|
X
k=1
|S(tk)|
X
l=1
tk·msp(|S(tk)|)
l
= 2 ·|P(S)|
X
k=1
|S(tk)|
X
l=1
MSP(S(tk))
l
≤2·|P(S)|
X
k=1
|S|
X
l=1
MSP(S)
l
≤2·|P(S)|·H|S|·MSP(S).
The time to compute ξ(S)is determined by the computation time of msp(i)for all
i∈[|S|](for identical jobs) which is accomplished by one run of LPT for |S|identical
jobs that takes time O(n·log m).
4.6.4 Efficiency Considerations for Identical Jobs
The results in this section were originally presented in [14] with the alternative
definition of β-BB. Whereas [14] performs a tight (but rather technical) analysis
for a variety of cases, we settle here for showing that in many cases, 2m
m+1 is an
upper bound on the social cost approximation achieved by the Moulin mechanisms
employing our cost-sharing methods for (Q|pi=1|Cmax)from Section 4.6.2.
Theorem 4.16. For each (Q|pi=1|Cmax)cost-sharing problem, let ξbe the cost-
sharing method as defined in the proof of Theorem 4.13. Fix v∈Rn
≥0. Let Q(v)be
the players selected by Moulinξ,µ:= |Q(v)|and λbe the largest cardinality of all
sets that minimize the social cost SC MSP(·,v). Then for all S⊆[n]it holds that
SC MSP(Q(v),v)≤γ·SC MSP(S, v), with
γ=
1if µ=λ
1 + 2m
m+1 ·Hnif λ>µand µ≤s([m])
2m
m+1 otherwise.
Proof. Fix s∈Nmand consider the cost-sharing problem (1,s).
Since negative bids have no impact on the social cost, we assume that v∈Rn
≥0.
The main idea of the proof is to order the players from 1to nsuch that v1≥. . . ≥vn.
40 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
Then, Q(v) = [µ], and [λ]is the maximum set of players that minimizes SC MSP(·,v).
Moulinξis obviously 1-EFF for µ=λ.
Let σ:= s([m]) and define x∈Rn
≥0by xk:= minj∈[k]
msp(j)
j. Then, for all S⊆[n]
with |S|=kand all i∈S, it is ξi(S) = 2m
m+1 ·xk. We frequently use that for all
i∈[µ]it is vi≥2m
m+1 ·xµ, as Moulinξserves [µ]. Furthermore, for all i>µit is
vi<2m
m+1 ·xi≤2m
m+1 ·msp(i)
i, as Moulinξdoes not serve players {µ+ 1, . . . , n}.
•Case (λ>µand µ≤σ):
By non-increasing makespan costs,
SC MSP([µ],v) = msp(µ) +
λ
X
i=µ+1
vi+
n
X
i=λ+1
vi
<msp(λ) + 2m
m+ 1 ·
λ
X
i=µ+1
msp(i)
i+
n
X
i=λ+1
vi
≤msp(λ) + 2m
m+ 1 ·Hn·msp(λ) +
n
X
i=λ+1
vi
≤1 + 2m
m+ 1 ·Hn· msp(λ) +
n
X
i=λ+1
vi!
=1 + 2m
m+ 1 ·Hn·SC MSP([λ],v).
In order to show γ=2m
m+1 we split the remaining case into three subcases:
•Case (λ>µand µ > σ):
By definition and (3.5) it holds that xσ≤msp(σ)
σ=1
σ. From (3.7) we conclude
that for all i≥σit holds that xi=1
σ. As µ > σ ≥mwe know by (3.6) that
msp(µ)≤µ
σ·2m
m+1 . Then together with λ
σ≤msp(λ)by (3.4),
SC MSP([µ],v) = msp(µ) +
λ
X
i=µ+1
vi+
n
X
i=λ+1
vi
<msp(µ) + 2m
m+ 1 ·λ−µ
σ+
n
X
i=λ+1
vi
≤2m
m+ 1 ·msp(λ) +
n
X
i=λ+1
vi
≤2m
m+ 1 · msp(λ) +
n
X
i=λ+1
vi!
=2m
m+ 1 ·SC MSP([λ],v).
•Case (λ<µand µ≥m+ 1):
(3.6),(3.4), and (3.7) imply msp(µ)≤µ
σ·2m
m+1 ,msp(λ)≥λ
σ, and xµ≥1
σ, thus
4.6 Applications To Scheduling 41
SC MSP([λ],v) = msp(λ) +
µ
X
i=λ+1
vi+
n
X
i=µ+1
vi
≥λ
σ+2m
m+ 1 ·(µ−λ)·xµ+
n
X
i=µ+1
vi
≥λ
σ+2m
m+ 1 ·µ−λ
σ+
n
X
i=µ+1
vi
≥µ
σ+
n
X
i=µ+1
vi
≥m+ 1
2m·
msp(µ) +
n
X
i=µ+1
vi
=m+ 1
2m·SC MSP([µ],v).
•Case (λ<µand µ < m + 1):
Let k∈[µ]be maximum with xµ=xk=msp(k)
k.
– It holds that λ > 0: If λ= 0 and thus [λ] = ∅, it holds that
SC MSP([k],v) = msp(k) +
n
X
i=k+1
vi
=k·xµ+
n
X
i=k+1
vi
≤k·2m
m+ 1 ·xµ+
n
X
i=k+1
vi
≤
n
X
i=1
vi
=SC MSP(∅,v),
contradicting the maximality of λ.
– It holds that λ≥k: If λ<k, by msp(k)
k=xk≤xλ≤msp(λ)
λ, we get
SC MSP([λ],v) = msp(λ) +
n
X
i=λ+1
vi
≥msp(λ) + 2m
m+ 1
k
X
i=λ+1
xk+
n
X
i=k+1
vi
≥λ
k·msp(k) + 2m
m+ 1 ·k−λ
k·msp(k) +
n
X
k+1
vi
≥msp(k) +
n
X
k+1
vi
=SC MSP([k],v),
a contradiction to λbeing maximum.
42 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
– It holds that 2λ>µ: Otherwise, µ≥2kand msp(2k)
2k≤msp(k)
kby (3.3) contra-
dicts the maximality of k.
– It holds that
µxµ=µ
k·msp(k)≥µ
2k·msp(µ)≥k+ 1
2k·msp(µ)≥m+ 1
2m·msp(µ),
which follows from msp(µ)≤2·msp(k)by (3.8), µ≥λ+1 ≥k+1, and m≥µ.
– It holds that Pµ
i=λ+1 vi≥1
m·msp(µ): This follows from µ−λ≥1,m≥µ,
and thus
µ
X
i=λ+1
vi≥(µ−λ)·2m
m+ 1 ·xµ
≥(µ−λ)·msp(µ)
µ
≥1
m·msp(µ).
Finally, applying 2·msp(λ)≥msp(2λ)≥msp(µ)(confer (3.3)), we get
SC MSP([λ],v) = msp(λ) +
µ
X
i=λ+1
vi+
n
X
i=µ+1
vi
≥1
2·msp(µ) + 1
m·msp(µ) +
n
X
i=µ+1
vi
=m+ 2
2m·msp(µ) +
n
X
i=µ+1
vi
≥m+ 1
2m·
msp(µ) +
n
X
i=µ+1
vi
=m+ 1
2m·SC MSP([µ],v).
4.6.5 Cost-Sharing Methods in the Approximate Core
Theorem 4.17. For each (Q||Cmax)cost-sharing problem, there is a cost-sharing
method in the 2m
m+1 -core for CLPT. For each S⊆[n], the vector ξ(S)of cost shares
can be computed in time O(nlog n+n·m).
Proof. Fix p∈Nnand s∈Nmand consider the cost-sharing problem (p,s). Fur-
thermore, fix a set S⊆[n]and consider running LPT on S. Let ˆ
Sbe the set of jobs
that LPT assigns until there is exactly one machine that has a completion time equal
to CLPT(S) = CLPT(ˆ
S). We let φ0denote the assignment that LPT computes for S.
For simplicity, let τ:= |mφ0(ˆ
S)|. To define the cost-sharing method ξ, we look at the
following cases:
4.6 Applications To Scheduling 43
•If τ < |ˆ
S|, we define
ξi(S):=
2m
m+1 ·pi
s(mφ0(ˆ
S)) if i∈ˆ
S
0otherwise.
•If τ=|ˆ
S|, we distinguish three cases:
– If τ≥3, let A:= CLPT(S)·s(mφ0(ˆ
S)) −p(ˆ
S)and define
ξi(S):=
2m
m+1 ·pi
s(mφ0(ˆ
S)) if i∈ˆ
Sand A < τ−1
τ+1 ·p(ˆ
S)
2m
m+1 ·pi
s(mφ0(ˆ
S))−sτif i∈ˆ
Sand A≥τ−1
τ+1 ·p(ˆ
S)
0otherwise.
– If τ= 2, assume without loss of generality that ˆ
S={1,2}. Let z:= p1+p2
s1−p2
s2
and define
ξi(S):=
p1
s1−z
2if i= 1
p2
s1−z
2if i= 2
0otherwise.
– If τ= 1, assume without loss of generality that ˆ
S={1}. Define ξ1(S) := p1
s1
and ξi(S) := 0 for all i∈S\{1}. It holds that ξ1(S) = MSP(S) = CLPT(S).
We continue to show that ξas defined above is in the 2m
m+1 -core for CLPT.
•If τ < |ˆ
S|, there is at least one machine that is assigned more than one job at
the time when ˆ
Sis assigned. The overall cost share is larger than the actual cost
CLPT(S), since by Equation (3.10) and, obviously, τ≤m,
X
i∈S
ξi(S) = 2m
m+ 1 ·p(ˆ
S)
s(mφ0(ˆ
S))
≥2m
m+ 1 ·τ+ 1
2τ·CLPT(S)
≥CLPT(S).
To bound the cost shares for a subset T⊆Sfrom above, consider an optimal
assignment φfor T∩ˆ
Swith mφ(T∩ˆ
S)⊆mφ0(ˆ
S). Such an assignment exists due
to (3.9). If we further utilize (3.2), we get that
X
i∈T
ξi(S) = 2m
m+ 1 ·p(T∩ˆ
S)
s(mφ0(ˆ
S))
≤2m
m+ 1 ·p(T∩ˆ
S)
s(mφ(T∩ˆ
S))
≤2m
m+ 1 ·MSP(T∩ˆ
S)
≤2m
m+ 1 ·MSP(T).
44 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
•If τ=|ˆ
S|,LPT has assigned at most one job to each machine at the time when
ˆ
Sis assigned. In this case, it is CLPT(S) = pτ
sτ.
– If τ≥3, consider two cases:
◦If A < τ−1
τ+1 ·p(ˆ
S), the overall share is larger than CLPT(S), since
X
i∈S
ξi(S) = 2m
m+ 1 ·p(ˆ
S)
s(mφ0(ˆ
S))
>2m
m+ 1 ·τ+ 1
2τ·CLPT(S)
≥CLPT(S).
Furthermore, Pi∈Tξi(S)≤2m
m+1 ·MSP(T)can be show just like for case
τ < |ˆ
S|.
◦If A≥τ−1
τ+1 ·p(ˆ
S), we first observe that pg+pτ≥CLPT(S)·sgfor all
g∈[τ−1]. This is due to the fact that the assignment computed by
LPT is a Nash equilibrium. Summing up over all g∈[τ−1], we conclude
p(ˆ
S)+(τ−2) ·pτ≥CLPT(S)·(s(mφ0(ˆ
S)) −sτ). Further applying that
p(ˆ
S)≥τpτ(confer Assumption 3.1), we get
2τ−2
τ·p(ˆ
S)≥CLPT(S)·(s(mφ0(ˆ
S)) −sτ)
and thus
X
i∈S
ξi(S) = 2m
m+ 1 ·p(ˆ
S)
s(mφ0(ˆ
S)) −sτ
≥2m
m+ 1 ·τ
2τ−2·CLPT(S)
≥CLPT(S).
To bound the cost share for a set T⊆Sfrom above, we first show that
Pi∈Sξi(S)≤2m
m+1 ·MSP(S)and second we show that for all T(S,
Pi∈Tξi(S)≤2m
m+1 ·MSP(T). If MSP(S) = CLPT(S),Acan be written as
A=MSP(S)·s(mφ0(ˆ
S)) −sτ−p(ˆ
S)−pτ.
Now, A≥τ−1
τ+1 ·p(ˆ
S)leads to
p(ˆ
S)≤τ+ 1
2τ·MSP(S)·s(mφ0(ˆ
S)) −sτ+pτ.
Additionally, τ≥3implies that s(mφ0(ˆ
S)) −sτ≥2sτand τ+1
2τ≤2
3. Thus,
X
i∈S
ξi(S) = 2m
m+ 1 ·p(ˆ
S)
(s(mφ0(ˆ
S)) −sτ)
≤2m
m+ 1 ·τ+ 1
2τ· MSP(S) + pτ
s(mφ0(ˆ
S)) −sτ!
≤2m
m+ 1 ·τ+ 1
2τ·3
2·MSP(S)
≤2m
m+ 1 ·MSP(S).
4.6 Applications To Scheduling 45
On the other hand, if MSP(S)< CLPT(S), an optimal assignment φfor
Scannot use machines {τ, . . . , m}(confer Assumption 3.1). Hence we get
mφ(ˆ
S)⊆mφ(S)⊆mφ0(ˆ
S)\{τ}. Together with (3.2), this leads to
X
i∈S
ξi(S) = 2m
m+ 1 ·p(ˆ
S)
(s(mφ0(ˆ
S)) −sτ)
≤2m
m+ 1 ·p(ˆ
S)
s(mφ(ˆ
S))
≤2m
m+ 1 ·MSP(ˆ
S)
≤2m
m+ 1 ·MSP(S).
Now consider a proper subset T(S. Note that there is an optimal as-
signment φfor T∩ˆ
Swith mφ(T∩ˆ
S)⊆mφ0(ˆ
S)\ {τ}, since an optimal
assignment for a proper subset of ˆ
Sdoes not have to use machine τany-
more. Again applying (3.2), we get
X
i∈T
ξi(S) = 2m
m+ 1 ·p(T∩ˆ
S)
(s(mφ0(ˆ
S)) −sτ)
≤2m
m+ 1 ·p(T∩ˆ
S)
s(mφ(T∩ˆ
S)
≤2m
m+ 1 ·MSP(T∩ˆ
S)
≤2m
m+ 1 ·MSP(T).
– If τ= 2, then
X
i∈S
ξi(S) = p2
s2
=MSP(S) = CLPT(S).
For T⊆S\ {1,2},Pi∈Tξi(S) = 0 ≤MSP(T). Consider T⊆S\ {1}with
2∈T. We have that Pi∈Tξi(S)<p2
s1≤MSP(T). Analogously, for T⊆S\{2}
with 1∈Tit holds that Pi∈Tξi(S)<p1
s1≤MSP(T).
– If τ= 1, the core conditions are trivially satisfied.
The time to compute the cost shares ξ(S)is in all cases determined by the running
time of LPT with input S.
46 4 Moulin Mechanisms and Cost-Sharing Methods in the Approximate Core
4.7 Conclusion and Open Problems
•We first argue that the good cases of O(1)-EFF in the fine-grained analysis for
(Q|pi=1|Cmax)cost-sharing problems might be likely to occur. Note that only if
Moulinξgives the service to less or equal than σ:= s([m]) players, the worst case
performance may arise. Now, if the provider announces that only bidding at least
1
σmay result in receiving the service and furthermore only maintains a certain
set of machines such that his cluster may operate at full capacity, we think it
likely that there are more than σplayers who each bid at least 1
σ. In our opinion,
our fine-grained analysis pursues an interesting direction. It is an open question
if such an analysis is promising for the problem of scheduling arbitrary jobs on
parallel machines and/or for other cost-sharing problems.
•With respect to social cost efficiency of our mechanisms for (Q||Cmax)cost-sharing
problems, it is still an open issue to improve the O(d·log n)-EFF or our methods
or to increase the lower bound of max{d, Hn}.
•Certainly, our scheduling model may not very well reflect real-world applications.
Players might not only be interested in their job being processed, but also in its
completion time. Furthermore, the makespan as the completion time of the whole
system may not reflect the provider’s cost properly. On the other hand, the result
by Brenner and Schäfer [16] rules out the existence of reasonable cost-sharing
methods for minimizing the sum of completion time. We thus consider our work
to be a first basic step for cost-sharing scheduling scenarios. We obtain first results
for more realistic scenarios in Chapter 6.
•Finally, one has to go beyond cross-monotonicity and develop alternatives to
Moulin mechanisms with better approximate budget-balance and/or social cost
efficiency. We take this action in Chapters 5 and 6, where we are able to improve
on some of the results in this chapter.
5
Group-Strategyproof Non-Moulin Mechanisms
5.1 Contribution
We consider the problem of computing GSP non-Moulin mechanisms with good
approximate budget-balance for symmetric cost functions. As symmetric costs only
depend on the cardinality of the served set, we approach this problem by defining
exactly one vector of cost shares for each cardinality and making the cost share of
a specific player solely dependent on his rank in the served set. This is realized by
applying preference-ordered cost-sharing methods as introduced in Section 5.3. We
generalize this approach for specific non-symmetric cost functions. In particular, our
results are as follows:
•For symmetric and subadditive costs, we investigate what can be achieved by
charging all players equally. We present GSP mechanisms that are 2-BB and show
that in general, this is the best possible. If costs are additionally submodular, we
even obtain 1-BB.
•We consider the case that for any set of served players, there are at most two
different cost shares and introduce two-price cost-sharing forms (2P-CSFs) that
define these cost shares. Furthermore, we present a mechanism MechCSFFand
identify a validity requirement such that for any valid 2P-CSF F,MechCSF Fis
GSP. This is analogous to Moulinξwhich is GSP if the cost-sharing method ξ
is cross-monotonic. The usefulness of our new technique lies in the fact that 2P-
CSFs do not necessarily represent cross-monotonic cost-sharing methods. Hence,
for certain classes of cost functions, 2P-CSFs allow for better budget-balance
approximations than cross-monotonic cost-sharing methods:
– For symmetric and subadditive costs, we give an algorithm to compute valid
2P-CSFs that are √17+1
4-BB, where √17+1
4≈1.28. We show that in general,
this is the best valid 2P-CSFs can yield. Yet, this significantly improves over
2-BB, which is generally the best possible for cross-monotonic cost shares
(confer Section 4.6.5). Any such 2P-CSF Fand the corresponding mechanism
MechCSFFare computable in polynomial time, given that the symmetric costs
can be evaluated in polynomial time.
– We apply this technique to (Q|pi=1|Cmax)cost-sharing problems obtaining
√17+1
4-BB and to (P|pi=1|Cmax)cost-sharing problems where we even achieve
1-BB. Interestingly, the bound of 2for cross-monotonic cost-shares is derived
from (P|pi=1|Cmax)costs (confer Theorem 4.7, p. 35).
48 5 Group-Strategyproof Non-Moulin Mechanisms
– Unfortunately, we show that the corresponding mechanisms can generally not
be better than Ω(n)-EFF.
•For the non-symmetric (Q||Cmax)cost-sharing problems, we use several 2P-CSFs
to obtain GSP and (√17+1
4·d)-BB mechanisms, where dis the number of dif-
ferent processing times. This beats the previously best-known (2d)-BB (confer
Section 4.6). Our mechanisms are computable in polynomial time.
•For the specific non-symmetric (P3|pi∈ {1,2}|Cmax)cost-sharing problems, we
extend our techniques to guarantee GSP and 1-BB by generalizing the notion of
a preference order and making cost shares dependent on the rank as well as the
cardinalities of both classes of served players. Unfortunately, the same approach
fails for 4 identical machines (this result is not part of this thesis; we refer to [12]).
The results we introduce in this chapter are published in [12].
5.2 Organization
Section 5.3 formally defines preference-ordered cost-sharing methods. Our results
for charging all players equally are presented in Section 5.4. Section 5.5 introduces
2P-CSFs together with our novel mechanisms and give the applications for the sym-
metric (Q|pi=1|Cmax)and especially (P|pi=1|Cmax)cost-sharing problems. The ap-
plications for our techniques extended to non-symmetric cost-sharing problems are
presented in Section 5.6.
5.3 Preference-Ordered Cost-Sharing Methods
We introduce a property of cost-sharing methods that ensures that the cost share of
a player iin a set of served players S⊆[n]only depends on the rank of iin Sand
the cardinality of S:
Definition 5.1. A cost-sharing method ξ: 2[n]→Rn
≥0is preference-ordered if there
are vectors ξk∈Rk
≥0for all k∈[n]such that for all S⊆[n]
ξi(S):=(ξ|S|
rank(i,S)if i∈S
0otherwise.
Example 5.2. Let n= 5,ξ2= (2,1), and consider sets S={2,4}and T={1,5}.
Then ξ(S) = (0,2,0,1,0) and ξ(T) = (2,0,0,0,1).
5.4 Symmetric and Subadditive Costs and One Price
This section discusses the (im)possibilities of charging all players equally. Lemma 5.3
gives the positive result that GSP and 2-BB can be achieved for any symmetric and
subadditive cost function whereas Lemma 5.4 establishes tightness of this result.
Lemma 5.3. For any symmetric and subadditive cost function C, there is a GSP
mechanism that is 2-BB for Cand always charges players equally. If Cis also sub-
modular, this mechanism is even 1-BB for C.
5.4 Symmetric and Subadditive Costs and One Price 49
Proof. Let Cbe a symmetric and subadditive cost function and c: [n]→R≥0be as
in Definition 2.11.
•We define
lk:= 2 ·min
j∈[k]
c(j)
jfor all k∈[n].
For all k∈[n]it holds that k·lk≤2·c(k). For the lower bound, fix an arbitrary
k∈[n]. Subadditivity implies for all j∈[n]that c(j)≤c(dj
2e)+c(bj
2c)≤2·c(dj
2e)
and especially c(2j)
2j≤c(j)
j. Thus, minj∈[k]c(j)
j=c(j0)
j0for a j0∈ {dk
2e, . . . , k}and
k·lk=k·2·c(j0)
j0≥2·c(dk
2e)≥c(k).
Define a cost-sharing method ξas follows: for all S⊆[n]let ξi(S) := l|S|for all
i∈Sand ξi(S) := 0 otherwise. Clearly, ξis cross-monotonic. Furthermore, we
have that for all S⊆[n]it holds that
C(S) = c(|S|)≤ |S|·l|S|=X
i∈S
ξi(S)≤2·c(|S|) = 2 ·C(S).
By Theorem 4.1, Moulinξis GSP and 2-BB for C.
•If Cis also submodular, it holds that c(j+1)
j+1 ≤c(j)
jfor all j∈[n−1], which can be
shown by an easy induction: Consider Definition 2.8 and without loss of generality,
let S:= {1}and T:= {2}. Then c(2) −c(1) ≤c(1) and thus c(1) ≥c(2)
2. Now
assume that c(k+1)
k+1 ≤c(k)
kfor all k∈[j−1] and without loss of generality, consider
Definition 2.8 with S:= [j]and T:= [j+ 1] \{1}. Then, utilizing the induction
hypothesis,
c(j+ 1) −c(j)≤c(j)−c(j−1) ≤c(j)−j−1
j·c(j)⇐⇒ c(j+ 1)
j+ 1 ≤c(j)
j.
It follows that k·lk= 2 ·c(k)for all k∈[n]. For l0
k:= 1
2·lk, the corresponding
(cross-monotonic) cost-sharing method ξ0where for all S⊆[n],ξ0
i(S) := l0
|S|if
i∈Sand ξ0
i(S) := 0 otherwise, is even 1-BB for C.
Lemma 5.4. For any ε > 0, there is a symmetric and subadditive cost function C
for which no GSP mechanism that charges all players equally is (2 −ε)-BB for C.
Proof. Fix 0< ε ≤1and let nbe large enough such that n−1>2−ε
ε. Define Cby
C(S) := 1 for all S([n]and C([n]) := 2. Assume that there is a GSP mechanism
M= (Q, x)that is (2 −ε)-BB for Cand charges all players equally. As Mis GSP, it
induces a unique cost-sharing method ξ(confer Theorem C.1). For each set S⊆[n],
let l(S):=ξ1(S)(by assumption, ξi(S) = l(S)for all i∈S).
We first observe that for each j∈[n], it is l([n]\{j})< l([n]). Assume that there
is a j∈[n]with l([n]\{j})≥l([n]). Then
(n−1) ·l([n]\{j}) = X
i∈[n]\{j}
ξi([n]\{j})≤(2 −ε)·C([n]\{j}) = 2 −ε .
50 5 Group-Strategyproof Non-Moulin Mechanisms
It follows that n·l([n]\{j})≤2−ε+l([n]\{j})≤2−ε+2−ε
n−1<2. However,
2 = C([n]) ≤Pi∈[n]ξi([n]) = n·l([n]) ≤n·l([n]\{j})<2yields a contradiction.
In particular, it holds that l([n]\{1})< l([n]) and l([n]\{2})< l([n]). Define the
vector of true valuations vby v1:= v2:= l([n]) and vj:= b+
jfor all j∈[n]\{1,2}.
It has to hold that Q(b+
1,v−1) = [n]\ {2}. Otherwise, if Q(b+
1,v−1) = [n], player
2may submit b2<0in order to decrease the cost-share of all other players while
maintaining zero utility. Analogously, Q(b+
2,v−2) = [n]\{1}.
We finish this proof by looking at the possible values of Q(v)and showing a
contradiction to GSP for all of them. If Q(v) = [n], player 1may bid b+
1in order
to receive the service for a strictly lower cost share. If Q(v)=[n]\{1}or Q(v) =
[n]\ {1,2}, player 1may bid b+
1in order to now receive the service for a strictly
positive utility. If Q(v) = [n]\{2}, the same holds for player 2.
5.5 Symmetric Costs and Two Prices
5.5.1 Two-Price Cost-Sharing Forms
In this section, we apply specific preference-ordered cost-sharing methods (confer
Definition 5.1) with the property that they charge each set of players at most two
different cost shares. The corresponding vectors ξ1, . . . ξnare such that each ξkfor
each cardinality k∈[n]is of the form ξk:= (hk, . . . , hk, lk, . . . , lk)with hk> lk. Play-
ers who are first in the order (the disadvantaged players) pay the higher cost share
hkand the remaining players (the advantaged players) pay the lower cost share lk.
We require that there is always at least one advantaged player and define the number
of disadvantaged players in vector ξkby dk:= |max{i∈[k−1] |ξk
i> ξk
k}|. We call
a contiguous range {s, s + 1, . . . , t} ⊆ [n]of cardinalities with ds=dt+1 = 0 (let
dt+1 := 0 if t=n), and dk>0for k∈ {s+ 1, . . . , t}asegment. That is, only at the
beginning of a segment there is no disadvantaged player paying the higher cost share.
In order to succinctly represent the vectors {ξk}k∈[n], we use two-price cost-sharing
forms (2P-CSFs):
Definition 5.5. Atwo-price cost-sharing form (2P-CSF) is a tuple
F= (n, h,l,d)with n∈N,h,l∈Rn
≥0,and d∈Nn
0.
We furthermore define β-BB of 2P-CSFs analogously to the β-BB of the cost-
sharing methods they induce:
Definition 5.6. Let F= (n, h,l,d)be a 2P-CSF, Cbe a symmetric cost function
and c: [n]→R≥0as in Definition 2.11. Fis β-BB for Cif for all k∈[n],
c(k)≤dk·hk+ (k−dk)·lk≤β·c(k).
5.5.2 Validity of Two-Price Cost-Sharing Forms
We give a validity requirement of 2P-CSFs that allows for defining GSP mechanisms,
as we show in Section 5.5.3.
5.5 Symmetric Costs and Two Prices 51
Definition 5.7. A 2P-CSF (n, h,l,d)is valid if (5.1)–(5.7) hold:
•If there is only one player, he pays the lower cost share:
d1= 0 (5.1)
•Higher cost shares are strictly larger than lower cost shares:
For all k∈[n] : hk> lk(5.2)
•Lower cost shares are non-increasing:
For all k∈[n]\{1}:lk≤lk−1(5.3)
•Lower cost shares stay the same within a segment:
For all k∈[n]\{1}: (lk< lk−1) =⇒dk= 0 (5.4)
•The number of disadvantaged players increases by at most one:
For all k∈[n]\{1}:dk≤dk−1+ 1 (5.5)
•Higher cost shares may only increase at the beginning of a segment:
For all k∈[n]\{1}: (hk> hk−1) =⇒dk= 0 (5.6)
•Higher cost shares may only decrease at the beginning of a segment or if there is
only one disadvantaged player:
For all k∈[n]\{1}: (hk< hk−1) =⇒dk≤1(5.7)
Note that (5.1) and (5.5) imply that there is always at least one advantaged player.
Example 5.8 gives a valid 2P-CSF. The segments are {1,2},{3, . . . , 11}and {12,13}.
Example 5.8. A valid 2P-CSF with resulting vectors ξ1, . . . , ξnfor n= 13:
k hklkdkξk
1 9 3 0 (3)
2 9 3 1 (9,3)
3 6 2 0 (2,2,2)
4 6 2 1 (6,2,2,2)
5 6 2 2 (6,6,2,2,2)
6 6 2 3 (6,6,6,2,2,2)
7 6 2 4 (6,6,6,6,2,2,2)
8 6 2 2 (6,6,2,2,2,2,2,2)
9 5 2 1 (5,2,2,2,2,2,2,2,2)
10 4 2 1 (4,2,2,2,2,2,2,2,2,2)
11 4 2 2 (4,4,2,2,2,2,2,2,2,2,2)
12 7 1 0 (1,1,1,1,1,1,1,1,1,1,1,1)
13 7 1 1 (7,1,1,1,1,1,1,1,1,1,1,1,1)
52 5 Group-Strategyproof Non-Moulin Mechanisms
5.5.3 GSP Mechanisms for Two-Price Cost-Sharing Forms
Before we present our mechanisms that are GSP for any valid 2P-CSF, we give an
intuition how the mechanism for the 2P-CSF in Example 5.8 works. For n= 13,
consider v= (−1,−1,3,2,2,6,6,2,2,2,2,5,5). As there are less than 12 players
with vi≥1, we can at most serve 11 players. On the other hand, we should serve at
least 3players, as there are more than 3players with vi≥2and serving less players
is not consistent with GSP:
•If we only serve 1or 2players with vi≥3, a group of 3players, composed of the
served players and a sufficient number of players with vi≥2, may bid b+
iand
all other players may bid −1. As a result, the players originally served for a cost
share of at least 3now pay a cost share of 2.
•If we serve no player, players 3,4and 5may bid b+
iand all others may bid −1in
order to provide player 3with a strictly positive utility.
Thus, the number of served players is somewhere in the segment {3,11}. Serving all
remaining players who exhibit positive valuations and charging cost shares according
to ξ11 would imply a cost share of 4for the disadvantaged players 3and 4. We first
try to reject a suitable subset of indifferent players with vi= 2, such that all players
with vi>2are served for the low cost share of 2. However, at this point, we need 8
indifferent players while there are only 6. Now, as the bid of the least preferred player
3is smaller than h11 = 4, he is rejected. In the same fashion, we reject players 4and
5as v4< h10 and v5< h9, and there are not enough indifferent players. Thereby,
the number of indifferent players reduces to 4.
In order to charge especially player 6the low cost share, we need 5indifferent
players. Nevertheless, as player 6can pay for the high cost share, he receives the
service for h8= 6. But now, the number of indifferent players we need in order to
serve all remaining players for the low cost share of 2is only 4. Thus, we reject players
8to 11 and serve S={6,7,12,13}, where ξ(S) = (0,0,0,0,0,6,2,0,0,0,0,2,2) as
induced by ξ4.
The intuition is that including the least preferred player for the current higher
cost share never harms the other players. Instead, it may even benefit in that more
players can be served for the low cost share afterwards. Once a player is included for
a higher cost share, this cost share remains fixed during the further execution.
We remark that sequential stand alone mechanisms for submodular costs use the
same approach (confer Appendix B.1). However, whereas subsequent players are ir-
relevant to a specific player for sequential stand alone mechanisms, they are highly
relevant for our mechanism. On the one hand, we account for rejecting indifferent
players, and on the other hand, it is not obvious that subsequent non-indifferent
player are incapable of helping a preceding player.
Before formally defining our mechanism, we give the auxiliary values ρ(k, j)for
all k∈[n]and all j∈[k]indicating how many players iwith rank(i, S)≥jin any
set Swith |S|=kare required to not receive the service such that at most the first
j−1players in Sare disadvantaged:
Definition 5.9. Let F= (n, h,l,d)be a valid 2P-CSF. For all k∈[n]and all
j∈[k], define
ρ(k, j):=(0if dk< j
min{i∈[k−j]|dk−i≤j−1}otherwise. (5.8)
5.5 Symmetric Costs and Two Prices 53
Note that the values ρ(k, j)are well defined. If j∈[dk], we know that especially
dj< j as by validity, there is always at least one advantaged player. Note that by
validity property (5.5), for j∈[dk]it holds that min{i∈[k−j]|dk−i≤j−1}=
min{i∈[k−j]|dk−i=j−1}. Here, ρ(k, j)even gives the number of required
players such that exactly the first j−1players are disadvantaged.
Example 5.10. For the 2P-CSF from Example 5.8 it is ρ(11,1) = 8, ρ(11,2) = 1, and
ρ(11, j) = 0 for all 2< j ≤11.
Now we are ready to state mechanism MechCSFFin Algorithm 5.1:
Algorithm 5.1 (computing MechCSFF(b) = (Q(b), x(b)) for F= (n, h,l,d)).
1: k:= max nj∈[n]0|{i∈[n]|bi≥lj}| ≥ jo.Starting cardinality
2: if k= 0 then return (∅,0)
3: l:= lk.Low cost-share
4: A:= {i∈[n]|bi≥l}.(potential) Advantaged players
5: I:= {i∈[n]|bi=l}.(alleged) Indifferent players
6: D:= ∅.Disadvantaged players
7: loop
8: `:= min A . Least preferred player
9: r:= rank(`, A ∪D)
10: if |I| ≥ ρ(|A|+|D|, r)then .Sufficient indifferent players?
11: A:= A\{ρ(|A|+|D|, r)largest elements of I}
12: break
13: if b`≥h|A|+|D|then D:= D∪{`}.Make disadvantaged
14: A:= A\{`};I:= I\{`}
15: xi:= h|A|+|D|for i∈D;xi:= lfor i∈A;xi:= 0 otherwise
16: return (A∪D, x)
It is easy to verify that for every β-BB 2P-CSFF,MechCSFFis β-BB and meets
NPT, VP, and strict CS. Lemma 5.11 will be helpful for finally proving GSP of
MechCSFFin Theorem 5.12:
Lemma 5.11. For any valid 2P-CSF F= (n, h,l,d)it holds for all k∈[n]and all
j∈[dk]that
ρ(k, j + 1) ≤ρ(k, j)−1and (5.9)
ρ(k−1, j) = ρ(k, j)−1.(5.10)
Proof. To see (5.9), let k0< k be maximum with dk0=j−1, i.e., ρ(k, j) = k−k0.
Furthermore, let k00 ≤kbe maximum with dk00 =j, i.e., ρ(k, j + 1) = k−k00. Either
k00 =kand consequently ρ(k, j +1) = 0 ≤k−k0−1 = ρ(k, j)−1, or k0< k00 < k (by
validity property (5.5)) implying that ρ(k, j + 1) = k−k00 ≤k−k0−1 = ρ(k, j)−1.
The second property (5.10) is straightforward to show as for k0< k being maximum
with dk0=j−1,ρ(k, j) = k−k0and ρ(k−1, j) = k−1−k0.
Note that Lemma 5.11 especially implies that adding a player `to set Din line 13
never harms subsequent players in A\{`}. To see this, look at the number of indif-
ferent players needed to evaluate line 10 to true. If `is added to D, by (5.9), this
number decreases by at least one for the next iteration. On the other hand, if `is
not added, by (5.10), it only decreases by exactly one. For an intuition, the 2P-CSF
from Example 5.8 yields ρ(11,1) = 8 and ρ(11,2) = 1, but ρ(10,1) = 7.
54 5 Group-Strategyproof Non-Moulin Mechanisms
Theorem 5.12. For any valid 2P-CSF F, MechCSF Fis GSP.
Proof. For any input b∈Rn, denote by k(b)and l(b)the values of kand lin
lines 1 and 3 of Algorithm 5.1. Moreover, set s(b) := max{j∈[k(b)] |dj= 0}to
the beginning of the segment that k(b)is in (if k(b) = 0, let s(b) = 0). Note that
s(b)≤ |Q(b)| ≤ k(b).
Assume that MechCSF Fis not GSP. Then, there exists a coalition K⊆[n]
with true valuations vK= (vi)i∈K∈R|K|and a bid vector b∈Rnsuch that
ui(b, vi)≥ui((vK,b−K), vi)for all i∈K, with at least one strict inequality. Without
loss of generality, let v:= (vK,b−K).
•Without loss of generality, we may assume that s(b) = s(v):
– If s(b)> s(v), it holds that s(b)> k(v), and thus |Q(b)|> k(v). By choice
of k(v), this means that the number of players receiving the service for bis
strictly larger than the number of players who can actually afford to pay at
least l(b). Formally, there is a player j∈Q(b)with vj< l(b)because k(v)
would not have been maximal in line 1 otherwise. Since j∈Q(b), it holds that
bj≥xj(b)≥l(b)> vj, so we have j∈Kand uj(b, vj)<0≤uj(v, vj), a
contradiction.
– If s(b)< s(v), it holds that k(b)< s(v)and thus |Q(b)|< s(v). Here, the
idea is to change the bid vector bto b0in such a way that exactly s(v)players
receive the service for b0while utilities can only improve. Formally, let L:=
{j∈[n]|bj≥l(v)}. Clearly, |L|< s(v)as k(b)would not have been maximal
in line 1 otherwise. Now, let Mbe a set of s(v)−|L|players j∈[n]\Lwith
l(v)≤vj. Such a set Mexists. Define a new bid vector b0∈Rnby b0
j:= l(v)
for j∈Mand b0
j:= bjotherwise. Then s(b0) = |Q(b0)|=k(b0) = s(v)and
uj(b0, vj)≥uj(b, vj)for all players j∈[n].
Let now l:= l(b) = l(v)and L:= {`1, . . . , `t}be the players that are considered for
vin Line 8, where player `kis examined in loop k∈[t].
•Without loss of generality, we may assume that K⊆ L:
For input v, players i /∈ L either receive the service for lor are rejected due to
vi=l. These players can neither strictly improve their utility for another outcome
in the segment, nor change the outcome for vby bidding bi6=viwithout strictly
decreasing their utility.
Let `jbe the smallest player in L∩Kthat strictly increases his utility and define
L<j := {`1, . . . , `j−1}and L>j := {`j+1, . . . , `t}.
•Without loss of generality, we may assume that K⊆ L>j ∪{`j}:
Players L<j ∩Q(v)do not harm subsequent players if they continue to receive
the service (confer Lemma 5.11), and players in L<j \Q(v)strictly decrease their
utility if they bid in order to receive the service.
•Without loss of generality, we may assume that vi> l for all i∈Kand that
bi=land i /∈Q(b)for all i∈K\{`j}:
The only way for player i∈K\{`j}to be beneficial for player `jis to bid in order
to not receive the service. Players with vi=lare considered to be rejected anyway,
thus vi> l for all i∈K. Players i∈K\{`j}may bid bi=las the mechanism
then rejects these players as desired. Without loss of generality we may consider
a coalition with minimum cardinality, thus i /∈Q(b)for all i∈K\{`j}.
5.5 Symmetric Costs and Two Prices 55
By the above assumptions, the runs of both mechanisms with inputs vand bare
the same up to consideration of player `j. Let ρbe the number of players needed in
Line 10 in loop jof MechCSFFfor inputs vand b, respectively. Define
I:= {i∈[n]|vi=l}\{i∈ L<j |vi=l}.
Figure 5.1 particularly illustrates the sets Kand Iand is helpful for understanding
the remaining part of the proof.
As there were not enough indifferent players for vto be rejected, it holds that
|I|< ρ. For input bhowever, there are just enough (alleged) indifferent players, thus
|K\ {`j}| +|I|=ρ. Define `kto be the player in K\ {`j}that is highest in the
order. Let R:= I∩{`j+1, . . . , `k−1}. By definition, |K\{`j}|+|R| ≤ k−j. Let ρ0be
the number of indifferent players needed for `kfor input v. It is ρ0≤ρ−(k−j)by
iteratively applying Theorem (5.11.) and thus ρ0≤ρ−|K\{`j}|−|R|=|I|−|R|.
It follows that `kreceives the service for vand pays l. A contradiction, as `k/∈Q(b)
and vk> l, and thus u`k(b, v`k) = 0 < v`k−l=u`k(v, v`k).
Fig. 5.1. Sketch for the proof of Theorem 5.12. Shows sequence of player `1,...,`tconsidered in
the run for v, and players i1, i2, i3still in the game for vafter deleting players iwith vi< l. Shaded
players iare those with vi=l.
`1`j−1`j`j+1 `j+2 `j+3 `j+4 `j+5 i1i2i3
|{z}
=`k=`t
K K K
I,R I,R I I
Lemma 5.13. For each 2P-CSF F that is computable in time O(t),MechCSF Fis
computable in time O(t+n2).
Proof. For the running time, consider Algorithm 5.1. In the worst case, it has to
compute the whole 2P-CSF F in time O(t). Given these values, observe that every
operation outside the loop takes time at most O(n2). The loop is executed at most
ntimes and each operation in the loop takes at most time O(n). An overall running
time of O(t+n2)results.
5.5.4 √17+1
4-BB Two-Price Cost-Sharing Forms for Subadditive Costs
Theorem 5.14 is the main theorem of this section. It is an immediate corollary of
Theorem 5.15, Theorem 5.16, Theorem 5.12, and Lemma 5.13.
Theorem 5.14. For any symmetric and subadditive cost function C, there is a 2P-
CSF F such that MechCSFFis GSP and √17+1
4-BB for C. Moreover, if Cis given
as an array of nfunction values, it can be computed in time O(n2).
Algorithm 5.2 computes a 2P-CSF for increasing cardinalities.
56 5 Group-Strategyproof Non-Moulin Mechanisms
Algorithm 5.2 (computing a 2P-CSF for subadditive costs).
Input: symmetric and subadditive cost function C(cas in Definition 2.11)
Output: valid 2P-CSF (n, h,l,d)
1: h:= l:= d:= 0;β:= √17+1
4
2: l1:= β·c(1);h1:= ∞;d1:= 0;f:= 1
3: for k:= 2, . . . , n do
4: if β·c(k)
k≤lfthen lk:= β·c(k)
k;hk:= ∞;dk:= 0 ;f:= k
5: else
6: lk:= lk−1;hk:= min{β·c(k)−(k−1) ·lk, hk−1}
7: if hk+ (k−1) ·lk< c(k)then
8: dk:= 2
9: else if hk+ (k−1) ·lk≥2·c(f)then
10: dk:= 1
11: else if hk≥(β2−β)·c(f)then
12: dk:= 1
13: if (β2−β)·c(f)+(k−1) ·lk≥c(k)and f≥4then
14: hk:= (β2−β)·c(f)
15: else
16: dk:= 0;hk:= ∞
17: return (n, h,l,d)
Here, “∞” is a placeholder for any “sufficiently large” value (a value strictly larger
than β·c(f)is sufficient) to simplify the presentation.
Theorem 5.15. For any symmetric cost function C, given as an array of nfunction
values, the 2P-CSF computed by Algorithm 5.2 is valid and computable in time O(n).
Proof. Fix a symmetric cost function Cand define c: [n]→R≥0according to Defi-
nition 2.11. Let (n, h,l,d)be the output 2P-CSF. Consider an arbitrary cardinality
k∈[n]. Clearly, di∈ {0,1,2}.
•Condition (5.1) (d1= 0) trivially holds by line 2.
•Condition (5.2) (hk> lk) is shown by induction: Obviously, h1=∞> β·c(1) = l1.
Assume that hj> ljfor all j∈[k−1]. In the case hk=∞there is nothing to
show. If hkis set to some different value, then especially line 4 evaluated to false.
We define f:= max{j∈[k−1] |β·c(j)
j=lk}to be the last cardinality previous
to kfor which the lower cost share was set in lines 2 or 4. Now, c(k)
k>c(f)
f=lf
β
and lk=. . . =lf. The value hkis set for the first time in line 6, and either
hk=hk−1> lk−1=lkor hk=β·c(k)−(k−1) ·lk> lk. If hkis set in line 14,
hk= (β2−β)·c(f) = (β−1) ·f·lk>0.28 ·4·lk> lk.
•Condition (5.3) (lk≤lk−1) and Condition (5.4.) (lk< lk−1=⇒dk= 0) hold since
lk6=lk−1only if lkwas set in line 4.
•Condition (5.5) (dk≤dk−1+ 1) has only be checked for dk= 2. Then hk<
c(k)−(k−1) ·lkbecause line 7 evaluated to true and thus hk= min{β·c(k)−
(k−1) ·lk, hk−1}=hk−1. Now assume dk> dk−1+ 1, i.e., dk−1= 0. Then,
hk=hk−1=∞, a contradiction to hk< c(k)−(k−1) ·lk.
•Condition (5.6) (hk> hk−1=⇒dk= 0) holds since line 6 ensures that from
dk>0it follows that hk≤hk−1.
5.5 Symmetric Costs and Two Prices 57
•Condition (5.7) (hk< hk−1=⇒dk≤1) holds as for dk= 2 it is hk=hk−1as
shown for Condition (5.5.).
Computability in O(n)is obvious.
Theorem 5.16. For any subadditive and symmetric cost function C, the 2P-CSF
computed by Algorithm 5.2 is √17+1
4-BB.
Proof. Fix a subadditive and symmetric cost function Cand let cbe as in Defin-
ition 2.11. For all k∈[n], we define γ(k) := dk·hk+ (k−dk)·lkand show that
c(k)≤γ(k)≤√17+1
4·c(k)for all k∈[n].
•If dk= 1,hkis either set in line 6 or in line 14. In both cases hk≤β·c(k)−(k−1)·lk
and thus γ(k) = hk+ (k−1) ·lk≤β·c(k). Furthermore, as line 7 evaluated to
false and hiis only reset to (β2−β)·c(f)if line 13 evaluates to true, γ(k)≥c(k).
For dk∈ {0,2}, we define f:= max{j∈[k]|β·c(j)
j=lk}to be the last cardi-
nality previous or equal to kfor which the lower cost share was set in lines 2 or
4. Furthermore, let g:= min({j∈ {k+ 1, k + 2, . . . , n} | β·c(j)
j≤lk} ∪ {n+ 1})
be the next such cardinality after k(or g=n+ 1 if fis the largest such cardi-
nality). It is f≤k < g ≤2f. Otherwise, if g > 2f,gwould not be minimum
due to c(2f)
2f≤2·c(f)
2f=c(f)
fbecause of subadditivity. Since cis non-decreasing,
c(k)≤c(2f)≤2·c(f). Set h0
k:= min{β·c(k)−(k−1) ·lk, hk−1}. We will make use
of the following property: for k≥3,
dk−1= 1 and γ(k−1) ≥2·c(f) =⇒ ∀j∈ {k, k + 1, . . . , g −1}:dj= 1 .(5.11)
Proof of (5.11): If h0
k=hk−1, then h0
k+ (k−1) ·lk=γ(k−1) + lk. If h0
k=
β·c(k)−(k−1) ·lk, then h0
k+ (k−1) ·lk=β·c(k)≥β·c(k−1) ≥γ(k−1). In
any case, h0
k+ (k−1) ·lk≥γ(k−1) ≥2·c(f)≥c(k)and line 9 evaluates to true.
Then hk=h0
kand dk= 1. Inductively, (5.11) follows.
•Consider dk= 2. Observe that d1= 0 and d2= 0 due to c(2)
2≤c(1)
1. Thus, by
validity, k≥4. We first show hk= (β2−β)·c(f). Define k0:= max{j∈[k]|dj=
1}. Let s:= max{j∈[k]|dk= 0}be the start of the segment that kis in. By
validity, f≤s < k0< k,ds+1 = 1, and hs+1 ≥hk=hk0.
Since line 9 evaluated to false for cardinality k0because of (5.11), line 11 must
have evaluated to true, implying hk=h0
k≥(β2−β)·c(f). Now assume ‘>’. From
ls+1 =ls=β
f·c(f)≥β
s·c(f)and the fact that line 13 evaluated to false for
cardinality (s+ 1) (as by assumption hs+1 ≥hk>(β2−β)·c(f)), we conclude
that β2·c(f)≤(β2−β)·c(f) + s·ls+1 < c(s+ 1).
Then, however, hs+1 +s·ls+1 =β·c(s+ 1) > β3·c(f)>2·c(f), a contradiction,
since line 9 must have evaluated to false due to (5.11). Hence, hk= (β2−β)·c(f)
and thus
γ(k)=2hk+ (k−2) ·lk
= 2hk+ (k−2) ·β·c(f)
f
≥(2β2−β)·c(f)
= 2 ·c(f)
≥c(k).
58 5 Group-Strategyproof Non-Moulin Mechanisms
Also, β·c(f)< β2·c(f) = hk+β·c(f)≤hk+s·ls+1 < hk+(k−1)·lk< c(k), where
the last inequality holds since line 7 evaluated to true. Applying the inequalities
above, we get
γ(k)=2hk+ (k−2) ·lk
< hk+c(k)
= (β2−β)·c(f) + c(k)
<(β−1) ·c(k) + c(k)
=β·c(k).
•Now let dk= 0. If k=f, then γ(k) = β·c(k). In the following, we consider
k > f. Since line 11 evaluated to false, h0
k<(β2−β)·c(f). We first show that
h0
k=β·c(k)−(k−1) ·lk. Assume otherwise. Then, h0
k=hk−1and dk−1= 1
since hk−1/∈ {∞,(β2−β)·c(f)}. Yet, line 9 evaluated to false for (k−1) because
of (5.11). Thus, h0
k=hk−1≥(β2−β)·c(f)by line 11 which contradicts h0
k<
(β2−β)·c(f).
Now β·c(k) = h0
k+ (k−1) ·lk<(β−1) ·β·c(f) + (k−1) ·lk. Furthermore,
γ(k) = k·lk=k·lf=k·β
f·c(f)> β ·c(f)and γ(k)>(k−1) ·lk. Putting
everything together gives
c(k) = β·c(k)
β≤(β−1) ·γ(k) + γ(k)
β=γ(k).
Moreover, as line 4 evaluated to false, γ(k) = k·lf< k ·β
k·c(k) = β·c(k).
Theorem 5.17 shows that √17+1
4-BB is in general the best that can be achieved with
our technique.
Theorem 5.17. For all ε > 0, there is a symmetric and subadditive cost function C
for which no valid √17+1
4−ε-BB 2P-CSF exists.
Proof. Fix β:= √17+1
4, let 0< ε ≤β−1, and set α:= β−ε. Additionally, let
j, l ∈Nwith l > lnββ−1
εand j > (l+1)·α
ε=(l+1)·β
ε−(l+ 1). Set m:= j+l+ 1 and
n:= m+ 1 and consider a symmetric cost function Cthat induces cdefined below:
k1··· j j + 1 j+ 2 ··· j+l m n
c(k) 1 ··· 1β−β−1
β1β−β−1
β2··· β−β−1
βlβ2
Clearly, Cis subadditive. Now assume there is a valid 2P-CSF (n, h,l,d)which
is α-BB. For all k∈[n], we again define γ(k):=dk·hk+ (k−dk)·lk.
The rough idea is the following: It can be shown that dm≥1and dn=dm+1. Let
s:= max{k∈[m−1] |dk= 0}be the start of the segment that cardinality nis in.
Clearly, s < m and hn≤hs+1. By case analysis, hs+1 ≤α·c(s+ 1) −c(s)< β2−β.
Thus γ(n)≤hn+α·β < 2·β2−β= 2 = c(n), a contradiction to α-BB. In detail,
•dm≥1: Otherwise we obtain a contradiction to α-BB:
γ(m) = m·lm≤m·lj≤m·α
j=α·1 + l+ 1
j< α ·1 + ε
α=β=c(m).
5.5 Symmetric Costs and Two Prices 59
•dn=dm+1 =dm+ 1: Otherwise we again obtain a contradiction to α-BB:
γ(m+ 1) ≤α·β+lm+1 < β2+α
j< β2+ε
l+ 1 ≤β2+ε
2≤β2+β−1
2<2.
•Bounds on hs+1: Due to α·c(s+ 1) ≥γ(s+ 1) = hs+1 +γ(s)≥hs+1 +c(s), it is
hs+1 ≤α·c(s+ 1) −c(s)for all s∈[n−1].
– If s∈[j−1] then
hs+1 ≤α·c(s+ 1) −c(s) = α−1< β2−β .
– If s∈ {j, j + 1, . . . , j +l−1}, let k:= s+ 1 −j. Then
hs+1 ≤α·c(s+ 1) −c(s)
=α·c(j+k)−c(j+k−1)
=α·β−β−1
βk−β−β−1
βk−1
< β2−β−1
βk−1−β−β−1
βk−1
=β2−β .
– If s=j+l=m−1then
hs+1 ≤α·c(m)−c(m−1) = α·β−β−β−1
βl
< α ·β−(β−ε) = α·(β−1) < β2−β .
5.5.5 Applications To Scheduling Identical Jobs
We like to remind the reader that for (Q|pi=1|Cmax)cost-sharing problems, an op-
timal solution can be computed in polynomial time, e.g., by the LPT algorithm in
time O(n·log m). Theorem 5.18 and Lemma 5.19 give our positive results with re-
spect to approximate budget-balance, and Lemma 5.20 gives the lower bound on the
approximation of social cost efficiency.
Theorem 5.18. For each (Q|pi=1|Cmax)cost-sharing problem there is a 2P-CSF F
such that mechanism MechCSFFis GSP, √17+1
4-BB for MSP, and computable in
O(n2).
Proof. As the cost function MSP for identical jobs is symmetric and subadditive, the
theorem follows by Theorem 5.14. The evaluation of msp : [n]→R≥0has no influence
on the asymptotic running time, as computing msp(i)for all i∈[n]is accomplished
by one run of LPT in time O(n·log m).
Lemma 5.19. For each (P|pi=1|Cmax)cost-sharing problem there is a 2P-CSF F
such that mechanism MechCSFFis GSP, 1-BB, and computable in O(n2).
Proof. Consider a (P|pi=1|Cmax)cost-sharing problem with nplayers and mma-
chines. We define a valid 1-BB 2P-CSF (n, h,l,d)that is computable in time O(n).
The lemma then follows by Theorem 5.12 and Lemma 5.13.
60 5 Group-Strategyproof Non-Moulin Mechanisms
Consider an arbitrary k∈[n]and let k=p·m+qfor p, q ∈N0and q < m.
•If k∈[m], define hk:= ∞,lk:= 1
k, and dk:= 0. It holds that msp(k) = 1.
•If k > m and q= 0, define hk:= ∞,lk:= 1
m, and dk:= 0. It holds that
msp(k) = p.
•Otherwise, define hk:= msp(k)−(k−1) ·1
m,lk:= 1
m, and dk:= 1. It holds that
msp(k) = p+ 1. In this case, hk=p+ 1 −(k−1)
m>1
m=lk.
We remark that the same 2P-CSF results by applying Algorithm 5.2 and dividing
all cost-shares by √17+1
4.
Lemma 5.20. There is a (P|pi=1|Cmax)cost-sharing problem for which MechCSFF
for the 2P-CSF Fcomputed by Algorithm 5.2 (or defined in the proof of Lemma 5.19)
is no better than Ω(n)-EFF.
Proof. Consider the (P|pi=1|Cmax)cost-sharing problem with n= 2m−1. The
corresponding symmetric costs are msp(k)=1for all k∈[m]and msp(k)=2
for all k∈ {m+ 1, . . . , 2m−1}. For the computed 2P-CSF Fby Algorithm 5.2
and β:= √17+1
4it holds that for all k∈[m],dk= 0 and lk=β
k, and for all
k∈ {m+ 1, . . . , 2m−1},dk= 1,lk=β
kand hk=β·2m−k+1
m(in particular, line 4
evaluates to true for all k∈[m]. Furthermore, for all k∈ {m+ 1, . . . , 2m−1}, lines
4 and 7 evaluate to false and line 9 to true, as hk+ (k−1) ·lk= 2 ·β, and f=m
in any case). For v:= (h2m−1−ε, . . . , hm+1 −ε, 1
m+ε, . . . , 1
m+ε),ε < 2m(β−1),
and MechCSF F= (Q, x)it holds that Q(v) = [m]and
SC MSP([m],v) = 1 +
2m−1
X
k=m+1
β·2m−k+ 1 −ε
m>1 +
m
X
k=2
k
m= 1 + m+ 1
2−1
m
However, SC MSP([2m−1],v) = 2. Note that we get the same lower bound for the
2P-CSF from the proof of Lemma 5.19.
5.6 Non-Symmetric Costs
In this section we apply our ideas for symmetric costs to settings with non-symmetric
costs in two flavors. First, for (Q||Cmax)cost-sharing problems we show that di-
viding the players in dsets with equal processing times and applying MechCSF
separately for every set yields GSP and (√17+1
4·d)-BB mechanisms. Second, for
(P3|pi∈ {1,2}|Cmax)cost-sharing problems, we generalize the notion of a preference-
ordered cost-sharing method and re-use the ideas of mechanism MechCSF to obtain
GSP and even 1-BB.
5.6.1 Applications To The General Scheduling Setting
Lemma 5.21. For each (Q||Cmax)cost-sharing problem there is a mechanism that
is GSP and (√17+1
4·d)-BB for CLPT, where dis the number of different processing
times. Furthermore, it is computable in O(n2).
5.6 Non-Symmetric Costs 61
Proof. Fix p∈Nnand s∈Nmand consider the cost-sharing problem (p,s). For
each set S⊆[n], we define P(S) := {pi|i∈S}as the set of different processing
times of the jobs in Sand let P(S) := {t1, . . . , t|P(S)|}. Define N:= [n]and Nt:=
{i∈N|pi=t}for all t∈ P(N).
For each input vector b∈Rn, the mechanism M= (Q, x)works as follows:
•For each t∈ P(N):
– Compute the 2P-CSF (|Nt|,h,l,d)from Algorithm 5.2 for costs induced by
|Nt|identical jobs, i.e., for cost function msp.
– Define a 2P-CSF Ft:= (|Nt|, t ·h, t ·l,d)by multiplying the cost shares by t.
– Use MechCSF Ftwith input bNtto decide on the set of served players Qt⊆Nt.
•Let ξt(Qt)be the cost-shares induced by Ftand set Q(b) := ∪t∈P(N)Qtand
xi(b):=ξpi
i(Qpi)for all i∈Q(b)and xi(b) := 0 otherwise.
This mechanism is clearly GSP. It is (√17+1
4·d)-BB for CLPT since with
X
i∈Q(b)
xi(b) = X
t∈P(Q(b)) X
i∈Qt
ξt
i(Qt)
it holds that
X
i∈Q(b)
xi(b)≤X
t∈P(Q(b))
t·√17 + 1
4·msp(|Qt|)
=√17 + 1
4·X
t∈P(Q(b))
MSP(Qt)
≤√17 + 1
4·|P(Q(b))|·MSP(Q(b)) ,
and moreover,
X
i∈Q(b)
xi(b)≥X
t∈P(Q(b))
t·msp(|Qt|)
=X
t∈P(Q(b))
MSP(Qt)
=X
t∈P(Q(b))
CLPT(Qt)
≥CLPT(Q(b)) .
The last inequality is due to Lemma 3.4.
Computing msp(i)for all i∈Ntakes time O(n·log m)by one run of LPT for n
identical jobs. For each t∈ P(N), the computation of Ftand each run of MechCSFFt
with |Nt|players takes O(|Nt|2)time. Since Pt∈P(N)|Nt|2≤n2, this algorithm is
computable in O(n2).
62 5 Group-Strategyproof Non-Moulin Mechanisms
5.6.2 Applications To Scheduling On Identical Machines
We consider (P3|pi∈ {1,2}|Cmax)cost-sharing problems. Here, we assume that
p1≤. . . , ≤pn. We adjust the definition of a preference ordered cost-sharing method:
Definition 5.22. A cost-sharing method ξ: 2[n]→Rn
≥0is 2-type preference ordered,
if for every (s1, s2)∈[n]2
0with s1+s2≤n, there is a vector ξ(s1,s2)∈R(s1+s2)
≥0such
that for all players i∈[n]and all sets of players S⊆[n]with s1:= |{i∈S|pi= 1}|
and s2:= |{i∈S|pi= 2}| we have
ξi(S):=(ξ(s1,s2)
rank(i,S)if i∈S
0otherwise.
Example 5.23. Let n= 6,ξ1,2= (2,3,3) and consider S={1,4,5}with p1= 1 and
p4=p5= 2. Then ξ(S) = (2,0,0,3,3,0).
Theorem 5.24. For each (P3|pi∈ {1,2}|Cmax)cost-sharing problem, there is a
mechanism that is GSP and 1-BB and runs in time O(n).
Proof. We first define a 2-type preference-ordered cost-sharing method according to
a matrix X. Subsequently, we define mechanism MX.
We first define vectors ξs1,s2for all s1, s2∈[n]0with s1+s2≤nthat induce
a 2-type preference-ordered cost-sharing method. There are fixed low cost shares of
1
3for players with pi= 1 and of 2
3for players with pi= 2. However, some least
preferred players have to pay higher cost shares which we define with help of matrix
X(here given for n= 9, but intuitively extendable to arbitrary n):
X:=
∗(−,2) (−,4
3)∗(−,2) (−,4
3)∗(−,2) (−,4
3)∗
(1,−) (1,1) (2
3,−) (1,−) (1,1) (2
3,−) (1,−) (1,1) (2
3,−) (1,−)
(2
3,−) (2
3,1) ∗(2
3,−) (2
3,1) ∗(2
3,−) (2
3,1) ∗(2
3,−)
∗(−,1) (−,11) ∗(−,1) (−,11) ∗(−,1) (−,11) ∗
(1,−)∗(−,1) (−,11) ∗(−,1) (−,11) ∗(−,1) (−,11)
(2
3,−) (1,−)∗(−,1) (−,11) ∗(−,1) (−,11) ∗(−,1)
∗(2
3,−) (1,−)∗(−,1) (−,11) ∗(−,1) (−,11) ∗
(1,−)∗(2
3,−) (1,−)∗(−,1) (−,11) ∗(−,1) (−,11)
(2
3,−) (1,−)∗(2
3,−) (1,−)∗(−,1) (−,11) ∗(−,1)
∗(2
3,−) (1,−)∗(2
3,−) (1,−)∗(−,1) (−,11) ∗
Note that we start indexing at 0. Each entry X(s1, s2)=(X1(s1, s2),X2(s1, s2))
consists of two vectors: the vector X1(s1, s2)of higher cost shares for players with
pi= 1 and the vector X2(s1, s2)of higher cost shares for players with pi= 2. All
vectors except (1,1) (which we abbreviate with ‘11’) consist of only one cost share.
To increase readability, we write ‘−’ for the empty vector and ‘∗’ for (−,−). The
resulting vector of cost-shares is defined as
ξ(s1,s2):=
X1(s1, s2),1
3, . . . , 1
3
| {z }
s1
,X2(s1, s2),2
3, . . . 2
3
| {z }
s2
Example 5.25. ξ(2,4) = (2
3,1
3,1,2
3,2
3,2
3),ξ(2,2) = (1
3,1
3,2
3,2
3), and ξ(0,2) = (4
3,2
3).
5.6 Non-Symmetric Costs 63
Algorithm 5.3 (computing MX(b) = (Q(b), x(b)) for matrix X).
1: Aj:= {i∈[n]|pi=jand bi≥j
3} ∀j∈[2] .(potential) Advantaged players
2: if |A1|= 1 and bmin A1<2
3then A1:= ∅
3: Ij:= |{i∈Aj|bi=j
3}| ∀j∈[2] .(alleged) Indifferent players
4: Dj:= ∅ ∀j∈[2] .Disadvantaged players
5: loop
6: (ρ1, ρ2):=lex. smallest element in (t1, t2)∈[|I1|]0×[|I2|]0with
7: Xj|D1|+|A1|−t1,|D2|+|A2|−t2=|Dj| ∀j∈[2]∪(n, n)
8: if (ρ1, ρ2) = (n, n)and loop is executed for the first time then
9: if X(|A1|,|A2|) = (1,−)and |I2| ≥ 1then (ρ1, ρ2) := (0,1)
10: if X(|A1|,|A2|) = (1,1) and |I2| ≥ 2then (ρ1, ρ2) := (0,2)
11: if (ρ1, ρ2)6= (n, n)then .Sufficient indifferent players?
12: Aj:= Aj\{ρjlargest elements of Ij} ∀j∈[2]
13: break
14: S:= A1∪D1∪A2∪D2;sj:= |Aj|+|Dj| ∀j∈[2]
15: Let `∈A1∪A2be minimum with ξ(s1,s2)
rank(`,S)>p`
3.Least preferred
16: if b`≥ξ(s1,s2)
rank(`,S)then Dp`:= Dp`∪{`}.Make disadvantaged
17: Ap`:= Ap`\{`};Ip`:= Ip`\{`}
18: x:= ξ(A1∪D1∪A2∪D2), where ξis the cost-sharing method induced by X
19: return (A1∪D1∪A2∪D2,x)
Initially, players with insufficient bids are rejected. Then, we try to remove indifferent
players such that all remaining players in A1∪A2pay their minimum cost-share. In
case of success, (ρ1, ρ2)6= (n, n)in line 11. Otherwise, the first player not in D1∪D2
who has to pay more than his minimum cost-share is accepted (for a cost-share that
stays fixed during the further execution) or rejected, according to his bid. In lines 9
and 10, special cases are considered in which a player with processing time 1improves
his cost-share from 1to 2
3(and not to the minimum cost-share). This check is only
required in the first iteration; otherwise, the existence of sufficient indifferent players
is already precluded by (ρ1, ρ2) = (n, n)after line 6.
Fig. 5.2. Sketch for Example 5.26
.
(−,4
3)∗
(2
3,−)(1,−)
(−,2)
(1,1)
(1)
(2)
(3)
Example 5.26. Let n= 6,p:= (1,1,2,2,2,2), and consider b:= (1
3−ε, 1,2
3,2,2,2).
Initially, A1={2},A2={3,4,5,6},I1=∅,I2={3}, and D1=D2=∅. It is
X(|A1|+|D1|,|A2|+|D2|) = (1,1), and to ‘move’ to the next (∗)entry (confer
arrow (1) in Figure 5.2), we need that |I1| ≥ 1and |I2| ≥ 1. As this is not the
case, (ρ1, ρ2)=(n, n)in line 8. Even after line 10, (ρ1, ρ2)=(n, n), as also entry
(2
3,−)cannot be reached due to |I2| 6≥ 2(confer arrow (2) in Figure 5.2). Thus, in
64 5 Group-Strategyproof Non-Moulin Mechanisms
line 15, `= 2. As b2≥1,D1={2}and 2is removed from A1. In the next iteration,
again X(|A1|+|D1|,|A2|+|D2|) = (1,1), and to ‘move’ to entry (1,−),|I2| ≥ 1
is required (confer arrow (3) in Figure 5.2). As this is possible, (ρ1, ρ2) = (0,1) in
line 6, player 3is removed from A2in line 12, and the set {2,4,5,6}is served for
cost-shares ξ(1,3) = (1,2
3,2
3,2
3).
Budget-Balance
1-BB follows by direct computation, e.g., for S⊆[n]with (s1, s2) = (2,4), it holds
that MSP(S)=4. For S⊆[n]with (s1, s2) = (2,2) or (s1, s2) = (0,2),MSP(S) =
2. Example 5.25 shows that these costs are exactly recovered. The straightforward
verification for the remaining cases is left to the reader.
Transitions
Showing GSP is rather intricate as there are many case differentiations. We define
transitions that are helpful for finally proving GSP. We look at all possible runs of MX
that consider line 15 at least 2times, i.e., there are at least two disadvantaged players
who are either rejected or receive the service for a high cost share. By definition of
X, line 15 is executed at most three times. This can be observed by the ‘∗’ entries
in Xthat act as a kind of ‘barriers’.
For one specific run for bid vector b∈Rnin which line 15 is executed t∈ {2,3}
times, we define a transition to be the sequence of the tentries X(s1, s2), where
s1, s2are defined in line 14. We let `kbe the player considered during the k-th call
of line 15. All possible transitions are given in Table 5.1. We write +, if `kis served
(i.e. accepted in line 16), and −otherwise (+/−if both cases may occur).
Table 5.1. All possible transitions of length 2and 3
transition `1`2`3p`1p`2p`3
(−,4
3)→(−,2) −+/−2 2
(1,1) →(1,1) + +/−1 2
(1,1) →(−,2) −+/−1 2
(2
3,−)→(1,−)−+/−1 1
(2
3,1) →(2
3,1) + +/−1 2
(−,11) →(−,11) + +/−2 2
(−,11) →(−,1) −+/−2 2
(2
3,1) →(1,1) →(1,1) −+ +/−1 1 2
(2
3,1) →(1,1) →(−,2) − − +/−1 1 2
Proof of GSP
Assume that there is a (P3|pi∈ {1,2}|Cmax)cost-sharing problem (p,1)for which
MX= (Q, x)is not GSP. Then, there exists a coalition K⊆[n]with true valuations
vK= (vi)i∈K∈R|K|and a bid vector b∈Rnsuch that ui(b, vi)≥ui((vK,b−K), vi)
for all i∈K, with at least one strict inequality. Without loss of generality, let
v:= (vK,b−K).
It can easily be verified that the mechanism is SP. This excludes the case |K|= 1.
Define A1, A2and I1, I2to be the initial sets computed in lines 1and 3for input v.
If one of the special cases from lines 9 and 10 occurs, then MXterminates. It follows
5.6 Non-Symmetric Costs 65
that min A1has to be the player in Kwho strictly improved his utility. However,
then there is at least one player i∈Q(v)with vi>pi
3=xi(v)≥bisuch that
i /∈Q(b)and 0 = ui(b, vi)< ui(v, vi); a contradiction. Furthermore, observe that
line 15 is executed at least once, as otherwise all players iwith vi>pi
3receive the
service and pay pi
3, hence no player may strictly improve his utility for b. Let tbe
the number of calls of line 15 for MXwith input vand let L:= {`1, . . . , `t}be the
disadvantaged players, where `kis considered for the k-th call.
Without loss of generality, we may assume that K⊆ L: For input v, players
i /∈ {`1, . . . , `t}either receive the service for pi
3or are rejected due to vi=pi
3. These
players can neither strictly improve their utility for another outcome, nor change the
outcome for vby bidding bi6=viwithout strictly decreasing their utility.
As K⊆ L and |K|>1,t∈ {2,3}. Let `j∈Kwith u`j(b, v`j)> u`j(v, v`j). In
particular, v`j>p`j
3. We derive contradictions for all possible transitions for v.
We start with the transitions of length 2. Here, if j= 1, we may assume that
v`2>p`2
3and b`2=−1, as the only way for `2to be beneficial for `1is to bid such
as not to receive the service beforehand, and players with vi=pi
3are considered to
be rejected anyway. On the other hand, if j6= 1 and `1does receive the service for
v,`1may only change the outcome for `2bidding such as not to receive the service;
we may then assume that b`1=−1. We do not have to consider the case that j6= 1
and `1does not receive the service for v, as player `1may only change the outcome
for `2by bidding such as to receive the service, resulting in a negative utility. Thus,
for all transitions where the entry for `1is ‘–’, it follows from j6= 1 that j6= 2.
•(−,4
3)→(−,2):
In particular, |I2\{`1}| <1. If j= 1, due to X(|A1|,|A2|−1) = (−,2), it holds
that |I2\{`1, `2}| ≥ 1, a contradiction.
•(1,1) →(1,1):
In particular, |I2|<1. If j= 1, due to X(|A1|,|A2|−1) = (1,−), it holds that
|I2\{`2}| ≥ 1, a contradiction. Thus j= 2, and due to X(|A1|−1,|A2|) = (−,2),
it is |I2\{`2}| ≥ 1, a contradiction.
•(1,1) →(−,2):
In particular, |I2|<1. If j= 1, due to X(|A1|,|A2|−1) = (1,−), it holds that
|I2\{`2}| ≥ 1, a contradiction.
•(2
3,−)→(1,−):
In particular, |I2|<1. If j= 1, due to X(|A1|−1,|A2|) = (1,−), it is |I2| ≥ 1, a
contradiction.
•(2
3,1) →(2
3,1):
In particular, |I2|<1. If j= 1, due to X(|A1|,|A2|−1) = (2
3,−), it holds that
|I2\{`2}| ≥ 1, a contradiction. Thus j= 2, and due to X(|A1|−1,|A2|) = (1,1),
it is |I2\{`2}| ≥ 1(there are four cases here: either (ρ1, ρ2) = (1,1) or (ρ1, ρ2) =
(0,2) in the first iteration for b, or otherwise, the least preferred player with pi= 1
is considered in line 15, thus accepted for a price of 1or rejected. In any case,
(ρ1, ρ2) = (0,1) in the next iteration, as `2strictly improves), a contradiction.
•(−,11) →(−,11):
It is |I1|<1and |I2|<1. If j= 1, due to X(|A1|,|A2| − 1) = (−,1), it holds
that |I1| ≥ 2or |I2\{`1, `2}| ≥ 1if |A1|>3, and |I2\{`1, `2}| ≥ 1if |A1|= 3, a
contradiction. Thus j= 2, where we get the same contradiction.
66 5 Group-Strategyproof Non-Moulin Mechanisms
•(−,11) →(−,1):
In particular, |I1|<1and |I2\{`1}| <1. If j= 1, due to X(|A1|,|A2|−1) = (−,1),
|I1| ≥ 2or |I2\ {`1, `2}| ≥ 1if |A1|>3, and |I2\ {`1, `2}| ≥ 1if |A1|= 3, a
contradiction.
For the two transitions with t= 3, if j= 1, we may assume that either v`2>p`2
3
and b`2=−1, or v`3>p`3
3and b`3=−1, or both. Having established that j6= 1,
as 1does not receive the service for any of the two transitions, `1is not capable of
helping `2or `3without obtaining negative utility. Thus, in case j= 2, we only have
to consider that `3drops out, i.e., v`3>p`3
3and b`3=−1. With the same arguments,
for j= 3 we only have to consider the transition (2
3,1) →(1,1) →(1,1) where `2
bids −1.
•(2
3,1) →(1,1) →(1,1):
In particular, |I2|<1. If j= 1 and only `2bids −1, due to X(|A1|−1,|A2|) =
(1,1),|I2| ≥ 2, a contradiction. If j= 1 and only `3bids −1, due to X(|A1|,|A2|−
1) = (2
3,−),|I2\ {`3}| ≥ 1, a contradiction. If j= 1 and `2and `3submit a
negative bid, due to X(|A1| −1,|A2|− 1) = (1,−)it holds that |I2\{`3}| ≥ 1,
a contradiction. If j= 2, due to X(|A1|,|A2|−1) = (2
3,−), the fact that 1does
not receive the service and therefore X(|A1|−1,|A2|−1) = (1,−), it holds that
|I2\ {`3}| ≥ 1, a contradiction. If j= 3, due to X(|A1| − 1,|A2|) = (1,1), the
fact that `1does not receive the service and therefore X(|A1|−2,|A2|) = (−,2),
it holds that |I2\{`3}| ≥ 1, a contraction.
•(2
3,1) →(1,1) →(−,2):
By the same arguments as for the previous transition, j /∈ {1,2}.
Running Time
For the running time, consider Algorithm 5.1. Given X, every operation outside
the loop takes time at most O(n). The loop is executed at most 3times and each
operation in the loop takes at most time O(n). An overall running time of O(n)
results.
5.7 Conclusion and Open Problems
We regard as the main asset of our work presented in this chapter that it is a
systematic first step for finding GSP mechanisms that perform better than Moulin
mechanisms. While symmetric costs are arguably of limited practical interest, we
yet transferred our techniques to the minimum makespan scheduling problem as an
application and also to a setting with non-symmetric costs. Clearly, there are many
open issues:
•For symmetric and/or subadditive costs, we still need an exact characterization
with respect to the best possible approximate budget-balance that GSP mecha-
nisms can achieve.
•For non-symmetric costs, we would like to answer if there are promising general-
izations of our techniques.
•Finally, a central question is if we can adjust our cost-sharing forms in some way
such that the better budget-balance (compared to Moulin mechanisms) does not
come at the price of increased social cost.
6
Egalitarian Mechanisms
6.1 Contribution
We introduce the new behavioral assumption that coalitions do not form if some
member would lose service. Yet, coalitions do already form if at least one player
wins the service. Being reminded of collectors, we call resistance against collective
collusion in the new sense group-strategyproof against collectors (CGSP). We further
introduce weak CGSP (WCGSP).
•We show that CGSP is strictly stronger than WGSP but incomparable to GSP.
Moreover, we prove that – contrary to WGSP – any WCGSP mechanism induces
unique cost-shares. This strictly improves on the result by Moulin [97] establishing
unique cost-shares for any GSP mechanism. Additionally, Moulin’s result is based
on strict CS while we solely require CS.
•We give an algorithm for computing CGSP mechanisms that we call ‘egalitarian’
due to being inspired by Dutta and Ray’s [34] ‘egalitarian solutions’.
•We show that our egalitarian mechanisms are CGSP by identifying them as a
subclass of acyclic mechanisms and by showing that all acyclic mechanisms are
CGSP (and thus remarkably stronger than WGSP). In addition, our mechanisms
are 1-BB for arbitrary costs and additionally 2Hn-EFF for the very natural class
of subadditive costs. As main tools, we use most cost-efficient set selection and
price functions.
•We give a (rather unrealistic) cost function for which approximation of social
cost efficiency by egalitarian mechanisms is infeasible. We nevertheless identify a
property of price functions that helps to establish efficiency guarantees for more
reasonable cost functions.
•We present a framework for coping with the computational complexity of egalitar-
ian mechanisms, especially for problems in which players are endowed with some
size (e.g., processing times). Besides the use of approximation algorithms, the key
idea here are ‘monotonic’ cost functions that must not increase when replacing a
player by another one with a smaller size.
•We give applications for scheduling and bin-packing that underline the power of
our new approach. For (Q||Cmax)cost-sharing problems and their subproblems,
our results are given in Table 1.1. Notably, our framework also allows for first
results on cost-sharing problems for more realistic scheduling models.
All result presented in this chapter are published in [11].
68 6 Egalitarian Mechanisms
6.2 Organization
In Section 6.3, we give the formal definitions of CGSP and WCGSP as well as their
relation to other notions of incentive-compatibility, and show that WCGSP mech-
anisms induce unique cost-shares. We introduce egalitarian mechanisms together
with set selection and price functions in Section 6.4. Section 6.5 proves CGSP of
egalitarian mechanisms via acyclic mechanisms, and Section 6.6 gives our results on
efficiency approximation. We present our computational framework in Section 6.7,
and its applications in Section 6.8.
6.3 Collectors’ Behavior
6.3.1 New Behavioral Assumptions
In the demand for GSP lies an implicit modeling assumption that is common to
most recent works on cost-sharing mechanisms: First, a player is only willing to be
untruthful and join a coalition of false-bidders if this does not involve sacrificing
his own utility. Second, a coalition always requires an initiating player whose utility
strictly increases.
Clearly, there are other reasonable behavioral assumptions on coalition formation.
We introduce and study the following: First, besides not giving up utility, a player
would not sacrifice service, either. (Although his utility is zero both when being served
for his valuation and when not being served.) Second, it is sufficient for coalition
formation if the initiating player gains either utility or service. While we consider
this behavior very human, it especially reminds us of collectors. We hence denote
a mechanism’s resistance against coalitions in this new sense as group-strategyproof
against collectors.
Definition 6.1. A mechanism M= (Q, x)is group-strategyproof against collectors
(CGSP) if for every coalition K⊆[n]and every true valuation vector vK= (vi)i∈K∈
R|K|there is no bid vector b∈Rnsuch that
•ui(b, vi)≥ui((vK,b−K), vi)and i /∈Q(vK,b−K)\Q(b)for all i∈Kand
•ui(b, vi)> ui((vK,b−K), vi)or i∈Q(b)\Q(vK,b−K)for at least one i∈K.
We remark that CGSP in a model with quasi-linear utilities is equivalent to GSP in
a changed model where a preference of being served for the price of valuation over
not being served is internalized in the utilities. To illustrate the interrelation between
CGSP and GSP, we introduce a property which is a relaxation of both, called weak
CGSP:
Definition 6.2. A mechanism M= (Q, x)is weakly CGSP (WCGSP) if for every
coalition K⊆[n]and every true valuation vector vK= (vi)i∈K∈R|K|there is no
bid vector b∈Rnsuch that
•ui(b, vi)≥ui((vK,b−K), vi)and i /∈Q(vK,b−K)\Q(b)for all i∈Kand
•ui(b, vi)> ui((vK,b−K), vi)for at least one i∈K.
Lemma 6.3. The following implications hold. They do generally not hold in the op-
posite directions.
GSP
=⇒
WCGSP =⇒WGSP =⇒SP
CGSP
=⇒
6.3 Collectors’ Behavior 69
Proof. The implications GSP ⇒WCGSP ⇒WGSP ⇒SP and CGSP ⇒WCGSP
hold by definition. In the following, we show that the converse directions do not hold
in general.
We have GSP 6⇐ WCGSP because on the one hand, we show in Theorem 6.14
that acyclic mechanisms are CGSP (and thus WCGSP), and on the other hand,
acyclic mechanisms are not GSP in general [94]. To show CGSP 6⇐ WCGSP, we give
a mechanism that is GSP (and thus WCGSP) but not necessarily CGSP. Consider
a modified version of Moulinξfor a cross-monotonic cost-sharing method ξ(confer
Algorithm 4.1). After the execution of Moulinξ, delete players who receive the service
for a cost share equal to their valuation, if this does not change the cost shares of the
remaining service-receiving players. This mechanism is still GSP. However, a player
deleted in the additional step may submit a bid strictly larger than his valuation in
order to receive the service. His utility in both outcomes is zero. This contradicts
CGSP.
We prove that WCGSP 6⇐ WGSP by considering the trivial mechanisms TrivC
proposed by Immorlica et al. [73] for any cost function C, and adapted to negative
bids in Algorithm 6.1. We show that TrivCis WGSP but not necessarily WCGSP.
Algorithm 6.1 (computing TrivC(b) = (Q(b), x(b))).
1: x:= 0;Q:= {i∈[n]|bi≥0};
2: while Q6=∅and bmin Q< C(Q)do
3: Q:= Q\{min Q};
4: if Q6=∅then
5: xmin Q:= C(Q)
6: return (Q, x)
To see that TrivCis WGSP, assume that it is not. Then, there exists a coalition
K⊆[n]with true valuations vK= (vi)i∈K∈R|K|and a bid vector b∈Rnsuch
that ui(b)> ui(vK,b−K)for all i∈K. Observe that for all i∈K, it has to hold
that i∈Q(b)and vi>0. Specifically, bi≥0. Furthermore, there has to be a j∈K
with j /∈Q(vK,b−K), as otherwise the outcomes for band (vK,b−K)are the same.
Consider the smallest such j∈Kand assume that jis deleted from set Qfor input
(vK,b−K), i.e., vj< C(Q). When running TrivCon input b, the computation is
exactly the same until jis considered (to pay for the same set Q). As j∈Q(b),
it is bj≥C(Q) = xj(b)and Q=Q(b), thus uj(b, vj) = vj−C(Q(b)) <0 =
uj((vK,b−K), vj), contradicting j∈K.
To see that TrivCis not WCGSP, consider an example with two players and
costs C({1,2})=2and C(1) = C(2) = 1. Assume that the true valuations are
v:= (2 −ε, ε), resulting in Q(v) = ∅. However, for vector b:= (2 −ε, −1), the
outcome is Q(b) = {1}and x(b) = (1,0).
Finally, WGSP 6⇐ SP can be observed from Example B.9 on page 117 that shows
that the SP marginal cost mechanisms are generally not WGSP.
We remark that the acyclicity of Moulinξaccording to [94] and Theorem 6.14 will im-
ply that Moulinξis both GSP and CGSP. Furthermore, already the sequential stand
alone mechanisms introduced by Moulin [97] achieve CGSP and 1-BB (Lemma B.4,
p. 113). Yet, they are only Ω(n)-EFF in general (Example B.2, p. 112).
70 6 Egalitarian Mechanisms
6.3.2 Sufficient Conditions for Unique Cost Shares
Interestingly, already WCGSP is sufficient for a mechanism to induce unique cost
shares. The proof of Theorem 6.4 uses ideas from an analogous result in [97] (confer
Theorem C.1). However, Theorem 6.4 is stronger since GSP and strict CS are relaxed
to WCGSP and CS. Conversely, WGSP mechanisms do not always induce unique
cost shares, even if we demand 1-BB, as stated by Lemma 6.5.
Theorem 6.4. Let M= (Q, x)be a WCGSP mechanism. Then, for any two b,b0∈
Rnwith Q(b) = Q(b0), it holds that x(b) = x(b0). This result holds even if we restrict
our model to non-negative bids and only require CS.
Proof. We first restrict attention to the more intricate case in which b∈Rn
≥0for all
bid vectors band strict CS is not required to hold.
For b∈Rn
≥0and S⊆[n], we define yb,S ∈Rn
≥0by setting
yb,S
i:=
b+
ifor all i∈S∩Q(b)
0for all i∈S\Q(b)
bifor all i /∈S
The main part of the proof consists of showing (6.1) and (6.2) for all b∈Rn
≥0, all
S⊆[n], all i∈[n], and T:= S\{j /∈Q(b)|bj= 0}:
i∈T=⇒i∈Q(b)⇔i∈Q(yb,S)and xi(b) = xi(yb,S)(6.1)
i /∈T=⇒ui(b, bi) = ui(yb,S, bi).(6.2)
Assume for now that (6.1) and (6.2) hold and consider two bid vectors b,b0∈Rn
≥0
with Q(b) = Q(b0). We have to show that x(b) = x(b0). Observe that x(b) = x(yb,[n])
since for the players j∈[n]with j /∈Q(b)and bj= 0 it holds that xj(b) = 0 and
0 = uj(b, bj) = uj(yb,[n], bj) = 0 −xj(yb,[n]). Analogously, x(b0) = x(yb0,[n]). Finally,
we utilize yb,[n]=yb0,[n]to conclude that x(b) = x(yb,[n]) = x(yb0,[n]) = x(b0).
To complete this proof, fix b∈Rn
≥0. We omit the superscript bfor yb,S and simply
write yS. Note that by CS and the definition of yS,i∈S∩Q(b)⇒i∈Q(yS)for
all i∈[n]. We prove (6.1) and (6.2) by induction on the cardinality of S.
•Let S:= {k}.
Then yS= (yS
k,b−k). If k /∈Q(b)and bk= 0, (6.2) holds for kwith b=yS.
Otherwise, we have to verify (6.1). If k∈Q(b), it follows from k∈Sthat
k∈Q(yS). Let k∈Q(yS)and assume that k /∈Q(b). From yS
k= 0, it follows
that xk(yS) = 0. However, bk>0already contradicts SP, because for band true
valuation bk, player kmay bid yS
kin order to receive the service for 0. Formally,
uk((yS
k,b−k), bk) = bk−0>0 = uk(b, bk).
If xk(b)> xk(yS), then k∈Q(b)and thus k∈Q(yS). For band true valuation
bk,kmay bid yS
kin order to pay a strictly lower cost share, already contradicting
SP. Formally,
uk((yS
k,b−k), bk) = bk−xk(yS)> bk−xk(b) = uk((bk,b−k), bk).
Analogously, if xk(b)< xk(yb), then k∈Q(yS)∩Q(b)and for ySand true
valuation yS
k,kmay bid bkin order to pay a strictly lower cost share, which again
contradicts SP.
6.3 Collectors’ Behavior 71
We continue to verify (6.2) for all players j /∈S. If uj(b, bj)> uj(yS, bj)for a
j6=k, we get a contradiction to WCGSP with coalition K={j, k}by considering
true valuations yS
j(= bj)and yS
k. Formally,
uk(b, yS
k) = uk(yS, yS
k) = uk((yS
{j,k},b−{j,k}), yS
k)
and uj(b, yS
j) = uj(b, bj)> uj(yS, bj) = uj(yS, yS
j) = uj((yS
{j,k},b−{j,k}), yS
j)
and k, j /∈Q(yS
{j,k},b−{j,k})\Q(b).
The last observation holds for kdue to (6.1). For jit is due to j∈Q(b)as by
assumption uj(b, bj)>0. Analogously, if uj(b, bj)< uj(yS, bj)for a j6=k, we
get a contradiction to WCGSP by considering true valuations yS
j(= bj)and bk.
•Induction hypothesis: (6.1) and (6.2) hold for all S([n]with |S|< m.
•Let S⊆[n]with |S|=m.
Then, yS= (yS
S,b−S). We first show (6.1). Let T:= S\ {j /∈Q(b)|bj= 0}.
For all i∈Twith i∈Q(b)it is i∈Q(yS)because of i∈S. Consider i∈T
with i∈Q(yS). Assume i /∈Q(b), thus bi>0and xi(b) = 0. Due to yS
i= 0, it
is xi(yS)=0. Define z:= (bi,yS
−i). Observe that yS= (yS
i,z−i). By induction
hypothesis, ui(z, bi) = ui(b, bi)=0. This already contradicts SP, since for zand
true valuation zi(= bi),imay bid yS
ito receive the service for 0. Formally,
ui((yS
i,z−i), zi) = ui(yS, zi) = zi−0>0 = ui(z, zi).
– Assume xi(b)> xi(yS)for an i∈T:
In this case, xi(b)>0, thus i∈Q(b)∩Q(yS). Again, let z:= (bi,yS
−i). By
induction hypothesis, ui(b, bi) = ui(z, bi). If i∈Q(z), we get xi(z) = xi(b),
already contradicting SP, since for bid vector zand true valuation zi(= bi),
player imay bid yS
iin order to pay a strictly smaller cost share. Formally,
ui((yS
i,z−i), zi) = ui(yS, zi) = zi−xi(yS)> zi−xi(z) = ui(z, zi).
If i /∈Q(z), it follows by ui(b, bi) = ui(z, bi)=0that bi=xi(b), which again
contradicts SP, since for zand true valuation zi(= bi), player imay bid yS
i
to receive the service for xi(yS)< bi. Formally,
ui((yS
i,z−i), zi) = ui(yS, zi) = zi−xi(yS)>0 = ui(z, zi).
– Assume xi(b)< xi(yS)for an i∈T:
For all j∈T, it is xj(b)≤xj(yS)and j∈Q(b)⇔j∈Q(yS). Now for bid
vector ySand true valuations {yS
j}j∈T,Tcan form coalition K=Tto strictly
improve the utility of at least one of its members, since (bT,yS
−T) = b. This is
a contradiction to WCGSP. Formally,
∀j∈T:uj((bT,yS
−T), yj) = uj(b, yj)
=yj·in(Q(b)) −xj(b)
=yj·in(Q(yS)) −xj(b)
≥yj·in(Q(yS)) −xj(yS)
=uj(yS, yj)
and ∃i∈T:ui((bT,yS
−T), yi)> ui(yS, yi)
and ∀j∈T:j /∈Q(yS)\Q(bT,yS
−T).
72 6 Egalitarian Mechanisms
Therefore, xi(b) = xi(yS)for all i∈T.
•Additionally, ui(b, bi) = ui(yS, bi)for all i /∈T:
If ui(b, bi)> ui(yS, bi)for an i /∈T, consider vector yS, coalition K=T∪{i}and
true valuations yS
T∪{i}. Note that yS
i=biand i∈Q(b). As (b{T∪i},yS
−{T∪i}) = b,
ui((b{T∪i},yS
−{T∪i}), yS
i)> ui(yS, yS
i)
and ∀j∈T:uj((b{T∪i},yS
−{T∪i}), yS
j) = yS
j·Q(b)−xj(b)
=yS
j·Q(yS)−xj(yS)
> uj(yS, yS
j)
and ∀j∈T∪{i}:j /∈Q(yS)\Q(b{T∪i},yS
−{T∪i}),
contradicting WCGSP. If ui(b, bi)< ui(yS, bi)for an i /∈T, we analogously get a
contradiction to WCGSP by considering vector band coalition K=T∪{i}with
true valuations bT∪{i}bidding yS
T∪{i}.
Thus, (6.1) and (6.2) hold.
In case that bids are allowed to be negative and thus strict CS comes ‘for free’,
we adjust the proof by letting yb,S
i:= −1for all i∈S\Q(b). It now directly holds
for all i∈Sthat i∈Q(b)⇔i∈Q(yb,S)and (6.1) and (6.2) hold even for replacing
Twith S, as can be verified along the lines of the above proof. This approach was
also used by Moulin [97] who assumed non-negative bids and strict CS, where yb,S
i
was set such that i∈S\Q(b)does not receive the service when bidding yb,S
i.
However, i∈Q(b)⇔i∈Q(yb,S)cannot be guaranteed for all i∈Swhen setting
yb,S
i:= 0 for all i∈S\Q(b). The main idea of our proof for non-negative bids is to
show that (6.1) holds for players in the restricted set T:= S\{j /∈Q(b)|bj= 0}
which still allows to derive contradictions to WCGSP.
Lemma 6.5. For any cost function C: 2[3] →R≥0, there is a WGSP and 1-BB
mechanism MC= (Q, x)such that there are bids b,b0∈Rn
>0with Q(b) = Q(b0), but
x(b)6=x(b0).
Proof. Consider mechanism MC,a with cost function Cand threshold a∈R>0from
Algorithm 6.2:
Algorithm 6.2 (computing MC,a(b) = (Q(b), x(b))).
1: x:= 0,Q:= {i∈[3] |bi>0}
2: if Q6=∅then i:= min Q
3: while Q6=∅do
4: if bi≥C(Q)then xi:= C(Q);return
5: else Q:= Q\{i}
6: if i= 1 then if b1≥athen i:= 2 else i:= 3
7: else if Q6=∅then i:= min Q
8: return (Q, x)
Obviously, 1-BB is met. Player 1may only strictly improve his utility, if 2or 3bids
in order to not receive the service for b. However, such a coalition would not form.
Therefore, 1is not part of any coalition. Furthermore, 2can only strictly improve
his utility, if 3bids in order to not receive the service for b. Vice versa, the same
6.4 Egalitarian Mechanisms 73
is true for player 3. Therefore, MC,a is WGSP. However, payments are not uniquely
determined by the served set. For a < C([3]),
b:= (a, C({2,3}), ε) =⇒Q(b) = {2,3}and x(b) = (0, C{2,3},0)
b0:= (a−ε, ε, C({2,3})) =⇒Q(b0) = {2,3}and x(b0) = (0,0, C({2,3})) .
6.4 Egalitarian Mechanisms
6.4.1 Set Selection and Price Functions
Egalitarian mechanisms borrow an algorithmic idea proposed by Dutta and Ray [34]
for computing the ‘egalitarian solution’ for a cost-allocation problem. Given a set of
players Q⊆[n]to be served, cost shares are computed iteratively: Find the most
cost-efficient subset Sof the players that have not been assigned a cost share yet.
That is, the quotient of the marginal cost for including Sdivided by |S|is minimal.
Then, assign each player in Sthis quotient as his cost share. If players remain who
have not been assigned a cost share yet, start a new iteration.
We introduce most cost-efficient set selection functions σCand corresponding
price functions ρCthat are used within this iterative process. Specifically, let Q⊆[n]
be the set of players to be served. For some fixed iteration, let N(Qbe the
subset of players already assigned a cost-share. Then, σC(Q, N)selects a non-empty
set of players from Q\Nwho are assigned the cost share ρC(Q, N)according to
Definition 6.6:
Definition 6.6. Let Cbe a cost function. The most cost-efficient set selection func-
tion σC: 2[n]×2[n]→2[n]and its corresponding price function ρC: 2[n]×2[n]→R≥0
are defined as
σC(Q, N) := lexicographic max in arg min
∅6=T⊆Q\NC(N∪T)−C(N)
|T|,
ρC(Q, N) := min
∅6=T⊆Q\NC(N∪T)−C(N)
|T|.
In general, we allow for other set selection and price functions that fulfill a certain
validity requirement introduced in Section 6.4.3 which is in particular met by σCand
ρC. Clearly, evaluating σCcan take exponentially many steps (in n). Furthermore,
evaluating Cmay be computationally hard. In Section 6.7 we thus study how to pick
‘suitable’ cost-efficient subsets in polynomial time.
6.4.2 Computing Egalitarian Mechanisms
Based on set selection function σand price function ρ, we define mechanism Egalσ,ρ
as computed by Algorithm 6.3. We require that for all N(Q⊆[n]it holds that
∅ 6=σ(Q, N)⊆Q\N.
Obviously, Egalσ,ρ meets NPT, VP, and strict CS. If the most cost-efficient set
selection and price functions σCand ρCfor an arbitrary cost function Care applied,
EgalσC,ρCis even 1-BB.
74 6 Egalitarian Mechanisms
Algorithm 6.3 (computing Egalσ,ρ(b) = (Q(b), x(b))).
1: Q:= [n];N:= ∅;x:= 0
2: while N6=Qdo
3: S:= σ(Q, N),a:= ρ(Q, N)
4: Q:= Q\{i∈S|bi< a}
5: if S⊆Qthen xi:= afor all i∈S;N:= N∪S
6: return (Q, x)
Example 6.7. Consider C: 2[4] →R≥0with C({1,2}) := 1,C({1,2,3}) := 1 and
C(S) = |S|otherwise. Let v:= (1
3,1
2,1
4,4). We compute EgalσC,ρC(v) = (Q(v), x(v))
for the most cost-efficient set selection an price function σCand ρCfrom Defini-
tion 6.6. Denote all variables in iteration kof Algorithm 6.3 with subscript k.
Then, S1={1,2,3}and a1=1
3. As v3<1
3,Q1={1,2,4}and N1=∅. In the
second iteration, S2={1,2},a2=1
2, and Q2={1,2,4}. Thus, N2={1,2}and
x2= (1
2,1
2,0,0). In the third iteration, S3={4},a3= 2, and Q3={1,2,4}. Thus,
N3={1,2,4}=Q(v)and x3= (1
2,1
2,0,2) = x(v).
We remark that the mechanisms given by Devanur et. al. [31] are not just acyclic
(see [94]) but also egalitarian. Using the terminology as in [31], they could be com-
puted by Algorithm 6.3 by letting σ(Q, N)be the next set that ‘goes tight’ after all
players in Nhave been ‘frozen’ and all in [n]\Qhave been dropped.
6.4.3 Validity of Set Selection and Price Functions
For a set selection function σand a price function ρwe demand the validity require-
ment from Definition 6.8 which is motivated with respect to Algorithm 6.3.
Definition 6.8. A set selection function σ: 2[n]×2[n]→2[n]and a price function
ρ: 2[n]×2[n]→R≥0are valid if, for all N(Q, Q0⊆[n]:
•Any player is assigned a cost-share only once and the algorithm terminates:
∅ 6=σ(Q, N)⊆Q\N(6.3)
•Players in Q\Nnot in the selected set have no influence on it’s selection:
σ(Q, N)⊆Q0⊆Q=⇒σ(Q, N) = σ(Q0, N)and ρ(Q, N) = ρ(Q0, N)(6.4)
•The assigned prices are non-decreasing throughout the iterations of the algorithm:
Q0⊆Q=⇒ρ(Q, N)≤ρ(Q0, N)and (6.5)
0≤ρ(Q, N)≤ρ(Q, N ∪σ(Q, N)) (6.6)
Lemma 6.9. For any cost function C, the corresponding most cost-efficient set se-
lection and price functions σCand ρC(from Definition 6.6) are valid.
Proof. It is a straightforward observation that σCand ρCfulfill properties (6.3)–
(6.5) of Definition 6.8. To see property (6.6), let N(Q⊆[n]. Define S:= σ(Q, N),
a:= ρ(Q, N)and S0:= σ(Q, N ∪S), a0:= ρ(Q, N ∪S). Then,
a≤C(N∪S∪S0)−C(N)
|S|+|S0|=C(N∪S∪S0)−C(N∪S) + |S|·a
|S|+|S0|,
thereby implying that
a≤C(N∪S∪S0)−C(N∪S)
|S0|=a0.
6.5 Egalitarian Mechanisms are CGSP 75
6.5 Egalitarian Mechanisms are CGSP
Theorem 6.10 follows as a direct corollary of Theorem 6.14 in Section 6.5.2 showing
that acyclic mechanisms are CGSP and of Theorem 6.17 in Section 6.5.3 showing
that egalitarian mechanisms (for valid set selection and price functions) are acyclic.
Theorem 6.10. For any valid set selection function σand price function ρ,Egalσ,ρ
is CGSP.
6.5.1 Acyclic Mechanisms
By introducing acyclic mechanisms, Mehta et al. [94] gave a framework for construct-
ing WGSP mechanisms. An acyclic mechanism Acycξ,τ makes use of a cost-sharing
method ξand an offer function τ: 2[n]→Rn
≥0which specifies a non-negative offer
time τi(Q)for every subset Q⊆[n]and every player i∈Q. The acyclic mechanism
Acycξ,τ is computed by Algorithm 6.4.
Algorithm 6.4 (computing Acycξ,τ (b) = (Q(b), x(b))).
1: Q:= [n]
2: while ∃i∈Qwith bi< ξi(Q)do
3: Let j∈arg mini∈Q{τi(Q)|bi< ξi(Q)}(use arbitrary tie breaking rule)
4: Q:= Q\{j}
5: return (Q, ξ(Q))
Mehta et al. [94] showed that if τsatisfies a certain validity requirement with
respect to ξ,Acycξ,τ is WGSP. We specify this requirement in Definition 6.11. For
every Q⊆[n]and every i∈Q, let
Ei(Q):={j∈Q|τj(Q) = τi(Q)}
Li(Q):={j∈Q|τj(Q)< τi(Q)}
Gi(Q):={j∈Q|τj(Q)> τi(Q)}
be the sets of players with equal, lesser, and greater offer time compared to i.
Definition 6.11. An offer function τis valid for cost-sharing method ξif for all
i∈[n]it holds that
•ξi(Q\T) = ξi(Q)for every subset T⊆Gi(Q)and
•ξi(Q\T)≥ξi(Q)for every subset T⊆Gi(Q)∪(Ei(Q)\{i}).
Consider the set Qat the beginning of iteration kand a player i∈Q. Let
p:= ξi(Q). Like in [94], we say that Acycξ,τ offers the price pto player iin iteration
kif either no player is deleted in iteration k(i.e., Acycξ,τ terminates) or it holds
for iand the deleted player jthat τi(Q)≤τj(Q)(it is possible that j=i). We use
this terminology in Lemma 6.12 and Lemma 6.13 needed for the proof that acyclic
mechanisms are CGSP.
Lemma 6.12 ([94]). Let Acycξ,τ = (Q, x)be an acyclic mechanism and Qbe the set
at the beginning of iteration k. If player i∈Qis offered price p1in iteration kand
price p2in some subsequent iteration, then p1≤p2. In particular, ξj(Q(b)) = ξj(Q)
for all j∈Li(Q)and therefore Li(Q)⊆Q(b).
Lemma 6.13. Let Acycξ,τ = (Q, x)be an acyclic mechanism. For any bid vector b
and any set T⊆[n]\Q(b), there is an i∈Twith bi< ξi(Q(b)∪T).
76 6 Egalitarian Mechanisms
Proof. Consider the first iteration kin which an i∈Tis deleted for the first time
from set Q, i.e., T⊆Q. For all j∈Q\Q(b), it holds that τj(Q)≥τi(Q). Otherwise,
j∈Q(b)by Lemma 6.12. In particular, Q\ {Q(b)∪T} ⊆ Gi(Q)∪(Ei(Q)\ {i}).
Therefore, by validity of τ,bi< ξi(Q)≤ξi(Q\{Q\{Q(b)∪T}}) = ξi(Q(b)∪T).
6.5.2 Acyclic Mechanisms are CGSP
Theorem 6.14. For any cost-sharing method ξand any offer function τthat is valid
for ξ,Acycξ,τ is CGSP.
Proof. Assume that Acycξ,τ is not CGSP. Then, there exists a coalition K⊆[n]with
true valuations vK= (vi)i∈K∈R|K|and a bid vector b∈Rnsuch that for all i∈K,
ui(b, vi)≥ui((vK,b−K), vi)and i /∈Q(vK,b−K)\Q(b)and for at least one i∈K,
ui(b, vi)> ui((vK,b−K), vi)or i∈Q(b)\Q(vK,b−K). Without loss of generality,
let v:= (vK,b−K).
First, we show that Q(b)⊆Q(v). Let m:= n−|Q(v)|. Consider the first non-
identical iteration k∈[m]for vand b. Denote Qto be the set of players at the
beginning of k. If Q=Q(v), there is nothing to show. Denote the player deleted for
vin iteration l∈[m]by dl. Let Dl:= {dk, . . . , dl}. Note that Q(v) = Q\Dm. We
prove by induction that dl/∈Q(b)for all l∈ {k, . . . , m}.
•l=k:
If Q=Q(b),bdk≥ξdk(Q)> vdk. Then, dk∈Kand udk(b, vdk)<0 = udk(v, vdk),
a contradiction. Thus, let d0
kbe the player deleted for bin iteration k. If τd0
k(Q)<
τdk(Q), then by Lemma 6.12, d0
k∈Q(v)and vd0
k≥ξd0
k(Q(v)) = ξd0
k(Q)> bd0
k.
But d0
k∈Kcontradicts d0
k/∈Q(b). Thus, τd0
k(Q)≥τdk(Q), implying that for
b,dkis offered price ξdk(Q)in iteration k. If dk∈Q(b), then by Lemma 6.12,
bdk≥ξdk(Q(b)) ≥ξdk(Q)> vdk. Then, dk∈Kand udk(b, vdk)<0 = udk(v, vdk),
a contradiction.
•l→l+ 1:
Assume dl+1 ∈Q(b). By induction assumption, Q(b)⊆Q\Dl. By ξdl+1 (Q\Dl)>
vdl+1 , we get ξdl+1 (Q(b)) < ξdl+1 (Q\Dl). Otherwise, it holds that dl+1 ∈Kand
udl+1 (b, vdl+1 )<0 = udl+1 (v, vdl+1 ), a contradiction. Thus, by validity of τ, there
exists i∈Q\Dlwith i∈Ldl+1 (Q\Dl)and i /∈Q(b). Let
T:= {i∈Q\Dl|i∈Ldl+1 (Q\Dl)and i /∈Q(b)}.
It is T∩K=∅, since T⊆Q(v)(confer Lemma 6.12). Note that Dl,Q(b), and T
are disjoint. Let T0:= Q\(Dl∪Q(b)∪T). Figure 6.1 shows the relations between
sets Q, Dl, Q(b), T, and T0. By definition of T, it holds that for all j∈T0that
τj(Q\Dl)≥τdl+1 (Q\Dl), as j /∈T. Thus, for all j∈T0and all i∈T, we have
that τj(Q\Dl)> τi(Q\Dl). Therefore,
for all i∈T:bi=vi≥ξi(Q\Dl) = ξi(Q\(Dl∪T0)) = ξi(Q(b)∪T).
However, this contradicts Lemma 6.13. Therefore, dl+1 /∈Q(b).
From Q(b)⊆Q(v), it now follows that K∩Q(v) = K∩Q(b). Hence, by assumption,
there exists j∈Kwith j∈Q(b)∩Q(v)such that uj(b, vj)> uj(v, vj), and specifi-
cally ξj(Q(b)) < ξj(Q(v)). Then, there has to be at least one player i∈Q(v)with
i∈Lj(Q(v)) and i /∈Q(b). Defining T:= {i∈Q(v)|i∈Lj(Q(v)) and i /∈Q(b)}
and T0:= Q(v)\(Q(b)∪T)leads again to a contradiction to Lemma 6.13.
6.5 Egalitarian Mechanisms are CGSP 77
Fig. 6.1. Set relations for the proof of Theorem 6.14
Q
Dl
T
Q(b)T0
6.5.3 Egalitarian Mechanisms are Acyclic
In order to show that egalitarian mechanisms are acyclic, we make the observation
that instead of using the tie-breaking rule in line 3 of Algorithm 6.4, we may very
well simultaneously delete all players approved for deletion at the same time:
Lemma 6.15. Lines 3 and 4 of Algorithm 6.4 can be replaced by
Q:= Q\arg min
i∈Q{τi(Q)|bi< ξi(Q)}
without changing the computed mechanism Acycξ,τ .
Proof. The proof uses the same technique as the proof of Theorem 6.14. Define
mechanism (Q0, x0)by replacing lines 3 and 4 of Algorithm 6.4 computing Acycξ,τ =
(Q, x)by ‘Q:= Q\arg mini∈Q{τi(Q)|bi< ξi(Q)}’. We show that (Q0, x0) = (Q, x).
It is an easy observation that Lemma 6.12 and Lemma 6.13 hold as well for
(Q0, x0). Fix an arbitrary bid vector b∈Rn. First, we show that Q(b)⊆Q0(b).
Consider the first non-identical iteration kfor (Q, x)and (Q0, x0). Let Qbe the
set of players at the beginning of k. If Q=Q0(b), there is nothing to show. Assume
computing (Q0(b), x0(b)) needs miterations and denote the players deleted by (Q0, x0)
in iteration l∈[m]by Dl. Let Dl:= Dk∪. . . ∪Dl. Note that Q0(b) = Q\Dm. We
prove by induction that Dl∩Q(b) = ∅for all l∈ {k, . . . , m}.
•l=k:
Assume there is a d∈ Dkwith d∈Q(b). Since bd(Q)< ξd(Q), there has to be
an i∈Ld(Q)with i /∈Q(b), a contradiction to Lemma 6.12.
•l→l+ 1:
Assume that there is a d∈ Dl+1 with d∈Q(b). By induction hypothesis, Q(b)⊆
Q\Dl. Since bd< ξd(Q\Dl), there has to be an i∈Ld(Q\Dl)with i /∈Q(b).
Define
T:= {i∈Q\Dl|i∈Ld(Q\Dl)and i /∈Q(b)}and T0:= Q\(Dl∪Q(b)∪T).
By definition of T, it holds that for all j∈T0, we have that τj(Q\Dl)> τi(Q\Dl)
for all i∈T. Hence, bi≥ξi(Q\Dl) = ξi(Q\(Dl∪T0)) = ξi(Q(b)∪T)for all i∈T.
However, this contradicts Lemma 6.13. Therefore, Dl+1 ∩Q(b) = ∅.
Furthermore, Q0(b)⊆Q(b)can be proven analogously. Then, x(b) = x0(b)directly
follows from Q(b) = Q0(b)and Theorems 6.14 and 6.4.
Finally, Theorem 6.17 showing that egalitarian mechanisms are acyclic, is based
on the fact that Algorithm 6.4 computes Egalσ,ρ for the cost-sharing method ξσ,ρ
and offer function τσ,ρ as defined by Algorithm 6.5.
78 6 Egalitarian Mechanisms
Algorithm 6.5 (computing ξσ,ρ(Q)and τσ,ρ(Q)).
Input: set selection and price functions σ, ρ; set of players Q⊆[n]
Output: cost-sharing vector ξσ,ρ(Q)∈Rn
≥0; offer-time vector τσ,ρ(Q)∈Rn
≥0
1: N:= ∅;ξ:= 0;τ:= 0
2: while N6=Qdo
3: S:= σ(Q, N),a:= ρ(Q, N)
4: ξi:= aand τi:= 1 + maxj∈Q{τj}for all i∈S;N:= N∪S
5: return (ξ,τ)
Example 6.16. Continuing Example 6.7, is is ξσC,ρC({1,2,3,4}) = (1
3,1
3,1
3,3) and
τσC,ρC({1,2,3,4}) = (1,1,1,2), furthermore ξσC,ρC({2,3,4}) = (0,1,1,1) and
τσC,ρC({2,3,4}) = (0,1,1,1), and for another example, ξσC,ρC({1,2,4}) = (1
2,1
2,0,2)
and τσC,ρC({2,3,4}) = (1,1,0,2).
Theorem 6.17. For any cost-sharing method ξand any offer function τthat is valid
for ξ,Egalξ,τ is acyclic.
Proof. We show that for any valid σand ρ,Egalσ,ρ can also be computed by the
acyclic mechanism from Algorithm 6.4 with cost-sharing method ξσ,ρ and offer func-
tion τσ,ρ as computed by Algorithm 6.5, and hence is acyclic.
Fix valid σand ρ. Furthermore, set ξ:= ξσ,ρ and τ:= ξσ,ρ.
•τis valid for ξ:
Denote all variables in iteration kof Algorithm 6.5 just before line 3 with a
subscript kand with the input player set in parentheses. Let Q⊆[n]and i∈Q
be arbitrary. Let kbe the iteration such that i∈Sk(Q).
– For all T⊆Gi(Q)it follows by induction that Nm(Q) = Nm(Q\T)and
Sm(Q) = Sm(Q\T)and am(Q) = am(Q\T)for all m∈[k]. This follows by
by Property (6.4) of Definition 6.8 because Sm(Q)⊆Q\T. Hence, ξi(Q) =
ak(Q) = ak(Q\T) = ξi(Q\T).
– For all T⊆Gi(Q)∪(Ei(Q)\{i}), it is Nm(Q) = Nm(Q\T)and Sm(Q) =
Sm(Q\T)and am(Q) = am(Q\T)for all m∈[k−1]. Since still Nk(Q) =
Nk(Q\T)and Q\T⊆Q, it is ak(Q)≤ak(Q\T)because of Property (6.5).
Furthermore, ak(Q\T)≤ξi(Q\T)since ais non-decreasing in Algorithm 6.5
by Property (6.6). Thus, ξi(Q) = ak(Q)≤ak(Q\T)≤ξi(Q\T).
•Egalσ,ρ =Acycξ,τ :
Whenever Algorithm 6.3 accepts a set S:= σ(Q, N)this means that players in
Shave the minimum offering time of those in Q\Nand bi≥a:= ρ(Q, N)for
all i∈S. Consequently, also the acyclic mechanism serves these players for the
same price. On the other hand, when Algorithm 6.3 rejects players from S, the
same players are also rejected by the acyclic mechanism (see Lemma 6.15).
6.6 Efficiency of Egalitarian Mechanisms
Theorem 6.18 is the main theorem of this section and directly follows from Theo-
rem 6.21 and Lemma 6.22. Specifically, Theorem 6.21 provides a way to determine
the efficiency approximation not only for the most cost-efficient, but for any valid
set of set-selection and price functions, via the β-average property of price functions
introduced in Definition 6.19.
6.6 Efficiency of Egalitarian Mechanisms 79
Theorem 6.18. For any subadditive cost function Cand most cost-efficient set se-
lection and price functions σC, ρC,EgalσC,ρCis 2Hn-EFF for C.
Definition 6.19. Let Cbe a cost function, ρbe a price function, and β > 0. Then,
ρis called β-average for Cif for all N(Q⊆[n]and all ∅ 6=S⊆Q\N, it holds
that ρ(Q, N)≤β·C(S)
|S|.
Lemma 6.20. Let Cbe a cost function and σand ρbe valid set selection and price
functions such that ρis β-average for C. Moreover, let S⊆[n]and b∈Rnbe a bid
vector with bi≥β·C(S)
|S|for all i∈S. Then, Egalσ,ρ = (Q, x)serves at least one
player i∈S, i.e., S∩Q(b)6=∅.
Proof. Assume that S∩Q(b) = ∅. Consider the first iteration kin which Algo-
rithm 6.3 rejects a player i∈S: This happens in line 4. We indicate all variables in
that iteration immediately before that line by subscript k. Since player iis dropped,
bi< ak=ρ(Qk, Nk)≤β·C(S)
|S|,
where the last inequality holds because of S⊆Qk\Nk. A contradiction.
Theorem 6.21. Let Cand C0be cost functions and σand ρbe valid set selection
and price functions such that ρis β-average for C. Let Egalσ,ρ = (Q, x). Then, if for
all b∈Rn,Pn
i=1 xi(b)≥C0(Q(b)),Egalσ,ρ is (2β·Hn)-EFF for Cand C0.
Proof. Let Egalσ,ρ = (Q, x)and fix a true valuation vector v∈Rn. Denote Q:=
Q(v),x:= x(v). Moreover, let P⊆[n]be a set that minimizes the optimal social
cost, i.e., P∈arg minT⊆[n]{SC C(T, v)}. Without loss of generality, we may assume
that vi≥0for all i∈[n]since negative bids have no impact on the social cost. We
have
SC C0(Q, v) = C0(Q) + X
i∈[n]\Q
vi
≤X
i∈Q∩P
xi+X
i∈Q\P
xi
|{z}
≤vi
+X
i∈[n]\Q
vi
≤X
i∈Q∩P
xi+X
i∈P\Q
vi+X
i∈[n]\P
vi,and thus
SC C0(Q, v)
SC C(P, v)≤Pi∈Q∩Pxi+Pi∈P\Qvi+Pi∈[n]\Pvi
C(P) + Pi∈[n]\Pvi
≤Pi∈Q∩Pxi+Pi∈P\Qvi
C(P).
The last inequality holds since the left fraction is at least 1. Now, consider the
iteration kwhen for the first time Algorithm 6.3 decides to accept a player i∈Q∩P
(line 5). Fix all variables just after line 3 in that iteration kand indicate them with
a subscript k. We have xi=ak=ρ(Qk, Nk)≤β·C(Q∩P)
|Q∩P|, because Q∩P⊆Qk\Nk.
With the same arguments, for the second player i∈Q∩P, we can bound his cost-
share xi≤β·C(Q∩P)
|Q∩P|−1, and so forth. Finally, Pi∈Q∩Pxi≤β·H|Q∩P|·C(Q∩P).
80 6 Egalitarian Mechanisms
On the other hand, in P\Q, there is at least one player iwith vi< β ·C(P\Q)
|P\Q|.
Otherwise, due to Lemma 6.20, we would have (P\Q)∩Q6=∅, a contradiction.
Inductively and by the same lemma, for every j= 1, . . . , |P\Q|−1, there has to be
a player i∈P\Qwith vi< β ·C(P\Q)
|P\Q|−j. Finally, Pi∈P\Qvi≤β·H|P\Q|·C(P\Q).
Since Cis non-decreasing, we get
SC C0(Q, v)
SC C(P, v)≤β·Hmax{|Q∩P|,|P\Q|} ·(C(Q∩P) + C(P\Q))
C(P)≤2β·Hn.
Lemma 6.22. For any subadditive cost function C, the most cost-efficient price func-
tion ρCis 1-average for C.
Proof. Let N(Q⊆[n]and ∅ 6=S⊆Q\N. Then,
ρC(Q, N) = min
∅6=T⊆Q\NC(Q∪T)−C(Q)
|T|≤C(Q∪S)−C(Q)
|S|≤C(S)
|S|.
We conclude by showing that our efficiency bound is tight up to a factor of 2 and
that the approximate efficiency of egalitarian mechanisms is unbounded for arbitrary
cost functions.
Lemma 6.23. For the cost function Cwith C(T) = 1 for all ∅ 6=T⊆[n],EgalσC,ρC
for most cost-efficient set selection and price functions σCand ρCis no better than
Hn-EFF for C.
Proof. Let v:= (1
i−)n
i=1 ∈Rn
>0be the true valuation vector, ∈(0,1
n). Then,
Q(v) = ∅because in Algorithm 6.3, line 4, one player after the other would be
dropped. However, SC C([n],v) = 1 while SC C(∅,v) = Hn−n·.
Lemma 6.24. For any γ > 1, there is a cost function Cfor which EgalσC,ρCfor most
cost-efficient set selection and price functions σCand ρCis no better than γ-EFF
for C.
Proof. Define C: 2[4] →R≥0as follows: Let C({i}) = 1 for all i∈[4]. Let
C({1,2}) := 2 and C(T) := 3 for any other T([4] with |T|= 2. Let C({1,2,3}) := 4
and C(T) := 5 for any other T([4] with |T|= 3. Furthermore, C([4]) := M∈R>0,
where Mis sufficiently large.
Let EgalσC,ρC= (Q, x)and let the true valuation vector be v= (1,1,2,M
2).
The algorithm first accepts {1,2}, each for a price of 1. Subsequently, it gives the
service to 3for a price of 2and in the next iteration, player 4is rejected. Therefore,
Q(v) = {1,2,3}. We have that SC C({1,2,3},v) = 4 + M
2and SC C({2,3,4},v) = 6.
6.7 Computational Framework
For a given optimization problem Π, the computational complexity of the corre-
sponding cost-sharing problems can be partially eased by resorting to approximate
solutions. Certainly, this does not yet remedy the need to iterate through all avail-
able subsets in order to pick the most cost-efficient one. The basic idea therefore
6.7 Computational Framework 81
consists of using an approximation algorithm ALG that induces monotonic costs
CALG (see, e.g., [100]): Seemingly favorable changes to the input must not worsen
the algorithm’s performance. In the problems considered here, every player is en-
dowed with a size (e.g., processing time in the case of scheduling) and replacing a
player with a player with smaller size must not increase the cost of the algorithm’s
solution. We can then simply number the players in the order of their size such that
CALG(min|U|T)≤CALG(U)for all U⊆T⊆[n]. Finding the most cost-efficient set
then only requires iterating through all possible cardinalities.
We generalize this basic idea such that only a (polynomial-time computable)
monotonic bound Cmono on CALG is needed whereas the costs CALG induced by ALG
itself do not need to be monotonic any more.
Definition 6.25. Let Πbe an optimization problem and (IS)S⊆[n]specify a Π-cost-
sharing problem. A tuple R:= (ALG, Cmono)is a β-relaxation for (IS)S⊆[n]if ALG is
an approximation algorithm for Πinducing cost CALG and Cmono is a cost function
such that the following holds:
•For all T⊆[n]:CALG(T)≤Cmono(T)≤β·C(T).
•For all U⊆T⊆[n] : Cmono(min|U|T)≤Cmono(U).
Note that Cmono does not necessarily have to be subadditive (as required for
2Hn-EFF in Section 6.6), even if Cis. Thus, some additional care is needed. Given
aβ-relaxation, we can adapt the most cost-efficient set selection and price functions
from Definition 6.6 to obtain valid and β-average set selection and price functions:
Definition 6.26. Given a β-relaxation R:= (ALG, Cmono), we define set selection
and price functions σRand ρRrecursively as follows: For N(Q⊆[n], let ξσR,ρR(N)
as computed by Algorithm 6.5. Furthermore, let
k:= max arg min
i∈[|Q\N|]Cmono(N∪MINi(Q\N)) −Pi∈NξσR,ρR
i(N)
i,
Cmono(MINi(Q\N))
i
and S:= MINk(Q\N). Then, σR(Q, N) := Sand
ρR(Q, N) := min Cmono(N∪S)−Pi∈NξσR,ρR
i(N)
k,Cmono(S)
k.
Note that this recursion is well-defined. Computing σR(Q, N)and ρR(Q, N)requires
ξσR,ρR(N)for which only σR(N, ·)and ρR(N, ·)is needed (unless N=∅). Yet, N(Q
by assumption.
Lemma 6.27. Let R= (ALG,Cmono)be a β-relaxation for some Π-cost-sharing
problem (IS)S⊆[n]. Then σRand ρRare valid, and ρRis β-average for C.
Proof. Let σ:= σRand ρ:= ρR.
•σand ρare valid (confer Definition 6.8):
Clearly, properties (6.3) and (6.4) are fulfilled. To see (6.5), let N(Q0⊆
Q⊆[n]. Define Σ(N) := Pi∈Nξσ,ρ
i(N)and S:= σ(Q, N), k := |S|and
S0:= σ(Q0, N), k0:= |S0|. Since 1≤k0≤ |Q0\N| ≤ |Q\N|,
82 6 Egalitarian Mechanisms
ρ(Q, N)≤Cmono(MINk0(Q\N))
k0
≤Cmono(MINk0(Q0\N))
k0
=Cmono(S0)
k0.
Furthermore,
ρ(Q, N)≤Cmono(N∪MINk0(Q\N)) −Σ(N)
k0
≤Cmono(N∪MINk0(Q0\N)) −Σ(N)
k0
=Cmono(N∪S0)−Σ(N)
k0.
Since ρ(Q0, N)is equal to one of these upper bounds, we have ρ(Q, N)≤ρ(Q0, N).
Finally, to see property (6.6), let N(Q⊆[n]and define S:= σ(Q, N), k := |S|
and N0:= N∪S, S0:= σ(Q, N0), k0:= |S0|. Then,
ρ(Q, N)≤Cmono(MINk0(Q\N))
k0≤Cmono(MINk0(Q\N0))
k0=Cmono(S0)
k0.
Moreover, it is MINk+k0(Q\N) = S∪S0. Also, it is easy to see that Σ(N0) =
Σ(N) + k·ρ(Q, N)by making use of property (6.4) (similarly as in first part of
the proof of Theorem 6.17). Consequently,
ρ(Q, N)≤Cmono(N∪S∪S0)−Σ(N)
k+k0
=Cmono(N0∪S0)−Σ(N)
k+k0
=Cmono(N0∪S0)−Σ(N0) + k·ρ(Q, N)
k+k0,
implying that
ρ(Q, N)≤Cmono(N0∪S0)−Σ(N0)
k0.
Again, ρ(Q, N0)coincides with one of the upper bounds, thus it follows that
ρ(Q, N)≤ρ(Q, N ∪σ(Q, N)).
•ρis β-average for C:
Let N(Q⊆[n]and S⊆Q\N. Then,
ρ(Q, N)≤Cmono(MIN|S|(Q\N))
|S|≤Cmono(S)
|S|≤β·C(S)
|S|.
With Theorem 6.10, we know that EgalσR,ρRis CGSP. To also be able to apply The-
orem 6.21, we have to check that the sum of the cost-shares computed by EgalσR,ρR
always covers the cost of some solution for the instance induced by the selected play-
ers. This can generally not be ensured for the solution computed by ALG. In order to
compute a proper solution, we need Definition 6.28 and some further notation: For a
solution Xfor instance ISof a cost-sharing problem (IS)S⊆[n], we denote by CX(S)
the value of solution X(i.e., CALG(S) = CALG(IS)(S)).
6.7 Computational Framework 83
Definition 6.28. For an optimization problem Π, consider a Π-cost-sharing problem
(IS)S⊆[n]. Then, (IS)S⊆[n]is called mergable if for all disjoint T, U ⊆[n],T∩U=∅,
and for all feasible solutions Xfor ITand Yfor IUwith costs CX(T)and CY(U),
respectively, there is a feasible solution Zfor IT∪Uwith cost CZ(T∪U)such that
CZ(T∪U)≤CX(T) + CY(T). We denote this operation by Z=X⊕Y.
Based on σRand ρR, Algorithm 6.6 solves all of the service provider’s tasks,
including computing a feasible solution of the underlying optimization problem. Re-
member that for a problem instance I, we let ALG(I)denote the solution computed
by ALG. We address the running time afterwards.
Algorithm 6.6 (computing EgalσR,ρRvia β-Relaxations).
Input: β-relaxation R= (ALG, Cmono); bid vector b∈Rn
Output: player set Q(b)∈2[n], cost-share vector x(b)∈Rn
≥0, solution Z(b)for IQ(b)
1: x:= 0,Q:= [n],N:= ∅,Z:= ‘empty solution’
2: while N6=Qdo
3: S:= σR(Q, N);a:= ρR(Q, N)
4: Q:= Q\{i∈S|bi< a}
5: if S⊆Qthen
6: Z:= (ALG(IN∪S)if Cmono(N∪S)−Pi∈Nxi≤Cmono(S)
Z⊕ALG(IS)otherwise
7: N:= N∪S;xi:= afor all i∈S
8: return (Q, x, Z)
Lemma 6.29. Let R= (ALG,Cmono)be a β-relaxation for a mergable Π-cost-
sharing problem (IS)S⊆[n].
1. At the end of each iteration of Algorithm 6.6, it holds that x=ξσR,ρR(N).
2. Line 3 of Algorithm 6.6 needs at most 2nevaluations of Cmono.
3. The mechanism computed by Algorithm 6.6 is β-BB (for costs induced by the
returned solution Z).
Proof. Let b∈Rnbe the input bid vector, and Q:= Q(b),x:= x(b), and Z:= Z(b)
be the output set of served players, cost-share vector, and solution. Assume there
are miterations. For any k∈[m], indicate the value of all variables at the end of
the k-th iteration (just after line 7) with a subscript k.
1. Let σ:= σRand ρ:= ρR. We show by induction over k∈[m]that ξσ,ρ(Ni) = xi,
by making frequent use of validity property (6.4).
The base case x1=ξσ,ρ(N1)holds trivially if line 5 evaluated to false in iteration
1. Otherwise, it holds since N1=σ(Q, ∅) = σ(N1,∅)and (x1)i=ρ(Q, ∅) =
ρ(N1,∅)for all i∈N1.
Now consider the induction step k−1→k. Without loss of generality, assume
that line 5 evaluated to true in iteration kas otherwise Nk=Nk−1and xk=
xk−1. Observe that Nk=Nk−1∪σ(Q, Nk−1). As above, S:= σ(Q, Nk−1) =
σ(Nk, Nk−1)and a:= ρ(Q, Nk−1) = ρ(Nk, Nk−1). Define ywith yi:= afor all
i∈Sand 0otherwise. Since ξσ,ρ(Nk) = ξσ,ρ(Nk−1) + yand xk=xk−1+y,
xk=ξσ,ρ(Nk)by induction hypothesis.
2. The claim directly follows from (1).
3. Define Σ(Nk) := Pi∈Nkxi. We show that CZk(Nk)≤Σ(Nk)≤Cmono(Nk)by
induction over k∈[m].
84 6 Egalitarian Mechanisms
The base case k= 1 holds because CZ1(N1) = CALG(N1)≤Cmono(N1) = Σ(N1).
Now consider the induction step k−1→k. Without loss of generality, assume that
line 5 evaluated to true in iteration kas otherwise Zk=Zk−1and Nk=Nk−1.
If Cmono(Nk−1∪Sk)−Σ(Nk−1)≤Cmono(Sk)then again
CZk(Nk) = CALG(Nk)≤Cmono(Nk) = Σ(Nk).
Otherwise,
CZk(Nk) = CZk−1⊕ALG(ISk)(Nk−1∪Sk)
≤CZk−1(Nk−1) + CALG(Sk)
≤Σ(Nk−1) + Cmono(Sk)
< Cmono(Nk−1∪Sk)
=Cmono(Nk).
Now by Σ(Nk−1) + Cmono(Sk) = Σ(Nk), the claim follows. Clearly, Z=Zm,
Q=Nm, and thus CZ(Q)≤Σ(Q)≤Cmono(Q)≤β·C(Q).
Theorem 6.30 (Corollary of Lemmata 6.27, 6.29, Theorems 6.10 and 6.21). For an
optimization problem Π, let (IS)S⊆[n]be a mergable Π-cost-sharing problem having a
β-relaxation (ALG, Cmono). Then the mechanism computed by Algorithm 6.6 is CGSP,
β-BB, and (2β·Hn)-EFF for C0induced by the solution computed by Algorithm 6.6.
Moreover, Algorithm 6.6 evaluates Cmono for no more than 2n2subsets of [n], makes
no more than n(direct) calls to ALG, and the number of merge operations is no more
than n. (Note: C0not necessarily coincides with CALG.)
6.8 Applications to Scheduling and Bin Packing
We use three approaches for obtaining β-relaxations that are polynomial-time com-
putable: Monotonic approximation algorithms (Theorem 6.32 and Lemma 6.33), non-
monotonic approximation algorithms with monotonic bounds Cmono (Theorems 6.34
and 6.35), and optimal costs that are monotonic and polynomial-time computable
(discussed at the end of this section).
Note here that we assume that each player is given a unique number in [n]in
advance (outside the scope of Algorithm 6.6) and that players are sorted according
to the respective monotonicity criterion.
Lemma 6.31. Each BinPacking or (Q||Cmax)cost-sharing problem is mergable in
time O(n).
Proof. For BinPacking with disjoint object/player sets Tand U, we obtain a bin
packing for T∪Uby taking both the bins with objects from Tand the bins with
objects from U. The costs (number of bins) simply add up. For (Q||Cmax)and disjoint
job/player sets Tand U, we obtain a schedule for T∪Uby assigning each job to the
machine assigned before. The resulting makespan doesn’t exceed the sum of the two
makespans.
First, we consider (P||Cmax)cost-sharing problems:
Theorem 6.32. For each (P||Cmax)cost-sharing problem it holds that (LPT, CLPT)
is a 4m−1
3m-relaxation and Algorithm 6.6 runs in time O(n3·(log n+ log m)).
6.8 Applications to Scheduling and Bin Packing 85
Proof. We show that CLPT is monotonic itself, i.e., for all U⊆T⊆[n] :
CLPT(min|U|T)≤CLPT(U). Fix p∈Nnand consider the cost-sharing problem (p,1).
Consider U, U0⊆[n]with |U|=|U0|such that pU0is obtained by reducing exactly
one entry of pU(and re-sorting). We show that CLPT(U0)≤CLPT(U).
Let U:= {u1, . . . , u|U|}and U0:= {u0
1, . . . , u0
|U|}such that u1< . . . < u|U|and
thus by Assumption 3.1, pu1≥. . . ≥pu|U|; analogously for U0. For each k∈[|U|],
denote Ukand U0
kto be the first kelements in Uand U0, respectively. In addition,
let φand φ0the LPT assignments for Uand U0. We will show by induction that
p({i∈U0
k|φ0(i) = φ0(u0
k)})≤p({i∈Uk|φ(i) = φ(uk)}),
which immediately implies CLPT(U0)≤CLPT(U). The case k= 1 is trivially satisfied
by pu0
1≤pu1. For k→k+ 1, by induction hypothesis, it holds that
p({i∈U0
k|φ0(i) = φ0(u0
k+1)}) = min
h∈[m]p({i∈U0
k|φ0(i) = h})
≤min
h∈[m]p({i∈Uk|φ(i) = h})
≤p({i∈Uk|φ(i) = φ(uk+1)}).
Thus,
p({i∈U0
k+1 |φ0(i) = φ0(u0
k+1)}) = p({i∈U0
k|φ0(i) = φ0(u0
k+1)}) + pu0
k+1
≤p({i∈Uk|φ(i) = φ(uk+1)}) + puk+1
=p({i∈Uk+1 |φ(i) = φ(uk+1)}).
Yet, LPT is not monotonic for general (Q||Cmax)cost-sharing problems. Consider
(p,s)with n= 5,m= 2,p:= (41,40,40,30,30), and speeds s:= (40,20 −ε)
(scale pand sto guarantee s∈Nm). Let U:= {1,2,4,5}and U0:= {2,3,4,5}. The
corresponding costs are CLPT(U) = 2.525 and CLPT(U0) = 2.75 (confer Figure 6.2).
Fig. 6.2. LPT is not monotonic in general
41 40
40 20 −ε
30
30
40
30
30
40
40 20 −ε
pU= (41,40,30,30) pU0= (40,40,30,30)
The following 2-relaxations for BinPacking cost-sharing problems will be the
key for obtaining 2-relaxations for (Q||Cmax)cost-sharing problems:
Lemma 6.33. For each BinPacking cost-sharing problem, there is a 2-relaxation
for which Algorithm 6.6 runs in time O(n3·log n).
86 6 Egalitarian Mechanisms
Proof. Let RFFD denote the following algorithm: Given ς∈(0,1], round each weight
up to the next power of 2, i.e., let object i’s rounded weight be ς0
i:= 2dlog2ςiefor all
i∈[n]. Then, run the FFD which is known to produce an optimal packing for this
modified instance s0([27]). Clearly, RFFD is a 2-approximation algorithm running in
time O(n·log n). Since RFFD is optimal for the rounded sizes it is monotonic. Thus,
(RFFD, CRFFD)is a 2-relaxation that can be computed in time O(n·log n).
We also remark that it is known that NFD is monotonic [100] and a 2-approximation
algorithm for BinPacking. Hence, also (NFD, CNFD)is a 2-relaxation.
We note that FFD is not monotonic. For n= 11, define the cost-sharing problem
ς:= ( 9
17 ,9
17 ,8
17 ,5
17 ,5
17 ,5
17 ,4
17 ,4
17 ,4
17 ,3
17 ,3
17 ). Let U:= [11] \3and U0:= [11] \ {1}.
The respective costs are CLPT(U) = 3 and CLPT(U0) = 4 (confer Figure 6.3).
Fig. 6.3. FFD is not monotonic in general
9
17 9
17
5
17
3
17
4
17
5
17
3
17
5
17
4
17
4
17
9
17
4
17
5
17
3
17
4
17
4
17
8
17
5
17
5
17 3
17
ςU= ( 9
17 ,9
17 ,5
17 ,5
17 ,5
17 ,4
17 ,4
17 ,4
17 ,3
17 ,3
17 )ςU0= ( 9
17 ,8
17 ,5
17 ,5
17 ,5
17 ,4
17 ,4
17 ,4
17 ,3
17 ,3
17 )
We obtain two different relaxations for (Q||Cmax)cost-sharing problems, both
based on applying binary search to the associated BinPacking decision problems.
The first result from Theorem 6.34 is a more practicable one with respect to com-
putability, obtained essentially by turning the RFFD algorithm from Lemma 6.33 into
a decision procedure; the second result from Theorem 6.35 is obtained by adapting
the PTAS for (Q|pi=1|Cmax)optimization problems by Hochbaum and Shmoys [68],
tweaking a bound computed inside the algorithm such that it becomes monotonic.
For details, we refer to [11].
Theorem 6.34 ([11]). For each (Q||Cmax)cost-sharing problem there is a 2-relaxation
for which Algorithm 6.6 runs in time O(n3·log m·log Pi∈[n]pi).
Theorem 6.35 ([11]). For each (P||Cmax)cost-sharing problem, there is a (1 + ε)-
relaxation for which Algorithm 6.6 runs in time O(n2+ 1
ε2·log Pi∈[n]pi).
There are several mergable scheduling problems for which optimal costs are
monotonic and computable in polynomial time. For instance,
Lemma 6.36. For each (Q|pi=1|Cmax)cost-sharing problem, (LPT, CLPT)is a 1-relaxation
and Algorithm 6.6 runs in time O(n3·log m).
In the following, we give a selection of further scheduling problems, taken from Bruck-
ner [17].
6.9 Conclusion and Open Problems 87
We restrict our attention to the cases in which only one of the properties pi,ri
and wiis variable and let the others be fixed with pi:= 1,wi:= 1, and ri:= 0. We
get that 1-relaxations exist for:
– Variable processing times: (Q|pmtn|Cmax),(Q||PiCi), and (Q|pmtn|PiCi)
– Variable weights: (P||PiwiCi)and (P|pmtn|PiwiCi)
– Variable release dates: (Q|pmtn, ri|Cmax)
The result for (Q||PiCi)especially implies 1-BB for (1||PiCi). This is a drastic
improvement over Moulin mechanisms, since by Brenner and Schäfer [16], no cross-
monotonic cost-sharing method for (1||PiCi)cost-sharing problems can generally
be better than n+1
2-BB.
6.9 Conclusion and Open Problems
The pivotal point of this chapter is our new modeling assumption on coalition forma-
tion. We believe that CGSP is a viable replacement for the often too limiting GSP
requirement. Besides this novel structural property, we consider the main asset of
our work to be threefold:
•Egalitarian mechanisms; showing existence of CGSP, 1-BB, and 2Hn-EFF mech-
anisms for any non-decreasing subadditive cost function.
•Our framework for polynomial-time computation that reduces constructing CGSP,
O(1)-BB, and O(log n)-EFF mechanisms to finding monotonic approximation al-
gorithms.
•Showing that acyclic mechanisms are CGSP and thus remarkably stronger than
was known before.
An immediate issue left often by our work is, of course, to find more applications
of our polynomial-time framework. For instance, it is easy to see that SteinerTree
problems are mergable and their costs non-decreasing and subadditive; but do they
allow for a β-relaxation?
7
Group-Strategyproof Mechanisms for General Demand
7.1 Contribution
The point of departure for this chapter is a rather obvious idea for generalizing Moulin
mechanisms: Start with the maximum allocation and iteratively reduce service levels
until every player can afford his remaining levels. The cost shares are extracted from
marginal cost-sharing methods χ(confer Definition 2.28). We term these mechanisms
MoulinGDχ.
•Whereas this idea of constructing a general demand cost-sharing mechanism has
already been used by Mehta et al. [94], it was not known how GSP could be
achieved. We identify three properties of marginal cost-sharing methods χthat are
sufficient for MoulinGDχto be GSP. It comes as no surprise that a generalization
of binary-demand cross-monotonicity is among them.
•We introduce natural mechanisms Levelχthat for each marginal cost-sharing
method χcompute the same output as MoulinGDχ. We prefer to work with
mechanisms Levelχsince they naturally give away service levels incrementally. In
our opinion, this is much more intuitive and easier to handle.
•We give marginal cost-sharing methods χfl for every FaultTolerantFL cost-
sharing problem and show that Levelχfl is GSP, (3L)-BB, (3L·(1 + Hn))-EFF,
and computable in polynomial time. This improves on the results of Mehta et
al. [94], since a stronger notion of incentive-compatibility is guaranteed (GSP in-
stead of WGSP) while reducing the budget-balance and efficiency approximations
by a factor of L. The general idea is to tweak the binary demand cost-sharing
mechanism for FacilityLocation by Pál and Tardos [106] in a clever way and
then show that iteratively invoking the tweaked version works.
•For each GenSteinerForest cost-sharing problem, we give marginal cost-
sharing methods χgs such that Levelχgs is GSP, (2·HL)-BB, O(log2n·log L)-EFF
and computable in polynomial time. Here, the idea is to iteratively apply the
binary demand cost-sharing mechanisms for SteinerForest cost-sharing prob-
lems proposed by Könemann et al [82]. Prior to this result, no GSP cost-sharing
mechanisms for GenSteinerForest cost-sharing problems had been known.
The results presented in this chapter are published in [13].
90 7 Group-Strategyproof Mechanisms for General Demand
7.2 Organization
We present mechanism MoulinGDχin Section 7.3. Specifically, the validity properties
of marginal cost-sharing methods that guarantee that MoulinGDχis GSP are stated
in Section 7.3.1. We introduce mechanism Levelχin Section 7.3.2. The applications to
FaultTolerantFL and GenSteinerForest cost-sharing problems are presented
in Section 7.4 and Section 7.5.
7.3 Generalized Moulin Mechanisms
Given a marginal cost-sharing method χ, we propose to generalize Moulin mecha-
nisms (confer Algorithm 4.1) as in Algorithm 7.1:
Algorithm 7.1 (computing MoulinGDχ(B) = (q(B), x(B))).
1: q:= (L1, . . . , Ln)
2: while there exists iwith qi>0and bi,qi< χi,qi(q)do
3: qj:= qj−1for an arbitrary jwith qj>0and bj,qj< χj,qj(q)
4: return (q,x) with xi:= Pqi
`=1 χi,`(q)for all i∈[n]
Clearly, if χis β-BB then so is MoulinGDχ.
Note that in line 2, we only check if player ican pay for the current largest level qi.
However, the marginal cost-sharing methods χthat guarantee GSP of MoulinGDχ
imply that χi,1(q)≤. . . ≤χi,qi(q)(see Lemma 7.5). As additionally bi,1≥. . . ≥bi,qi
by Assumption 2.2, this check guarantees that all remaining levels can be paid for.
7.3.1 Validity of Marginal Cost-Sharing Methods
We now state the three properties of χthat are sufficient for MoulinGDχto be GSP.
The first is a generalization of binary demand cross-monotonicity and states that
the marginal cost share of a player for a certain service level can only decrease if the
service levels of other players are increased:
Definition 7.1. A marginal cost-sharing method χis cross-monotonic if for all al-
locations a∈ A, all players i∈[n]and j∈[n]\{i}with aj< Lj, and all service
levels `∈[Li], it holds that χi,`(a)≥χi,`(a+ej).
The second property ensures that the marginal cost-share χi,`(a)of player iwith
ai≥`is exactly the marginal cost-share χi,`(a≤`)(confer Section 2.2), i.e., this
marginal cost share for `is independent of service levels larger than `:
Definition 7.2. A marginal cost-sharing method χis level-restricted if for all allo-
cations a∈ A, for all players i∈[n], and for all service levels `∈[Li], it holds that
χi,`(a) = χi,`(a≤`).
The third property together with cross-monotonicity and level-restriction implies
that the marginal cost-share of a player is non-decreasing in the number of levels, as
we show in Lemma 7.5.
Definition 7.3. A marginal cost-sharing method χis non-decreasing if for all allo-
cations a∈ A, for service level `:= maxi∈[n]{ai}, and for all players i∈[n]with
ai=` < Li, it holds that χi,`(a)≤χi,`+1(a+Pj∈[n]:aj=`<Ljej).
We merge the three properties into the central term validity:
7.3 Generalized Moulin Mechanisms 91
Definition 7.4. A marginal cost-sharing method χis valid if it is level-restricted,
cross-monotonic, and non-decreasing.
Lemma 7.5. If χis valid, it holds for all allocations a∈ A, for all service levels
`∈[L], and for all players i∈[n]with ai> ` that χi,`(a)≤χi,`+1(a).
Proof. Fix allocation a, service level `and player iwith ai> `. It is
χi,`(a) = χi,`(a≤`)(χlevel restricted)
≤χi,`+1(a≤`+X
j∈[n]:a≤`
j=`<Lj
ej)(χnon-decreasing)
≤χi,`+1(a≤`+1)(χcross-monotonic)
=χi,`+1(a).(χlevel-restricted)
Together with Assumption 2.2, Lemma 7.5 states that
bi,`+1 −χi,`+1(a)≤bi,` −χ(i, `)(a)(7.1)
for all B∈ R, all, a∈ A, all `∈[L], and all i∈[n]with ai> `. The results in this
section heavily rely on this observation.
Lemma 7.6. MoulinGDχmeets NPT, VP and strict CS, given any valid marginal
cost-sharing method χ.
Proof. Fix B∈ R. NPT holds by definition of χ. VP follows from the fact that
bi,qi(B)≥χi,qi(B)(q(B)) and thus Pqi(B)
`=1 bi,` ≥Pqi(B)
`=1 χi,`(q(B)) = xi(B)by (7.1).
To receive service level `∈[Li]0, player imay submit biwith bi,m := Mfor m≤`
and a sufficient large Mand bi,m := −1otherwise. Thus, strict CS holds as well.
To gain insight on the importance of each of the three validity requirements, we
first show that if exactly one of the validity properties required for χdoes not hold,
MoulinGDχis not GSP. For all examples, let n= 2,L1=L2= 2, and χ2,`(a) := 2
for all `∈[2] and all a∈ A with a2≥`. We always assume that v2= (2,2).
Example 7.7. Consider χwith χ1,`(a) := 1 for all `∈[2] and all a∈ A with a1≥`,
with the only exception that χ1,2(2,2) := 2. Obviously, χis level-restricted and
non-decreasing, but not cross-monotonic since χ1,2(2,1) < χ1,2(2,2). For the case
that v1= (2,2), both players get service level 2, where u1((v1,v2),v1)=1and
u2((v1,v2),v2) = 0. Player 2may then bid b2= (−1,−1) in order to not receive the
service with the result that player 1receives level 2with utility u1((v1,b2),v1) = 2.
Example 7.8. Consider method χwith χ1,1(a) := 1 for all a∈ A with a1= 1, and
χ1,1(a) := 2 and χ1,2(a) := 3 for all a∈ A with a1= 2. Here, χis cross-monotonic
and non-decreasing but fails to be level-restricted due to χ1,1(2,2) > χ1,1(1,1). If
now v1= (3,3), then player 1receives level 2and u1((v1,v2),v1) = 1. However, for
b1= (3,−1), player 1receives only level 1, and u1((b1,v2),v1) = 2.
On the other hand, we get χ1,1(2,2) < χ1,1(1,1) when we change χsuch that
χ1,1(a) = 2 for all a∈ A with a1= 1 and χ1,1(a) = 1 for all a∈ A with a1= 2. If
v1= (3,3−ε), then player 1receives only one level and u1((v1,v2),v1) = 1. However,
he may bid b1= (3,3) to receive both levels such that u1((b1,v2),v1) = 2 −ε.
92 7 Group-Strategyproof Mechanisms for General Demand
Example 7.9. Consider χwith χ1,1(a) := 2 for all a∈ Awith a1≥1and χ1,2(a) := 1
for all a∈ A with a1≥2. Now we have the case that χis cross-monotonic and level-
restricted, but not non-decreasing. For v1= (1,1), player 1receives level 2and has
a utility of u1((v1,v2),v1) = −1. However, bidding b1= (−1,−1) ensures a utility
of zero.
7.3.2 Level Mechanisms
Given any valid marginal cost-sharing method χ, we define a general demand mech-
anism Levelχin Algorithm 7.3 and show in Theorem 7.10 that it computes the same
output as MoulinGDχ. Subsequently, Theorem 7.11 states that validity of χimplies
GSP of Levelχ. We consider the runtime of Levelχin Lemma 7.12.
Although it is possible to prove GSP of MoulinGDχdirectly, we prefer this
workaround for two reasons: whereas MoulinGDχiteratively decrements arbitrary
service levels that cannot be paid for, Levelχiteratively increases only service levels
of those players that have been granted the previous level as well. We think that this
is a much more intuitive way to determine an allocation. Furthermore, we may use
results for binary demand Moulin mechanisms, as these are employed within Levelχ.
For a consistent notation, we adjust cost-sharing methods ξand Moulinξto work
with vectors p∈ {0,1}ninstead of sets. We give the adjusted mechanism Moulinξ
below:
Algorithm 7.2 (computing Moulinξ(b) = (q(b), x(b))).
1: p:= 1;
2: while there exists iwith pi= 1 and bi,1< ξi(p)do
3: pj:= 0 for an arbitrary jwith pj= 1 and bj,1< ξj(p)
4: return (p, ξ(p))
Specifically, given a marginal cost-sharing method χ,Levelχgives away service
levels incrementally. In iteration `, only players that were given service level `−1are
considered as potential receivers of service level `. In order to determine the actual
receivers of level `, when previously having computed allocation qwith qi≤`−1for
all iin iterations 1to `−1, we apply the binary demand Moulin mechanism Moulinξ
with a binary demand cost-sharing method ξ:= χq:{0,1}n→Rn
≥0defined by
χq
i(p):=χi,qi+1(q+p)if qi< Li
0otherwise .
Moulinχqgets as an argument the bids bi,` from players iwith `−1 = qi< Lifor ser-
vice level `. For the other players, negative bids are simulated to ensure that they do
not receive service level `. Our new mechanism Levelχis computed by Algorithm 7.3:
Algorithm 7.3 (computing Levelχ(B) := (q(B), x(B))).
1: (q,x) := (0,0);`:= 1
2: b0:= (b1,1, . . . , bn,1).bids for the first service level
3: repeat
4: (q,x) := (q,x) + Moulinχq(b0)
5: `:= `+ 1; Define b0by b0
i:= (bi,` if `−1 = qi< Li
−1otherwise
6: until b0
i=−1for all i∈[n]
7: return (q,x)
7.3 Generalized Moulin Mechanisms 93
The validity of χhas a very descriptive interpretation for Levelχ. Cross-mono-
tonicity implies that in each iteration, it can only be beneficial for all players if more
players receive the current level. Level-restriction states that the marginal cost-share
of a level due to which a player was selected in line 4 to receive this level remains fixed
during further computation. Finally, if χis non-decreasing, the marginal cost-shares
for higher levels can only increase.
Theorem 7.10. For each bid matrix B∈ R and each valid marginal cost-sharing
method χ,Levelχand MoulinGDχcompute the same output.
Proof. Fix a valid marginal cost-sharing method χand a bid matrix B∈ R. Let
qLand xLbe the allocation and the vector of cost-shares computed by Levelχ, and
qMand xMbe the corresponding values computed by MoulinGDχ. We first show
by induction that (qL)≤`= (qM)≤`for all `∈[L].
•`= 1
Assume that T:= {i∈[n]|(qL
i)≤1= 0 and (qM
i)≤1= 1} 6=∅. Consider the
first iteration of Levelχ. Within Moulinχ0, called by Levelχin line 4, consider the
first time that Moulinχ0rejects a player j∈T, i.e., sets pjto 0in line 3. Let p
denote the corresponding vector right before this event. The way that jand p
were chosen, it holds that pj= 1 and (qM)≤1≤p. Hence, by validity of χ,
bj,1< χ0
j(p) = χj,1(p)≤χj,1((qM)≤1) = χj,1(qM).
Using (7.1) we get that
xM
j=
qM
j
X
k=1
χj,k(qM)>
qM
j
X
k=1
bj,k ,
contradicting VP of MoulinGDχ.
Similarly, it is T0:= {i∈[n]|(qL
i)≤1= 1 and (qM
i)≤1= 0}=∅, by considering
the first time that MoulinGDχsets qjto 0for a j∈T0in line 3. Let qbe the
corresponding allocation vector right before this event. Here, it is qj= 1 and
(qL)≤1≤qand hence by validity of χ,
bj,1< χj,1(q)≤χj,1((qL)≤1) = χ0
j((qL)≤1).
Then, however, in the first iteration of Levelχ,Moulinχ0would have rejected
player jwhich contradicts qL
j≥1.
•`−1→`
Assume that T:= {i∈[n]|(qL
i)≤`=`−1and (qM
i)≤`=`} 6=∅. For the `-th
iteration of Levelχ, consider the first time that Moulinξwith ξ=χ(qL)≤`−1sets pj
to 0for a j∈T. Let pbe the corresponding vector right before this event. By the
choice of p, we have that pj= 1 and p≥(qM)`. With (qL)≤`−1= (qM)≤`−1by
induction assumption, it holds that (qL)≤`−1+p≥(qM)≤`−1+(qM)`= (qM)≤`.
Now, by (7.1) and validity of χ,
bj,qM
j≤bj,` < χ(qL)≤`−1
j(p) = χj,`((qL)≤`−1+p)
≤χj,`((qM)≤`) = χj,`(qM)
≤χj,qM
j(qM).
94 7 Group-Strategyproof Mechanisms for General Demand
However, this contradicts the termination condition of MoulinGDχ.
Now assume T0:= {i∈[n]|(qL
i)≤`=`and (qM
i)≤`=`−1} 6=∅. For MoulinGDχ,
consider the first iteration that sets qjto `−1for a j∈T. Let qbe the cor-
responding allocation vector right before this event. Specifically, together with
(qL)≤`−1= (qM)≤`−1by induction assumption, it is q≥(qL)≤`. By (7.1) and
validity of χ,
bj,qL
j≤bj,` < χj,`(q)≤χj,`((qL)≤`) = χj,`(qL)≤χj,qL
j(qL),
a contradiction to the fact that Levelχhas given service level qL
jto j.
For the computed cost-shares, we have with q:= qL=qMthat for all i∈[n]
xL
i=
qi
X
`=1
χi,`(q≤`) =
qi
X
`=1
χi,`(q) = xM
i.
The proof of Theorem 7.11 uses the main idea from [99], which shows that Moulinξ
is GSP if ξis cross-monotonic (confer the proof of Theorem 4.1).
Theorem 7.11. For any valid marginal cost-sharing method χ,Levelχis GSP.
Proof. Assume that Levelχis not GSP. Then, there exists K⊆[n]with true valu-
ations VK= (vi)i∈K, and a bid matrix B∈ R such that ui(B)≥ui(VK,B−K)
for all i∈K, with at least one strict inequality. Without loss of generality, let
V:= (VK,B−K).
We first prove by induction that q(B)≤`≤q(V)≤`for all `∈[L].
•`= 1
Assume that T:= {i∈[n]|qi(B)≤1= 1 and qi(V)≤1= 0} 6=∅. Consider the
first iteration of Levelχwith input V. Within Moulinχ0, called by Levelχin line 4,
consider the first time that Moulinχ0rejects a player j∈T, i.e., sets pjto 0in
line 3. Let pdenote the corresponding vector right before this event. It is pj= 1
and p≥q(B)≤1. By cross-monotonicity,
vj,1< χ0
j(p) = χj,1(p)≤χj,1(q(B)≤1) = χj,1(q(B)) .
Thus, j∈K. Furthermore, by (7.1) we get
xj(B) =
qi(B)
X
k=1
χj,k(q(B)) >
qi(B)
X
k=1
bj,k .
Hence, uj(V,vj) = 0 > uj(B,vj)which contradicts j∈K.
•`−1→`
By induction hypothesis, for all players iwith qi(B)≤`> qi(V)≤`we have that
qi(B)≤`−1=qi(V)≤`−1=qi(V)≤`=`−1and qi(B)≤`=`. Assume that
T:= {i∈[n]|qi(B)≤`=`and qi(V)≤`=`−1} 6=∅and obtain j∈Tand p
as in case `= 1, for Moulinξwith ξ=χq(V)≤`−1. It holds that p≥(qi(B))`. By
induction hypothesis and validity of χ,
7.4 Applications to Fault Tolerant Facility Location 95
vj,` < χq(V)≤`−1
j(p) = χj,`(q(V)≤`−1+p)
≤χj,`(q(B)≤`−1+ (qi(B))`) = χj,`(q(B)≤`).
Thus, j∈K. However, by (7.1), induction hypothesis, and qj(V) = `−1, we get
a contraction to j∈K, namely,
uj(B,vj) =
qj(B)
X
k=1 vj,k −χj,k(q(B)≤k)
<
`−1
X
k=1 vj,k −χj,k(q(B)≤k)
≤
qj(V)
X
k=1 vj,k −χj,k(q(V)≤k)=uj(V,vj).
As an immediate consequence of q(V)≥q(B), we get a contraction to the as-
sumption that Levelχis not GSP, since no player strictly improves his utility for B:
For all i∈[n], by cross-monotonicity,
ui(B,vi)≤
qi(B)
X
`=1 vi,` −χi,`(q(V)≤`)
≤
qi(V)
X
`=1 vi,` −χi,`(q(V)≤`))
=ui(V,vi).
In order to determine the runtime of Levelχfor a specific χ, we introduce
Lemma 7.12:
Lemma 7.12. For each marginal cost-sharing method χsuch that for every a∈ A
and each `∈[L],χ∗,`(a)is computable in time O(t),Levelχis computable in time
O(L·n·t).
Proof. A call of Moulinq
χin iteration `∈[L]of Levelχwith the current allocation
q∈[`−1]n
0takes at most time O(n·t), as it has to eventually compute χ∗,`(q+p)for
ndifferent values for p∈ {0,1}(psimulates the set of served agents for this level).
7.4 Applications to Fault Tolerant Facility Location
In the examples within this section, we define instances of FaultTolerantFL via
networks. Facilities are illustrated as houses, where the roofs are labeled with the
opening cost. Players are circles labeled with the player’s identity. Edges (i, j)are
labeled with d(i, j). If iand jare not directly linked, d(i, j)is defined as the cost of
a shortest path between iand jin the network.
96 7 Group-Strategyproof Mechanisms for General Demand
Theorem 7.13 is the main theorem of this section:
Theorem 7.13. For each FaultTolerantFL cost-sharing problem there is a mar-
ginal cost-sharing method χfl and an approximation algorithm ALG such that Levelχfl
is GSP, 3L-BB, and 3L·(1+Hn)-EFF for CALG. Furthermore, Levelχfl is computable
in time O(n2·L·F·(n+ log |F|)) and ALG in time O(n·|F|·(n·L+ log |F|).
Guideline of Proof: Fix a FaultTolerantFL cost-sharing problem (F, o, d)(for the
remaining Section 7.4). In order to proof Theorem 7.13, we define a valid marginal
cost-sharing method χfl (together with an auxiliary method χ) in Section 7.4.1
and show that for every a∈ A and each `∈[L],χfl
∗,`(a)is computable in time
O(n·|F|·(n+ log |F|)). Section 7.4.2 gives the approximation algorithm ALG and
also discusses its polynomial-time computability. Section 7.4.3 then shows 3L-BB and
3L·(1 + Hn)-EFF for CALG. Theorem 7.13 then follows directly from Theorem 7.11
and Lemma 7.12.
7.4.1 The Marginal Cost-Sharing Method
In this section, we explain how to define χfl. To this end, we define a marginal cost-
sharing method χand obtain χfl by multiplying χby 3. We make this detour as χ
is consistent with the methods by Pál and Tardos [106] and Mehta et al. [94] and
allows for simpler comparisons. For a∈ {0,1}n, the computation of χreduces to the
method by Pál and Tardos [106] for binary demand facility location.
Fix a∈ A and `∈[L]. We only need to determine χi,`(a)for all players iwith
i∈A`:= {j∈[n]|aj≥`}. Simultaneously, every player iin A`uniformly grows
a ball with iat its center to infinity. This ball, the ghost of i, has radius tat time
t. We say that the ghost of itouches facility fat time tif d(i, f)≤t. If the ghost
of itouches fit starts filling f, contributing t−d(i, f)at time t≥d(i, f). Facility
fis said to be full, if all such contributions sum up to its opening cost of. Let t(f)
denote the time when fbecomes full, and Sf:= {i∈[n]|d(i, f)< t(f)}the set of
players that contributed to filling f. It holds that
X
i∈Sf
(t(f)−d(i, f)) = of.(7.2)
Definition 7.14. For all a∈ A, all `∈[L]and all i∈A`:= {i∈[n]|ai≥`}, we
define χi,`(a)to be the time that it takes for the ghost of ito touch `full facilities when
all players in the set A`grow their ghosts. For all other cases, we let χi,`(a) := 0.
Furthermore, we let χfl
i,`(a) := 3 ·χi,`(a)for all a∈ A, all i∈[n], and all `∈[Li].
Example 7.15. Given the instance in Figure 7.1, we look at allocation a= (2,2,2,1)
and explain how to compute χ(a). To denote certain events that occur during the
ghost-growing process, we introduce some notation. For the event that itouches f,
but fis not full yet, we write i◦f. The event that fbecomes full is denoted by f,
and the event that itouches a full facility fis written as i•f.
The marginal cost shares for level 1are determined by growing the ghosts of player
set A1={1,2,3,4}, where a1= (1,1,1,1). For level 2, we grow the ghosts of player
set A2={1,2,3}, where a2= (1,1,1,0). The events occuring at certain time steps t
are illustrated in Figure 7.2. Note that i∈Sfiff event i◦foccurs at a strictly smaller
time step than event i•f. Thus, for level 1,S1={1}, S2={2}and S3={3,4}.
For level 2,S1={1}, S2={2}, S3={3}and S4={1,2}. The cost shares are
χ∗,1(a) = χ∗,1(1,1,1,1) = (2,2,3
2,3
2)and χ∗,2(a) = χ∗,2(2,2,2,0) = (3,3,4,0). The
final cost shares are thus (5,5,11
2,3
2).
7.4 Applications to Fault Tolerant Facility Location 97
Fig. 7.1. Example instance for FaultTolerantFL
2
f11
1
14
5
2
2
f2
f4f3
1
2
1
1
1
314
Fig. 7.2. Events during the computation of χ(2,2,2,1) for the instance in Figure 7.1
Level 1: Level 2:
tEvent
1 1 ◦f1,2◦f2,3◦f3,4◦f3
3
2f3,3•f3,4•f3
2f1,f2
1•f1,2•f2,1◦f4,2◦f4
tEvent
1 1 ◦f1,2◦f2,3◦f3
2f1,f2,f3
1•f1,2•f2,3•f3,1◦f4,2◦f4
3f4,1•f4,2•f4
4 3 •f4
Note that χ∗,`(a)only depends on A`={j∈[n]|aj≥`}; it therefore holds that
χ∗,`(a) = χ∗,`(`·a`). This is even stronger than level-restriction.
Lemma 7.16. χand χfl are valid.
Proof. We only show validity of χ. Validity of χfl obviously follows. By construction,
χis level-restricted. Cross-monotonicity is met, since facilities are filled faster for a
larger set of players: For all allocations a∈ A, all players i∈[n]and j∈[n]\{i}
with aj< Lj, and all levels `∈[Li], it is
χi,`(a) = χi,`(`·a`)≥χi,`(`·(a+ej)`) = χi,`(a+ej).
To show that χis non-decreasing, observe that if the cost shares for level `and
level `+ 1 are determined for the same set of players, then the cost shares for level
`+ 1 cannot be strictly smaller. Fix aand let `:= maxi{ai}. For all i∈[n]with
ai=` < Li, it is
χi,`(a) = χi,`(`·a`)≤χi,`(X
j∈[n]:
aj=`<Lj
`·ej)≤χi,`+1(X
j∈[n]:
aj=`<Lj
(`+ 1) ·ej)
=χi,`+1((`+ 1) ·(a+X
j∈[n]:
aj=`<Lj
ej)(`+1)) = χi,`+1(a+X
j∈[n]:
aj=`<Lj
ej).
We now discuss computability. Fix a∈ A and `∈[L]. For each facility f∈F,
we can determine t(f)by ordering the players in A`={i∈[n]|ai≥`}by non-
decreasing distance to f. Assume without loss of generality that A`={1, . . . , |A`|}
98 7 Group-Strategyproof Mechanisms for General Demand
and that d(1, f)≤ ··· ≤ d(|A`|, f). For j∈[|A`|], let tj(f)be the time that facility
fgets full if only the ghosts in the set {1, . . . , j}grow their ghosts. It holds that
tj(f) = d(j, f)+ 1
jof−Pj−1
k=1 k·(d(k+ 1, f)−d(k, f)). Compute tj(f)iteratively
until j=|A`|or tj(f)≤d(j+ 1, f). Then t(f) = tj(f). Thus, the set {t(f)}f∈Fcan
be computed in time O(n2·|F|).
Now, for a player i∈A`, order the set {d(i, f), t(f)}f∈Fby non-decreasing values.
Let the corresponding ordered values be s1, . . . , s2|F|. Find the smallest j∈[2|F|]
such that |{f∈F|d(i, f)≤sjand t(f)≤sj}| =`. Then χi,`(a) = sj. The vector
χ∗,`(a)can thus be computed in time O(n·|F|·(n+ log |F|)).
7.4.2 The Approximate Solution
Given an allocation a∈ A, we give an approximation algorithm ALG that constructs
a solution with cost CALG(a). We decide which facilities to open during the iterative
computation of χi,`(a)for all i∈A`={j∈[n]|aj≥`}for `= 1, . . . , maxi{ai}.
Fix an iteration `. Let t(f)and Sfbe the values obtained for all f∈Fby growing
the ghosts of A`.
Facilities are opened in iteration `according to the following rule: Let F`−1be
the set of the already opened facilities in iterations 1, . . . , ` −1. If in iteration `, a
facility f /∈F`−1becomes full, we open fif and only if conditions O1and O2hold:
O1)There is no facility gthat was already opened in iteration `and for which
d(g, f)≤2·t(f).
O2)There are no `distinct facilities g1, . . . , g`∈F`−1for which d(gk, f)≤2·t(f)
for all k∈[`].
If facilities become full at the same time, we break ties arbitrarily. Finally, we connect
each player i∈[n]to aidistinct closest open facitities. It is straightforward to see
that this solution is computable in time O(n·|F|·(n·L+ log |F|).
For the sake of simplifying the analysis, we specify other connection rules that
may only increase the final cost: In each iteration `, we connect every player i∈A`
to one (more) open facility according to the following rules:
C1)If i∈Sffor an fopened in iteration `, we connect ito f.
C2)Otherwise, if at time χi,`(a)the ghost of itouches an open facility fto which
iis not connected yet, we connect ito f.
C3)Otherwise, let fbe a full but closed facility that the ghost of itouches at time
χi,`(a);fwas not opened in iteration `, because O1or O2do not hold:
a) If O1does not hold because of facility g, connect ito g.
b) If O2does not hold because of facilities g1, . . . , g`, connect ito a facility
g∈ {g1, . . . , g`}to which iis not connected yet.
If there are ties in C3aand C3b, break them arbitrarily.
Example 7.17. We continue Example 7.15. Let us now determine which facilities to
open and how to connect players to open facilities (for analysis). For Level 1, we
open f3, since there is no other already open facility. Then f1and f2become full.
We open f1, since 8 = d(f1, f3)>2·t(f1)=4. We also open f2, since d(f2, f3) =
d(f1, f3)>2·t(f1) = 2·t(f2)and 5 = d(f2, f1)>2·t(f2) = 4. Due to C1, we connect
players 3and 4to f3, player 1to f1, and player 2to f2. For Level 2, the first (and
only) facility that becomes full and is not opened yet is facility f4with t(f4)=3.
7.4 Applications to Fault Tolerant Facility Location 99
However, 3 = d(f4, fi)≤2·t(f4)=6for i∈ {1,2}. Thus, f4stays closed due to
O2. All players in {1,2,3}are connected due to C3b, i.e., 1is connected to f2,2is
connected to f1, and 3is connected to f1or f2.
Deleting O2and C3b, we get the opening and connection rules of [106]. However,
Example 7.18 shows that rule O2is crucial for a reasonable BB approximation:
Example 7.18. We continue Example 7.15 and Example 7.17. Our constructed solu-
tion has cost 24. The optimum solution has cost 17, with all facilities being opened.
So what is the reason that opening rule O2forbids to open f4? Consider the instance
in Figure 7.18 which consist of Mcopies of the above instance, where of4is replaced
by M. We assume that these copies are sufficiently far away from each other. To
ensure that f4(and the corresponding facilities in the copies) is filled before all cost
shares are determined, we introduce an additional construct (also sufficiently far away
from the others), where player 4M+ 1 has to fill facilities f4M+1 with of4M+1 = 1
and f4M+2 with of4M+2 =M, when his cost-share for level 2is computed. Consider
awith ai= 1 if i∈ {4,8, . . . , 4M}and ai= 2 otherwise. If we would open f4
and the corresponding facilities in the copies in the second iteration, the cost of the
constructed solution is larger than M·(M+ 1). However, the optimum cost for this
instance is 24·M+M+3. Thus, without rule O2, the BB factor would be unbounded.
Fig. 7.3. Unbounded budget-balance in case that rule O2is absent
M Copies
2
1
1
1
4
5
2
2
f2
f3
1
1
1
314
f1
1
f4
M
4M+ 1
11
1M
f4M+1 f4M+2
7.4.3 Budget-Balance and Efficiency
Theorem 7.19. χfl is (3L)-BB for CALG.
Proof. We show for χ, that for any a∈ A and X(a):=PL
`=1 Pi∈A`χi,`(a), it holds
that 1
3·CALG(a)≤X(a)≤L·FTFL(a).
Fix a∈ A. We first show the upper bound. Consider an arbitrary facility set
F0⊆Fwith |F0| ≥ maxi{ai}. Fix `∈[maxi{ai}]and i∈A`:= {i∈[n]|ai≥l}.
Let t(f)and Sfbe the values obtained for all f∈Fby growing the ghosts of A`.
Let F0
i⊆F0be an arbitrary set of aidistinct closest facilities in F0to i. We show:
∃f∈F0
i:χi,`(a)≤(t(f)if i∈Sf
d(i, f)otherwise .(7.3)
100 7 Group-Strategyproof Mechanisms for General Demand
Assume that (7.3) does not hold. Then for all f∈F0
i,χi,`(a)> t(f)> d(i, f)if i∈Sf
and χi,`(a)> d(i, f)≥t(f)otherwise. Thus, at time t:= maxf∈F0
i{t(f), d(i, f)},
the ghost of itouches at least ai≥`full facilities, a contradiction to t<χi,`(a).
Note that (7.3) especially holds for F0=F∗, when F∗is an optimal facility set for
a. Then,
X
i∈A`
χi,`(a)≤X
i∈[n]
X
f∈F∗
i:i∈Sf
t(f) + X
f∈F∗
i:i/∈Sf
d(i, f)
=X
f∈F∗X
i∈Sf:f∈F∗
i
(t(f)−d(i, f)) + X
i∈[n]X
f∈F∗
i
d(i, f)
≤X
f∈F∗
of+X
i∈[n]X
f∈F∗
i
d(i, f) = FTFL(a).
For the last inequality, confer (7.2). Thus, X(a) = PL
`=1 Pi∈A`χi,`(a)≤L·FTFL(a).
Now we show the lower bound, i.e., that 1
3·CALG(a)≤X(a). Fix an iteration
`. Let F`be the newly opened facilities in `, and F`−1be the facilities opened in
iterations 1. . . , ` −1. We show that the marginal cost shares of A`for level `recover
at least 1
3of the new opening and connection costs. Due to rule O1, for {g, f} ⊆ F`,
it holds that Sg∩Sf=∅, as shown in [106].
•Each player set Sffor an f∈F`pays at least 1
3of of+Pi∈Sfd(i, f):
Due to C1, all i∈Sffor an f∈F`are connected to f. By (7.2), it holds that
of=Pi∈Sf(t(f)−d(i, f)) and thus of+Pi∈Sfd(i, f) = |Sf|·t(f). We show that
for all i∈Sfit holds that χi,`(a)≥1
3·t(f).
Assume otherwise and consider i∈Sfwith χi,`(a)<1
3·t(f). Let {f1, . . . , f`}
be the full facilities that itouches at time χi,`(a). For all m∈[`],d(fm, f)≤
d(fm, i)+d(i, f)≤2·χi,`(a)<2
3·t(f)<2·t(f). If all facilities in {f1, . . . , f`}are in
F`−1, we get a contradiction to f∈F`by O2. Thus, there exists f0∈ {f1, . . . , f`}
with f0/∈F`−1. By d(f0, f)<2·t(f)and O1,f0is not open at time t(f)(but
full, since t(f0)≤χi,`(a)< t(f)). Facility f0was not opened at t(f0), since one
of the following holds:
– There is a facility gthat was already opened in `and d(g, f0)≤2·t(f0). This
contradicts f∈F`, since d(f, g)≤d(f, i)+d(i, f0)+d(f0, g)< t(f)+χi,`(a)+
2·t(f0)≤t(f)+3·χi,`(a)<2·t(f).
– There are facilities {g1, . . . , g`} ⊆ F`−1with d(gm, f0)≤2·t(f0)for all m∈[`].
Again, for all m∈[`],d(f, gm)<2·t(f), contradicting f∈F`.
•Each player not in Sffor f∈F`pays at least 1
3of his connection cost:
If iis connected to g∈F`−1∪F`due to C2,χi,`(a)≥d(i, g). If it is connected
due to C3aor C3b, it is d(i, g)≤d(i, f) + d(f, g)≤χi,`(a) + 2 ·t(f)≤3·χi,`(a)
in both cases.
Theorem 7.20. LevelχF L is (3L·(1 + Hn))-EFF for CALG.
We first show a property of χsimilar to (binary demand) summability in Lemma
7.21, which constitutes the main part of the proof.
7.4 Applications to Fault Tolerant Facility Location 101
Lemma 7.21. For any a∈ A and any ordering s1, . . . , s|S|of S:= {i∈[n]|ai>0}
with sj:= in({s1, . . . , sj})for all j∈[|S|], it is
|S|
X
j=1
χsj,asj(asj·sj)≤Hn·FTFL(a).
Proof. Roughly speaking, given a∈ A, the main idea of the proof is a ‘reduction’ to
the summability of a (binary demand) cost-sharing method ξthat we define according
to Pál and Tardos [106] for a (binary demand) FacilityLocation cost-sharing
problem (G, o, e)derived from (F, o, d)and an optimal solution for a.
Fix a∈ A and an ordering s1, . . . , s|S|of S:= {i∈[n]|ai>0}. Fix j∈[|S|]
and look at χsj,asj(asj·sj), computed for the original instance. For all f∈F, let
t(f)and Sfbe the corresponding values for growing the ghosts of set {s1, . . . , sj}.
In the original problem, let F∗be an optimal facility set for a, and F∗
sjbe the
facilities that sjis connected to in an optimal solution. It is F∗⊆ Fmaxi{ai}. In the
proof of Theorem 7.19, we have already shown that there exists gj∈F∗
sj, such that
χsj,asj(asj·sj)is at most t(gj)if i∈Sgj, or d(sj, gj)otherwise.
Let the new facility set be G:= {g1, . . . , g|S|}. For a each pair in {(sj, gj)}j∈[|S|],
let e(sj, gj) := d(sj, gj). Furthermore, for all j, j0such that gj=gj0, let e(sj, sj0) :=
d(sj, sj0). All other distances are defined to be sufficiently large, while ensuring that
eis a metric. For an illustration, see Example 7.22.
By construction of the new problem, it is χsj,asj(asj·sj)≤ξsj(sj)for all j∈[|S|],
where ξsj(sj)is computed on the new instance. Additionally, FL(S)≤FTFL(a). We
further use the fact that ξis Hn-SUM [115] in order to obtain
|S|
X
j=1
χsj,asj(asj·sj)≤|S|
X
j=1
ξsj(sj)≤Hn·FL(S)≤Hn·FTFL(a).
Example 7.22. Let a= (2,2,2,1) and look at S={1,2,3,4}for the upper network
below.
f11
1
14
5
2
2
f2
f4
1
1
1
1
14
2f3
2
3
114
f3
4
2
2
f4
1
2
2
3∞
The new problem defined in Lemma 7.21 is illustrated in the lower network, where
gray parts correspond to the unchanged distances. Particular values are:
-χ1,a1(a1·s1) = χ1,2(2 ·(1,0,0,0)) = 4,χ2,a2(a2·s2) = χ2,2(2 ·(1,1,0,0)) = 3,
-χ3,a3(a3·s3) = χ3,2(2 ·(1,1,1,0)) = 4,χ4,a4(a4·s4) = χ4,1(1 ·(1,1,1,1)) = 1.5,
-F∗={f1, f2, f3, f4},F∗
1={f1, f3}, F∗
2={f2, f3},,F∗
3={f3, f4}, F∗
4={f4},
-g1=g2=g3=f3, g4=f4.
102 7 Group-Strategyproof Mechanisms for General Demand
Proof of Theorem 7.20:
Fix V∈ R and let q:= q(V). Let a∈ A be a service vector with optimal social
cost. We have to upper bound the value:
γ:= SC CALG (q,V)
SC FTFL(a,V)=CALG(q) + Pn
i=1 PLi
`=qi+1 max{0, vi,`}
FTFL(a) + Pn
i=1 PLi
`=ai+1 max{0, vi,`}.
Let r∈ A with ri:= min{qi, ai}for all i∈[n]. Let R`:= {i∈[n]|ri≥`}and
Q`:= {i∈[n]|qi≥`}for all `∈[L]. Let r`:= in(R`)and q`:= in(Q`). Obviously,
R`⊆Q`, i.e., r`≤q`. We have that
Xfl(q) :=
n
X
i=1
ri
X
`=1
χfl
i,`(`·q`) +
n
X
i=1
qi
X
`=ri+1
χfl
i,`(`·q`)
≤
n
X
i=1
ri
X
`=1
χfl
i,`(`·r`) +
n
X
i=1
qi
X
`=ri+1
χfl
i,`(`·q`)
≤3L·FTFL(r) +
n
X
i=1
qi
X
`=ri+1
χfl
i,`(`·q`)
≤3L·FTFL(a) +
n
X
i=1
qi
X
`=ri+1
vi,` .
With CALG(q)≤Xfl (q), we get
γ≤3·L·FTFL(a) + Pn
i=1 Pqi
`=ri+1 vi,` +Pn
i=1 PLi
`=qi+1 max{0, vi,`}
FTFL(a) + Pn
i=1 PLi
`=ai+1 max{0, vi,`}
≤3·L+Pi:ai>qiPai
`=qi+1 max{0, vi,`}
FTFL(a).
Here, the last inequality follows because the fraction is at least 1, so that we get
another upper bound when subtracting Pn
i=1 PLi
`=ai+1 max{0, vi,`}in both the nu-
merator and the denominator.
We finally show that Pi:ai>qiPai
`=qi+1 max{0, vi,`} ≤ 3·L·Hn·FTFL(a). Consider
the players in S:= {i∈[n]|ai> qi}in the reverse order in which they are deleted
by Levelχfl within the Moulin mechanism in line 4. Let this order be s1, . . . , s|S|.
For j∈[|S|], let pj∈ {0,1}nbe the vector right before pj
sjis set to 0by the
Moulin mechanism. Let sj:= in({s1, . . . , sj}). By Assumption 2.2 and a pretty rough
estimate, we get
X
i:ai>qi
ai
X
`=qi+1
max{0, vi,`} ≤ L·X
i:ai>qi
max{0, vi,qi+1}.
For player sj∈Sat position j, let q0∈[qsj]nbe the vector of service levels already
determined by Levelχfl when line 4 is invoked and in this call of Moulinξwith
ξ= (χfl)q0,sjis rejected due to vsj,q0
sj+1 <(χfl)q0
sj(pj). It is q0
sj=qsj. By definition,
(χfl)q0
sj(pj) = χfl
sj,qsj+1(q0+pj).
Furthermore, by construction,
7.4 Applications to Fault Tolerant Facility Location 103
χfl
sj,qsj+1(q0+pj) = χfl
sj,qsj+1((qsj+ 1) ·pj).
Using cross-monotonicity and the fact that pj≥sjyields
L·X
i:ai>qi
max{0, vi,qi+1}< L ·|S|
X
j=1
χfl
sj,qsj+1((qsj+ 1) ·pj)
≤L·|S|
X
j=1
χfl
sj,qsj+1((qsj+ 1) ·sj)
We define tby ti:= qi+ 1 for i∈Sand 0otherwise. By Lemma 7.21,
L·|S|
X
j=1
χfl
sj,qsj+1((qsj+ 1) ·sj) = 3 ·L·|S|
X
j=1
χsj,qsj+1((qsj+ 1) ·sj)
≤3·L·Hn·FTFL(t)
≤3·L·Hn·FTFL(a).
7.4.4 Comparison to the Method of Mehta et al.
We restate an example from Mehta et al. [94], showing that the cost shares of their
acyclic mechanisms are in general not cross-monotonic. The main difference to χ
(and χfl, respectively) is that χi,`(a)is independent of connections computed in
iterations 1to `−1.
We shortly describe the marginal cost-sharing method χmused in the acyclic
mechanism introduced by Mehta et al. [94]. The mechanism itself is essentially Mech-
anism MoulinGDχfrom Algorithm 7.1, where line 2 is replaced by ‘while there exists
iwith bi,` < χi,`(q)for an `∈[qi]’.
The marginal cost shares χm
i,`(a)for all i∈A`are computed iteratively for
`= 1, . . . , L. In each iteration `, the cross-monotonic cost-sharing method of Pál
and Tardos [106] is invoked for all players with ai≥`. The instance is changed in
such a way that opening costs are set to 0for already opened facilities. In order to
ensure that each player iis connected to aidistinct facilities, the distance d(i, f)for
already existing connections between iand fis set to infinity. All other distances stay
the same. This destroys the metric property of dand, according to Mehta et al. [94],
complicates the analysis. We give the example from [94] which shows that this mar-
ginal cost-sharing method is not cross-monotonic. In contrast, we compute the cross-
monotonic cost shares introduced in this work for this example. The instance is given
in Figure 7.4. We first consider the method from Mehta et al. [94]. For a= (2,2,2,2),
the cost shares for the first level are χm
∗,1((2,2,2,2)) = (3ε, 1,1+2ε, 3), and for the
second level, χm
∗,2((2,2,2,2)) = (2 + ε, 1 + 2ε, 1 + 2ε, 5+5ε). For a= (0,2,2,2), we
get χm
∗,1((0,2,2,2)) = (0,1+2ε, 1+2ε, 3), and χm
∗,2((0,2,2,2)) = (0,1+3ε, 1+3ε, 3).
Cross-monotonicity is violated by χm
4,2((2,2,2,2)) = 5 + 5ε > 3 = χm
4,2(0,2,2,2).
Essentially, this happens due to that fact that for (2,2,2,2), facilities 1and 3are
opened in the first iteration, player 4is connected to 3, and c(4,3) is set to infinity.
Thus, in the second iteration, his ghost has to grow longer to connect to a full fa-
cility, namely, facility 2. However, for (0,2,2,2), only facility 2is opened in the first
iteration due to the opening rule (To ensure that there is no tie between opening f1
104 7 Group-Strategyproof Mechanisms for General Demand
Fig. 7.4. Instance for which the method in [94] is not cross-monotonic
f1f2f3
0
12
3
4
11 + e
1+2e
1+3e
3
e
e
3e
and f2in the first iteration, we have slightly changed the original example, letting
of1be 3εinstead of 2ε). Therefore, in the first and second iteration, d(4,3) is 3.
Now, let us look at our method χ. The cost shares only differ at two entries: it
is χ3,2(2,2,2,2) = 1 + 4εand, interestingly, χ4,2(0,2,2,2) = 5 + 5ε. We state the
events for both a= (2,2,2,2) and a= (0,2,2,2) in Figure 7.5. Note that for both
vectors, is is sufficient to consider growing the ghosts of players in A2, since A1=A2
and the cost shares for level 1can be deduced from this process as well.
Fig. 7.5. Events for computing χ(a)for a= (2,2,2,2) (left) and for a= (0,2,2,2) (right)
A2={1,2,3,4}A2={2,3,4}
t Event
0 1 ◦f1
3εf1,1•f1
1 2 •f1
1 + ε2◦f2
1 + 2εf2,2•f2,3•f2
1 + 3ε3◦f3
1 + 4εf3,3•f3
t Event
2 + ε1•f2
3 4 •f3
3 + 3ε3•f1
3 + 6ε2•f3
4 + 6ε1•f3
5 + 5ε4•f2
t Event
1 2 ◦f1
1 + ε2◦f2
1 + 2εf2,2•f2,3•f2
1 + 3εf1,2•f1,3◦f3
1 + 4εf3,3•f3
3 4 •f3
3 + 3ε3•f1
3 + 6ε2•f3
5 + 5ε4•f2
7.5 Applications to Generalized Steiner Forest
Theorem 7.23 is the main theorem of this section:
Theorem 7.23. For each GenSteinerForest cost-sharing problem there is a
marginal cost-sharing method χgs and an approximation algorithm ALG such that
Levelχgs is GSP, (2·HL)-BB, and O(log2n·log L)-EFF for CALG. Furthermore, both
Levelχgs and ALG are computable in time polynomial in |L|,|V|,|E|, and n.
Guideline of Proof: Fix a GenSteinerForest cost-sharing problem (G, w,s,t)(for
the remaining Section 7.5). In order to proof Theorem 7.23, we define a valid marginal
7.5 Applications to Generalized Steiner Forest 105
cost-sharing method χgs in Section 7.5.1 and show that for every a∈ A and each
`∈[L],χgs
∗,`(a)is computable in time polynomial in |V|,|E|, and n. Section 7.5.2
gives a polynomial-time computable approximation algorithm ALG. Section 7.5.3
then shows (2 ·HL)-BB and O(log2n·log L)-EFF for CALG. Finally, Theorem 7.23
follows directly from Theorem 7.11 and Lemma 7.12.
7.5.1 The Marginal Cost-Sharing Method
We utilize the binary demand cost-sharing method ξkfor SteinerForest cost-
sharing problems by Könemann et al. [82], which is known to be 2-BB [82] and
O(log2n)-SUM [19]. We define χgs simply by
χgs
∗,`(a):=ξk(a`)for all allocations a∈ A .
Lemma 7.24. χgs is valid.
Proof. Let χ:= χgs and ξ:= ξk.
Level restriction follows from χi,`(a) = χi,`(`·a`)for all allocations a∈ A, all
players i∈[n], and all `∈[Li]. Cross-monotonicity is met, since for all allocations
a∈ A, all players i∈[n]and j∈[n]\{i}with aj< Lj, and all levels `∈[Li], it
follows by the cross-monotonicity of ξthat
χi,`(a) = ξi(a`)≥ξi((a+ej)`) = χi,`(a+ej).
To show that χis non-decreasing, fix a∈ A and let `:= maxi{ai}. For all iwith
ai=` < Li,
χi,`(a) = ξi(a`)≤ξi((a+X
j∈[n]:
aj=`<Lj
ej)`+1)≤χi,`+1(a+X
j∈[n]:
aj=`<Lj
ej).
The binary demand cost-sharing method ξkcan essentially be computed by an
approximation algorithm of Agrawal et al. [1] for SteinerForest with only a small
modification which is crucial for cross-monotonicity (confer [82]). It goes beyond the
scope of this thesis to discuss this computation in detail. We refer to [1] and [82] to
verify polynomial time computability of ξk(S)for all S⊆[n]in time polynomial in
|V|,|E|and n.
7.5.2 The Approximate Solution
During the computation of ξk(a`)for allocation a∈ A and each `∈[maxiai], a
Steiner forest for the set induced by a`is computed in time polynomial in |V|,|E|,
and n(essentially by the AKR algorithm of Agrawal, Klein, and Ravi; confer [1]).
We let the cost of the Steiner forest for a`be CAKR(a`). We construct a solution
for athat simply consists of the union of the Steiner forests from each iteration,
where multiple edges count as copies. This straightforward solution construction for
GenSteinerForest was also suggested by Goemans and Bertsimas [55]. We denote
the induced cost by CALG, where CALG(a) = Pmaxiai
`=1 CAKR(a`)for all allocations
a∈ A.
106 7 Group-Strategyproof Mechanisms for General Demand
7.5.3 Budget-Balance and Efficiency
Theorem 7.25. χgs is (2 ·HL)-BB for CALG.
Proof. Let χ:= χgs and ξ:= ξk. Fix a∈ A and let X(a):=Pn
i=1 Pai
`=1 χi,`(a). We
let A`:= {i∈[n]|ai≥`}for all `∈[maxiai]. It is
X(a) =
maxiai
X
`=1 X
i∈A`
ξi(a`)≥
maxiai
X
`=1
CAKR(a`) = CALG(a),
where the inequality holds as ξkis 2-BB for CAKR [82]. We again utilize this property
to show that
X(a) =
maxiai
X
`=1 X
i∈A`
ξi(a`)≤
maxiai
X
`=1
2·SF(a`)
= 2 ·
maxiai
X
`=1
1
`·GSF(`·a`)≤2·HL·GSF(a).
Theorem 7.26. Levelχgs is O(log2n·log L)-EFF for CALG.
Proof. Let χ:= χgs and ξ:= ξk. Fix V∈ R and let q:= q(V). Let a∈ A be a
service vector with optimal social cost. We show that
X
i:ai>qi
ai
X
`=qi+1
max{0, vi,`} ≤ O(log2n+ log L)·GSF(a).
The rest of the proof is along the lines of the proof of Theorem 7.20, and as a result,
we get O(log2n·log L)-EFF.
We consider the players in S:= {i∈[n]|ai> qi}in the reverse order they
are deleted within Levelχfrom vector pin the Moulin mechanism in line 4. Let this
order be s1, . . . , s|S|. For j∈[|S|], let pj∈ {0,1}nbe the vector in which the entry
pj
sjis set to 0by the Moulin mechanism. Furthermore, let S`:= {i∈S|ai≥`},
and s`:= in(S`). Finally, let s`,j ∈ {0,1}nbe such that s`,j
i:= 1 ⇔(s`
i= 1 and
pj
i= 1). Note that s`,j indicates the first jelements of S`(according to the order
s1, . . . , s|S|).
By Assumption 2.2, Lemma 7.5, and definition of the vectors pj,
X
i:ai>qi
ai
X
`=qi+1
max{0, vi,`}=X
sj∈S
asj
X
`=qsj+1
max{0, vsj,`}
≤X
sj∈S
asj
X
`=qsj+1
vsj,qsj+1 ≤
L
X
`=1 X
sj∈S`
ξsj(pj).
Furthermore, by cross-monotonicity and O(log2n)-SUM of ξ(for function SF),
L
X
`=1 X
sj∈S`
ξsj(pj)≤
L
X
`=1 X
sj∈S`
ξsj(s`,j)≤O(log2n)·
L
X
`=1
SF(S`).
Finally,
L
X
`=1
SF(S`)≤
L
X
`=1
1
`·GSF(a)≤HL·GSF(a).
7.6 Conclusion and Open Problems 107
7.6 Conclusion and Open Problems
Among the few works for general demand cost sharing, we regard our work to be a
substantial contribution to the development of GSP mechanisms for general demand.
•Central open questions are whether the provided budget-balance and efficiency
approximations are tight under the validity requirement or even for the GSP
requirement.
•In addition, it remains an open problem whether validity of marginal cost-shares
is not only sufficient, but also necessary for GSP mechanisms.
•Observe, that we did not fully exploit the potential of Levelχ. For both applica-
tions, the cost shares of level lonly depend on the players that receive at least
level l, while validity allows to make them dependent on the levels of the other
players as well. It is an interesting question if this degree of freedom may lead to
better approximations.
A
Budget-Balance and Efficiency Bounds for Moulin
Mechanisms and Acyclic Mechanisms
Existing Moulin mechanisms are summarized in Table A.1. Budget-balance and effi-
ciency approximations together in one line imply that there are Moulin mechanisms
that fulfill these approximations simultaneously.
Table A.1. Best known budget-balance and social cost efficiency approximations of Moulin
mechanisms (n.k. for not known)
Problem from BB EFF confer
general1[114] β β +αThm. 4.4, p. 33
submodular costs2[114] 1 Hn
SpanningTree [76, 81] 1 n.k.
SteinerTree3[76, 81, 114] 2 O(log2n)
SteinerForest [19, 82] 2 O(log2n)
FacilityLocation [106, 115] 3 O(log n)
PriceCollectingSteinerForest [66] 3 O(log2n)
ConnectedFacilityLocation4[86] 30 n.k.
VertexCover [73] 2·√nn.k.
EdgeCover [73] 2 n.k.
SingleSourceRentOrBuy4[106] 15 n.k.
[67, 86, 115] 4·(1 + ε)O(log2n)
MulticommodityRentOrBuy [115] O(1) log2n
TravelingSalesman3[76, 81] 2 n.k.
(Q||Cmax)5[14] 2·d O(d·log n)Thm. 4.14, p. 38
(Q|pi=1|Cmax)6[14] 2m
m+1 O(log n)Thm. 4.12, p. 36
(P||Cmax)6[10] 2m
m+1 O(n)
[16] 2−1
mO(log n)
1α, β: smallest numbers such that ξ(·)is α-SUM and β-BB
2via the Shapley value [116]
32-BB, as a minimum spanning tree is a 2-approximation
4randomized, sharing of expected cost
5d: number of different processing times
6m: number of machines
110 A Budget-Balance and Efficiency Bounds for Moulin Mechanisms and Acyclic Mechanisms
Table A.2 gives lower bounds on the approximate budget-balance and the approx-
imate efficiency of Moulin mechanisms.
Table A.2. Budget-balance and social cost efficiency lower bounds of Moulin mechanisms
Problem from BB EFF confer
general1[114] max{α, β}max{α, β}Thm. 4.4, p. 33
constant costs [114] 1 HnLemma 4.5, p. 35
SteinerTree [76, 83] 2
[115] Ω(log2n)
SteinerForest22Ω(log2n)
SingleSourceRentOrBuy22Ω(log2n)
MulticommodityRentOrBuy22Ω(log2n)
FacilityLocation [73] 3
[115] Ω(log n)
VertexCover6[73] Ω(n1/3)Ω(n1/3)
SetCover6[73] Ω(n)Ω(n)
EdgeCover7[73] 2−ε Hn
(1||
P
Ci)3,6 [16] n
2
n
2
(Q||Cmax)4,6,7 [10] d d Thm. 4.9, p. 36
Hn
(P|pi=1|Cmax)5,7 [10] 2m
m+1 HnThm. 4.7, p. 35
1α, β: smallest numbers such that ξ(·)is α-SUM and β-BB
2follows from bounds for Steiner tree
3already for a single machine
4d: number of different processing times
5m: number of machines
6efficiency entry due to the first line
7efficiency entry due to the second line
Table A.3 summarizes known approximation results for acyclic mechanisms com-
pared to the performance of Moulin mechansims.
Table A.3. Approximation results from [31, 94] for acyclic mechanisms com-
pared to lower bounds of Moulin mechanisms (confer Table A.2)
Moulin Lower Bounds Acyclic Upper Bounds
Problem BB EFF BB EFF
VertexCover Ω(n1/3)Ω(n1/3) 2 O(log n)
SetCover Ω(n)Ω(n)O(log n)O(log n)
FacilityLocation 3Ω(log n) 1.61 O(log n)
SteinerTree 2Ω(log2n)2O(log2n)
B
Incremental and Groves Mechanisms
B.1 Incremental (Sequential Stand Alone) Mechanisms
Incremental (or sequential stand alone) mechanisms for binary demand cost sharing
process the players in a fixed order and decide to serve a player if he can pay for
the pivotal cost of adding him to the set of already served players. His cost share
is exactly this pivotal cost. Let Cbe a cost function and π: [n]→[n]specify
an ordering such that π([n]) = [n],π(i)is the position of player iand π−1(i)is
the player at position i. The corresponding mechanism MechIncC,π is computed by
Algorithm B.1:
Algorithm B.1 (computing MechIncC,π(b) = (Q(b), x(b))).
1: Q:= ∅;x:= 0
2: for i:= π−1(1), . . . , π−1(n)do
3: if bi≥C(Q∪{i})−C(Q)then
4: xi:= C(Q∪{i})−C(Q);Q:= Q∪{i}
5: return (Q, x)
It is straightforward to see that for any cost function Cand any permutation π
MechIncC,π meets SP, 1-BB, VP, NPT, and strict CS. In addition, MechIncC,π is
GSP if the cost function is submodular: the cost share of a player computed for the
true valuations can only decrease if more players prior to him in the order bid such
as to receive the service. Especially for the first such player this results in a negative
utility. Note that it can be easily observed that the cost-shares by MechIncC,π are
cross-monotonic and due to Corollary C.4, MechIncC,π is welfare equivalent to a
Moulin mechanism.
If Cis supermodular, replacing ‘≥0in line 3 by ‘>0yields a GSP mechanism:
Here, the cost share of a player computed for true valuations can only decrease
if preceding players bid in order to be rejected. According to the replacement, in
particular the first such player had a positive utility when bidding truthfully.
In general, however, GSP cannot be guaranteed, as Example B.1 illustrates. Fur-
thermore, Example B.2 yields unsatisfying performance with respect to efficiency.
Example B.1. Consider C: 2[3] →Nwith
C(S):=
0if S=∅
2if |S|= 1
3if |S|= 2
5if |S|= 3 .
112 B Incremental and Groves Mechanisms
Cis neither sub- nor supermodular. Without loss of generality, look at πwith π(i) = i
for all i∈[n]. Let v:= (2,2,2). For MechIncC,π = (Q, x)as in Algorithm B.1, it
is Q(v) = [3] and x(v) = (2,1,2). For b:= (0,2,2) we have Q(b) = {2,3}and
x(b) = (0,2,1). Thus, K={1,3}can successfully form a coalition as u1(v, v1) =
u1(b, v1) = u3(v, v3)=0and u3(b, v3)=1. For mechanism MechInc0
C,π = (Q0, x0)
obtained by replacing ‘≥0with ‘>0, it is Q0(v) = ∅. For b:= (3,2,2) however, we
have Q(b) = {1,2}and x(b) = (2,1,0). Here, K={1,2},u1(b, v1) = u1(v, v2) =
u2(v, v2) = 0 and u2(b, v2) = 1.
Example B.2. Let C: 2[n]→Nwith C(S) := 1 for all S⊆[n]and consider πwith
π(i) = ifor all i∈[n]without loss of generality. Furthermore, consider v∈Rnwith
vi:= 1 −εfor all i∈[n]. For MechIncC,π = (Q, x)as in Algorithm B.1, the social
welfare of the computed solution Q(v) = ∅is SW C(∅,v) = 0, while the optimal
social welfare is SW C([n],v) = n·(1 −ε)−1. For the social cost efficiency measure,
we have SC C(∅,v) = n·(1 −ε), while the optimal social cost is SC C([n],v)=1.
The same holds for replacing ‘≥0with ‘>0in Algorithm B.1.
In the following, we show how to adapt incremental mechanisms to general de-
mand cost sharing in Algorithm B.2. For given maximum service levels L1, . . . , Ln,
we let sbe a sequence of players such that each player iappears exactly Litimes in
the sequence.
Algorithm B.2 (computing MechIncGDC,s(B) := (q(B), x(B))).
1: x:= 0;q:= 0;i:= first(s)
2: while i6=null do
3: if bi,qi+1 < C(q+ei)−C(q)then
4: delete ifrom the whole sequence s
5: else
6: qi:= qi+ 1;xi:= xi+C(q+ei)−C(q)
7: delete ifrom sonly at current position
8: i:= first(s)
9: return (q,x)
The sequential stand alone mechanisms are incremental mechanisms which only
allow sequences in which all entries ifor a player ioccur consecutively. In particular,
MechIncC,π is a sequential stand alone mechanism.
Moulin [97] investigated incremental mechanisms for general demand by looking
at generalizations of sub- and supermodularity. Let
δijC(q):=C(q+ei+ej)−C(q+ei)−C(q+ej) + C(q),
defined only if qi≤Li−1,qj≤Lj−1if i6=j, and qi≤Li−2if i=j.
Under assumption of diminishing marginal returns (confer Assumption 2.2),
Moulin [97] showed that incremental cost-sharing mechanisms are GSP for two cases.
First, if costs are submodular (δijC(q)<0for all i, j, qwith i6=j) and marginal
costs are decreasing (δiiC(q)<0for all i, q); second, if costs are supermodular
(δijC(q)>0for all i, j, qwith i6=j) and marginal costs are increasing (δiiC(q)>0
for all i, q).
If costs are submodular and marginal costs are decreasing, he showed a negative
result already for 2 players with Li≥2for i∈ {1,2}. In this case, only the two
possible sequential stand alone mechanisms yield 1-BB and GSP mechanisms. If
B.1 Incremental (Sequential Stand Alone) Mechanisms 113
costs are supermodular and marginal cost are increasing, he showed that essentially
all GSP and 1-BB mechanisms satisfying a certain continuity property (see [97]) are
incremental mechanisms.
Example B.3 shows that the social cost efficiency of incremental mechanisms is
in general no better than Pi∈[n]Li.
Example B.3. Let C:A → Nwith C(q)=1for all q∈ A and, without loss
of generality, πwith π(i) = ifor all i∈[n]. Furthermore, consider V∈ R with
vi,l := 1 −εfor all i∈[n]and all l∈[Li]. The social cost of the computed solution
q(V) = 0is Pi∈[n](1−ε)·Li, while the optimal social cost is 1, obtained by allocation
(L1, . . . , Ln).
Incremental Mechanisms are not GSP for our Problems
In general, there are no GSP incremental mechanisms for (Q||Cmax)cost-sharing
problems, as already for a (P2|pi=1|Cmax)problem with three players, we get sym-
metric costs Cwith c(1) = c(2) = 1 and c(3) = 2 (which are neither sub- nor
supermodular). Now let v:= (1,2,1). For mechanism MechInc = (Q, x)from Al-
gorithm B.1 with input b:= (1,−1,1), it is Q(v) = [3], x(v) = (1,0,1), Q(b) =
{1,3}, x(b) = (1,0,0). Thus, K={2,3}can successfully form a coalition. For mech-
anism MechInc0= (Q0, x0)obtained by replacing ‘≥0with ‘>0and b0:= (2,2,1), we
get that Q0(v) = (0,1,1), x0(v) = (0,1,0), Q0(b0) = (1,1,−1) and x0(b0) = (1,0,0).
Here, a successful coalition is {1,2}.
In addition, the FaultTolerantFL and GenSteinerForest instances in Fig-
ure B.1 induce the costs from Example B.1.
Fig. B.1. FaultTolerantFL and GenSteinerForest instances with costs from Example B.1
1
1
f1
1
f2
1
f3
1
3
1
2
1
1
1
1
1
s3
s2
s1
t1=t2=t3
1
1
1
1
1
1
1
1
1
FaultTolerantFL instance GenSteinerForest instance
Binary Demand Sequential Stand Alone Mechanisms are CGSP
We show acyclicity; CGSP then follows from Theorem 6.14.
Lemma B.4. For every cost function Cand every ordering π,MechIncC,π is acyclic.
Proof. Define ξ: 2[n]→Rn
≥0and τ: 2[n]→Rn
≥0as follows. For S⊆[n]and i∈[n],
let ξi(S) := C({j∈S|π(j)≤π(i)})−C({j∈S|π(j)< π(i)})if i∈Sand 0
otherwise. Furthermore, τi(S):=π(i).
114 B Incremental and Groves Mechanisms
B.2 Groves Mechanisms
Applying the celebrated family of Groves mechanisms [25, 63, 131] is essentially the
only way to achieve social welfare efficiency and strategyproofness at the same time.
As Groves mechanisms are defined for a rather general scenario, we first discuss them
in their original setting and afterwards apply them to our cost-sharing model for bi-
nary demand.
In the original setting, a choice dependent on players’ preferences has to be made
from some set of social alternatives. An alternative consists of an element k∈ K,
the set of project choices, and a payment ti∈Rfor each player. A negative tiis
interpreted as a bonus given to player i.
The preference of player i∈[n]is modeled by i’s true type θ∗
ithat he privately
observes from the set of his possible types Θiprior to the choice. Only the true
types θ∗
iare private information, everything else is assumed to be common knowledge
among the players. We let Θ:= Θ1×. . .×Θnbe the space of all possible type profiles
θ:= (θ1, . . . , θn). Depending on his type, player iappraises a project choice by a
valuation function νi:K×Θi→R, and a social alternative (k, t)by a quasi-linear
utility function ui:K×Rn×Θi→Rwith ui((k, t), θi) = νi(k, θi)−ti.
In order to choose an alternative (k, t)we consider a mechanism Γ:= (k, t), where
k:Θ→ K and t:Θ→Rn.
According to Definition 2.5, Γis strategyproof if for every profile vector θ∈Θ
there is no player iwith true valuation θ∗
i∈Θisuch that
ui((k(θi,θ−i), t(θi,θ−i)), θ∗
i)> ui(k((θ∗
i,θ), t(θ∗
i,θ), θ∗
i).
Furthermore, Γis called efficient if, assuming truthful bids, it always selects a project
choice that maximizes Pn
i=1 νi(k, θ∗
i)over all k∈ K. Note that this definition of
efficiency seems to deviate from our notion of efficiency from Definitions 2.13 and
2.14. However, we will fit the cost-sharing scenario into this model in such a way
that these definitions are equivalent.
Definition B.5. A mechanism M= (k, t)is called a Groves mechanism, if for all
inputs θ∈Θand all k∈ K it holds that
n
X
i=1
νi(k(θ), θi)≥
n
X
i=1
νi(k, θi)and (B.1)
ti(θ) = hi(θ−i)−X
j∈[n]\{i}
νj(k(θ), θj),(B.2)
where hiis an arbitrary function of θ−i.
By Equation (B.1), Groves mechanisms are efficient. Moreover, it is a simple
observation that Groves mechanisms are strategyproof:
Lemma B.6. Groves mechanisms are strategyproof.
Proof. Let Γ= (k, t)be a Groves mechanism and assume that Γis not strategyproof.
Then there exists a type profile θ∈Θ, a player i∈[n], and a true type θ∗
i∈Θisuch
that
B.2 Groves Mechanisms 115
ui((k(θi,θ−i), t(θi,θ−i)), θ∗
i)> ui((k(θ∗
i,θ−i), t(θ∗
i,θ−i)), θ∗
i).
Substituting the payments according to (B.2) an eliminating function h−i(θ−i)on
both sides leads to
νi(k(θi,θ−i), θ∗
i) + X
j∈[n]\{i}
νj(k(θi,θ−i), θ∗
j)
> νi(k(θ∗
i,θ−i), θ∗
i) + X
j∈[n]\{i}
νj(k(θ∗
i,θ−i), θ∗
j),
contradicting Equation (B.1).
Independently, Clarke [25] and Vickrey [131] proposed members of the family of
Groves mechanisms for which
hi(θ−i) = X
j∈[n]\{i}
νj(k−i(θ−i), θj),
where for all θ−i∈Θ−iand all k∈ K,k−isatisfies
X
j∈[n]\{i}
νj(k−i(θ−i), θj)≥X
j∈[n]\{i}
νj(k, θj).
Player i’s payment reflects the pivotal change of the joint valuation the others
have for k(θ)compared to the joint valuation the others have for k−i(θ−i). Hence,
next to Vickrey-Clarke-Groves mechanisms, these mechanisms are also called pivotal
mechanisms.
A result by Green and Laffont [60] states that essentially all efficient and strat-
egyproof mechanisms have to compute the payments as in (B.2), under the premise
that the set of possible types for each player is sufficiently rich. In other words, Groves
mechanisms characterize the only mechanisms that are efficient and strategyproof.
Unfortunately, the famous Green and Laffont impossibility theorem [61] rules out
the existence of Groves mechanisms that simultaneously meet the budget-balance
condition of Pn
i=1 ti(θ) = 0 for all θ∈Θ.
Groves (Marginal Cost) Mechanisms for Cost Sharing
In order to relate Groves mechanisms to the cost-sharing scenario, observe that the
set of possible types Θifor player iis given by R, and that the set of project choices
Kis given by 2[n]. The true type θ∗
iof a player iis simply his true valuation vi, the
actual input θis the bid vector b. For S⊆[n], we let νi(S, bi) := biif i∈Sand 0
otherwise. In order to relate the efficiency condition implied by (B.1) to our notion
of efficiency in Definitions 2.13 and 2.14, we have to introduce the service provider
as an additional player, referred to as ‘player 0’. For S⊆[n], we define the valuation
function of the provider to be ν0(S, b0) := −C(S)(independent of his dummy bid
b0). Then, for any vector vof true valuations,
max
S⊆[n] n
X
i=0
νi(S, vi)!= max
S⊆[n] X
i∈S
vi−C(S)!
= max
S⊆{i∈[n]|vi≥0} X
i∈S
vi−C(S)!,
116 B Incremental and Groves Mechanisms
implying that efficiency in this model is equivalent to efficiency (1-EFF) in the cost-
sharing model. According to (B.1), the function Qbecomes
Q(b) := lexicographic largest set in
S⊆[n]|SW C(S, b)≥SW C(S0,b)for all S0⊆[n]
Naturally, we would have to define the payments x(b)in such a way that they
meet (B.2) to account for SP and ensure that x0(b) = −C(Q(b)), as this is the
bonus the provider has to receive in order to cover his cost. Furthermore, we have
to ensure that the constructed Groves mechanism meets VP, NPT, and strict CS.
However, even though we can achieve all of these requirements, 1-BB requires
x0(b) + Pn
i=1 xi(b) = Pn
i=1 xi(b)−C(Q(b)) = 0, which by the Green and Laf-
font impossibility theorem is infeasible. Still, for the sake of illustration, we define
adequate corresponding payments and give examples with poor budget-balance.
We define the payment of the provider by letting
h−0(b−0) := X
j∈[n]0\{0}
νj(Q(b), bj)−C(Q(b)) .
Note that Q’s choice is independent of the dummy bid b0, thus Q(b) = Q(b−0).
We define the payments for players 1to naccording to the pivotal mechanism.
Let Q−ibe the function corresponding to k−i, i.e.,
Q−i(b−i) := lexicographic largest set in
S⊆[n]\{i} | SW C(S, b)≥SW C(S0,b)for all S0⊆[n]\{i}.
Furthermore, let S∗
−i:= Q−i(b−i), and S∗:= Q(b). Then define
xi(b) := X
j∈[n]0\{i}
νj(Q−i(b−i), bj)−X
j∈[n]0\{i}
νj(Q(b), bj)
=X
j∈S∗
−i
bj−C(S∗
−i)−X
j∈S∗\{i}
bj+C(S∗)
≤X
j∈S∗
bj−C(S∗)−X
j∈S∗\{i}
bj+C(S∗)
=bi.
Obviously, NPT holds due to xi(b)≥0for all i∈[n]. VP follows from the fact that
for all i∈[n],xi(b)≤biand xi(b)=0if i /∈S∗(in this case, S∗
−i=S∗). For any
bid that is strictly larger than maxS⊆[n]\{i}{C(S∪{i})−C(S)},i∈[n]will be in
the set maximizing social welfare. Furthermore, a negative bid will always result in
not being in the set maximizing social welfare. Thus, this Groves mechanism meets
strict CS. Moulin and Shenker [99] refer to this specific Groves mechanism as the
marginal cost mechanism.
Example B.7 shows that in general the provider cannot hope for a reasonable cost
recovery:
Example B.7. For M > 0, define C(S) := M· |S|for all S([n]and C([n]) :=
M·(n−1). Let the true valuations be vi:= Mfor all i∈[n]. Social welfare efficiency
requires to serve all players at cost M·(n−1), as for all S([n],SW C(S, v)=0
and SW C([n],v) = M. However, using the pivotal mechanism, every player pays 0.
B.2 Groves Mechanisms 117
On the other hand, Example B.8 shows that in general the players are not guaranteed
to not being overcharged:
Example B.8. For a sufficiently large M > 0, let C(S) := εfor all S([n]and
C([n]) := M. Furthermore, let the true valuations be vi:= M+εfor all i∈[n]. It
holds that for all S([n]that SW C(S, v) = (n−1) ·(M+ε)−ε < (n−1)·M+n·ε
and SW C([n],v) = (n−1)·M+n·ε. The set [n]maximizes social welfare. However,
the payments computed by the pivotal mechanism are M−εfor each player.
Besides not being able to guarantee reasonable budget-balance approximations,
Groves mechanisms are vulnerable to collusion. For Groves mechanisms, Green and
Laffont [62] considered a model of coalition forming in which a coalition forms if
players can strictly improve the sum of their utilities (and may then distribute their
surplus via side-payments). They show that averting this kind of coalition forming
is impossible for Groves mechanisms. On the positive side, they can show that for a
fixed coalition the expected gain to cheating compared to telling the truth decreases
with the number of players in the population. Example B.9 shows that even WGSP
cannot be guaranteed by Groves mechanisms in general:
Example B.9. Let n= 2 and for all ∅ 6=S⊆[2], let C(S) := 3. In addition,
v:= (2,2). Maximizing social welfare efficiency requires to serve both players. The
payments of the pivotal mechanisms are 1for each player. However, for b:= (3,3),
both players are served for a cost-share of 0.
Once we restrict attention to submodular costs, Moulin and Shenker [99] showed
that marginal cost mechanisms are essentially the only NPT, VP, and strict CS
mechanism from the class of Groves mechanisms. Furthermore, they showed that
submodularity ensures that the overall cost share never exceeds the service cost, i.e.,
the scenario from Example B.8 does not occur. In addition, Moulin and Shenker
[99] discussed that even for submodular costs, Groves mechanisms are in general not
group-strategyproof.
Presumably due to their bad performance with respect to budget-balance, there
are only few works that consider Groves mechanisms (and especially the marginal
cost mechanism) within cost-sharing scenarios (see, e.g., [4, 36, 37, 99]). For instance,
marginal cost mechanisms have been considered for Multicast cost-sharing prob-
lems with fixed transmission trees inducing submodular costs. Feigenbaum et al. [37]
investigated the distributed computation of the marginal cost mechanism and showed
that it can be computed by sending 2 messages per link in the multicast tree. Con-
trary they showed that computing the Shapley value for these problems requires a
quadratic total number of messages.
C
Characterization of GSP Mechanisms for Submodular
Costs
We have discussed at the beginning of Chapter 4 that there always exists 1-BB
and cross-monotonic cost-sharing methods (and thus 1-BB and GSP (Moulin) mech-
anisms) if cost are submodular. In this chapter, we introduce Theorem C.3 from
Moulin [97] that completes the characterization of 1-BB and GSP under submod-
ular costs, showing that every 1-BB and GSP mechanism for submodular costs is
equivalent to a Moulin mechanism.
For the proof of Theorem C.3, we need the notion of strong sets given in Defini-
tion C.2. We also remind the reader of the result by Moulin [97], showing that for a
GSP mechanism, cost-shares are uniquely defined by the set of served agents:
Theorem C.1. [97] Let M= (Q, x)be a GSP mechanism. Then, for any two vectors
b,b0∈Rnwith Q(b) = Q(b0), it holds that x(b) = x(b0).
Proof. Let b,b0∈Rnwith Q(b) = Q(b0). First consider vector b. Fix an arbitrary
S⊆[n]and define das:
di:=
b+
ifor all i∈S∩Q(b)
−1for all i∈S\Q(b)
bifor all i /∈S
Moulin shows that it holds that
for all i∈S:i∈Q(b)⇔i∈Q(d)and xi(b) = xi(d)(C.1)
for all i /∈S:ui(b, bi) = ui(d, bi).(C.2)
Then, considering b0instead of byields the same vector dand (C.1) – (C.2) hold
with breplaced by b0. For S:= [n], we get that xi(b) = xi(d) = xi(b0)for all i∈[n],
thus x(b) = x(b0).
We omit the proof of (C.1) and (C.2) as we have shown a more general result in
Theorem 6.4 in Chapter 6, restricting GSP to CGSP and strict CS to CS.
Clearly, every GSP mechanism M= (Q, x)induces a unique cost-sharing method ξ,
by setting ξ(S):=x(b)where bi<0if i /∈Sand bi=b+
iif i∈S.
Definition C.2. A set S⊆[n]is a strong set for a cost-sharing method ξand a
vector b∈Rnif there are no sets K, T ⊆[n]such that
for all i /∈K:i∈T⇔i∈S(C.3)
for all i∈K:bi·ini(T)−ξi(T)≥bi·ini(S)−ξi(S),(C.4)
with at least one strict inequality.
120 C Characterization of GSP Mechanisms for Submodular Costs
Theorem C.3. [97] For any GSP mechanism Mthat is 1-BB with respect to sub-
modular costs there exists a cross-monotonic cost-sharing method ξsuch that Moulinξ
is welfare equivalent to M.
Proof. Let M= (Q, x). Due to Theorem C.1 we may define a cost-sharing method
ξby setting ξ(S) := x(b)for all S⊆[n], where bi<0if i /∈Sand bi=b+
iif i∈S.
In particular, x(b) = ξ(Q(b)) for all b∈Rn.
1. First, we show that for every b∈Rn,Q(b)is a strong set for ξand b.
Fix b∈Rnand assume that Q(b)is not strong. Thus, there exist subsets K, T ⊆
[n]meeting
for all i /∈K:i∈T⇔i∈Q(b)(C.5)
for all i∈K:bi·ini(T)−ξi(T)≥bi·ini(Q(b)) −ξi(Q(b)) ,(C.6)
with at least one strict inequality.
Let d:= ((−1)−T,b+
T). By (C.1), (C.2) of Theorem C.1 (where S= [n]\K)
and (C.5) it holds that
for all i∈[n] : ui(b, bi) = ui((d−K,bK), bi).(C.7)
By (C.6) and (C.7), for all i∈K:
ui(d, bi) = bi·ini(Q(d)) −ξi(Q(d))
=bi·ini(T)−ξi(T)
≥bi·ini(Q(b)) −ξi(Q(b))
=ui(b, bi)
=ui((d−K,bK), bi).
with at least one strict inequality, contradicting GSP for true valuations bifor
all i∈K.
2. Second, we show that for every bid vector b∈Rnand any two strong sets
S, S0⊆[n]for ξand bit holds that
for all i∈[n] : bi·ini(S)−ξi(S) = bi·ini(S0)−ξi(S0).(C.8)
We show (C.8) by induction on κ(b):=|{i∈[n]|bi=b+
ior bi=−1}|.
•Consider b∈Rnwith κ(b) = n. By (1.),Q(b)is a strong set for ξand b. We
show now that Q(b)is the unique strong set for ξand b.
Assume that S6=Q(b)is another strong set.
– If there exists i∈Swith i /∈Q(b), it is bi=−1<0. Let K:= {i}and
T:= S\{i}. Then bi·ini(T)−ξi(T) = 0 > bi·ini(S)−ξi(S)contradicts
Sbeing strong.
– If there exists i∈Q(b)with i /∈S,bi=b+
i. Define sets K:= {i}and
T:= S∪{i}. Now, bi·ini(T)−ξi(T)>0 = bi·ini(S)−ξi(S)yields again a
contradiction to Sbeing strong. (Note to ensure that bi·ini(T)−ξi(T)>0
we can simply assume that b+
iis large enough.)
•Induction hypothesis: (C.8) holds for all b∈Rnwith κ(b)> k.
C Characterization of GSP Mechanisms for Submodular Costs 121
•Now choose bwith κ(b) = k. Without loss of generality, assume bi∈ {−1, b+
i}
for all i∈[k]. We know by (1.)that Q(b)is a strong set for ξand b. Consider
another strong set Sfor ξand b. We show
for all i∈[n] : bi·ini(Q(b)) −ξi(Q(b)) = bi·ini(S)−ξi(S).(C.9)
Due to the choice of b, it is ini(Q(b)) = ini(S)for all i∈[k]. Otherwise, we
can obtain a contradiction to Sbeing strong, by the same reasoning as in the
induction base case.
For an arbitrary j∈ {k+ 1, . . . , n}, we define a new vector das
d:= ((b+
j,b−j)if j∈S
(−1,b−j)otherwise. (C.10)
We show that Sis also strong for ξand d, i.e., there are no sets K, T ⊆[n]
such that
for all i /∈K:i∈T⇔i∈S(C.11)
for all i∈K:di·ini(T)−ξi(T)≥di·ini(S)−ξi(S)(C.12)
with at least one strict inequality
If there are T, K with j /∈Kthat meet the conditions, we get a contradiction
to Sbeing strong for ξand b, since di=bifor all i6=j. Now assume that
there are T, K with j∈Kthat meet the conditions. If jstrictly improves,
i.e., dj·inj(T)−ξj(T)> dj·inj(S)−ξj(S), it follows that dj=b+
j,j∈S∩T
and ξj(T)< ξj(S). This again yields a contradiction to Sbeing strong for ξ
and b. Thus, Sis also strong for ξand d. By (1.),Q(d)is another strong set
for ξand d, and by induction assumption
for all i∈[n] : di·ini(Q(d)) −ξi(Q(d)) = di·ini(S)−ξi(S).(C.13)
By definition of d,j∈Q(d)⇔j∈S. By (C.13), ξj(Q(d)) = ξj(S)and
therefore bj·inj(Q(d))−ξj(Q(d)) = bj·inj(S)−ξj(S). Furthermore, if it holds
that bj·inj(Q(b)) −ξj(Q(b)) < bj·inj(Q(d)) −ξj(Q(d)) then it follows that
uj(b, bj)< uj(d, bj), contradicting SP with true valuation bj. Since our choice
for jwas arbitrary from the set {k+1, . . . , n}, it holds for all j∈ {k+1, . . . , n}
that
bj·inj(Q(b)) −ξj(Q(b)) ≥bj·inj(S)−ξj(S).(C.14)
Now assume that there exists j∈ {k+ 1, . . . , n}such that (C.14) holds with
a strict inequality. With T=Q(b)and K={k+ 1, . . . , n}we obtain a
contradiction to Sbeing strong for ξand b. Thus (C.14) holds with equality
for all j∈ {k+ 1, . . . , n}. We are left with showing that (C.9) holds for all
i∈[k].
If there exists i∈[k]with bi·ini(Q(b)) −ξi(Q(b)) < bi·ini(S)−ξi(S), we
get by (C.13) that bi·ini(Q(b)) −ξi(Q(b)) < bi·ini(Q(d)) −ξiQ(d)), for an
arbitrary j∈ {k+1, . . . , n}and das defined in C.10. Thus, ui(b, bi)< ui(d, bi)
and uj(b, bj) = bj·inj(Q(b))−ξj(Q(b)) = bj·inj(Q(d))−ξj(Q(d)) = uj(d, bj).
With true valuations bi, bj, we obtain a contradiction to GSP.
Thus, for all i∈[k],bi·ini(Q(b)) −ξi(Q(b)) ≥bi·ini(S)−ξi(S), and for all
i∈ {k+ 1, . . . , n},bi·ini(Q(b)) −ξi(Q(b)) = bi·ini(S)−ξi(S). There cannot
be an i∈[k]with bi·ini(Q(b)) −ξi(Q(b)) > bi·ini(S)−ξi(S), since K:= [n]
and T:= Q(b)yield a contradiction to Sbeing strong for ξand b.
122 C Characterization of GSP Mechanisms for Submodular Costs
3. Third, we show that for all i, j ∈[n]with i6=jand all sets S⊆[n]\{i, j}, at
least one of the conditions (C.15) – (C.17) holds.
ξi(S∪{i, j}) = ξi(S∪{i})(C.15)
ξj(S∪{i, j}) = ξj(S∪{j})(C.16)
ξi(S∪{i, j})< ξi(S∪{i})and ξj(S∪{i, j})< ξj(S∪{j}).(C.17)
Assume that there is a set S⊆[n]\{i, j}such that none of the conditions (C.15)
– (C.17) holds for S. Then one of the following has to hold:
ξi(S∪{i, j})< ξi(S∪{i})and ξj(S∪{i, j})> ξj(S∪{j})(C.18)
ξi(S∪{i, j})> ξi(S∪{i})and ξj(S∪{i, j})< ξj(S∪{j})(C.19)
ξi(S∪{i, j})> ξi(S∪{i})and ξj(S∪{i, j})> ξj(S∪{j})(C.20)
Assume that (C.18) holds. Define dby dk:= b+
kfor all k∈S,dk:= −1for all
k /∈S∪{i, j}, and choose di, djsuch that ξi(S∪{i, j})< di< ξi(S∪{i})and
ξj(S∪{i, j})> dj> ξj(S∪{j}). By (1.)we know that Q(d)is strong for ξand
d.
•If i∈Q(d)and j∈Q(d), then with K:= {j}and T:= Q(d)\{j}, we have
for all k6=jthat k∈T⇔k∈Q(d)and
dj·in(j, T)−ξj(T) = 0 > dj−ξj(S∪{i, j}) = dj·in(j, Q(d)) −ξj(Q(d)) .
However, this contradicts that Q(d)is strong for ξand d.
•If i∈Q(d)and j /∈Q(d), then with K:= {i}and T:= Q(d)\{i}, we have
for all k6=ithat k∈T⇔k∈Q(d)and
di·ini(T)−ξi(T) = 0 > di−ξi(S∪{i}) = di·ini(Q(d)) −ξi(Q(d)) .
This poses a contradiction to Q(d)being strong for ξand d. The case i /∈Q(d)
and j∈Q(d)analogously leads to this contradiction.
•If i /∈Q(d)and j /∈Q(d)then with K:= {i, j}and T:= Q(d)∪{i, j}, we
have for all k /∈ {i, j}that k∈T⇔k∈Q(d)and
di·ini(T)−ξi(T) = di−ξi(S∪{i, j})>0 = di·ini(Q(d)) −ξi(Q(d)) ,
analogously for j. Again, this is a contradiction to Q(d)being strong for ξ
and d.
As a consequence (C.18) cannot hold. By exchanging the roles of iand j, also
(C.19) cannot hold. Thus, we are left to investigate (C.20). Define dfor all k /∈
{i, j}as before and let di, djbe such that ξi(S∪{i, j})> di> ξi(S∪{i})and
ξj(S∪{i, j})> dj> ξj(S∪{j}). It is not hard to verify that S∪{i}and S∪{j}
are both strong for ξand d. However,
di·ini(S∪{i})−ξi(S∪{i})>0 = di·ini(S∪{j})−ξi(S∪{j}),
a contradiction to (C.8) in (2.).
Thus, at least one of the conditions (C.15) – (C.17) holds.
4. Forth, we show that ξis cross-monotonic, by induction on n.
C Characterization of GSP Mechanisms for Submodular Costs 123
•Let n= 2. We first show that for all S⊆[n]with 1∈S, it holds that
ξ1(S∪ {2})≤ξ1(S). In case that S={1,2}, equality holds. Let S={1}.
Due to (C.15)–(C.17), either ξ1({1,2}) = ξ1({1}), or ξ2({1,2}) = ξ2({2}), or
ξ1({1,2})< ξ1({1})and ξ2({1,2})< ξ2({2}). The only case to consider is
ξ2({1,2}) = ξ2({2}). We utility the submodularity of Cand 1-BB of M to
obtain
C({1,2})−C({1})−C({2}) + C(∅) = 0
⇒ξ1({1,2}) + ξ2({1,2})−ξ1({1})−ξ2({2}) = 0
⇒ξ1({1,2}) = ξ1({1}).
The fact that ξ2(S∪{1})≤ξ2(S)for all S⊆[n]with 2∈S, holds analogously.
•Induction hypothesis: ξis cross-monotonic for all k < n.
•Assume without loss of generality that ξ2([n]) > ξ2([n]\{1}). We will obtain a
contradiction to the submodularity of C, thereby proving cross-monotonicity.
First assume that there is a j∈[n]\{1,2}with ξj([n]) < ξj([n]\{1}). Define
true valuations vby setting v1:= ξ1([n]) and vi:= b+
ifor all i∈[n]\{1}. It is
Q(v)∈ {[n],[n]\{1}}. If Q(v) = [n], then u1(((−1)1,v−1), v1) = 0 = u1(v, v1)
and u2(((−1)1,v−1), v2)> u2(v, v2)contradicts GSP. If Q(v)=[n]\ {1},
then u1((b+
1,v−1), v1) = 0 = u1(v, v1)and uj((b+
1,v−1), vj)> uj(v, vj), a
contradiction to GSP as well. Thus, for all j∈[n]\{1,2},ξj([n]) ≥ξj([n]\{1})
leading to
C([n]\{1}) =
n
X
i=2
ξi([n]\{1})<
n
X
i=2
ξi([n]) = C([n]) −ξ1([n]) .(C.21)
Since one of the cases (C.15) – (C.17) has to hold and we assume that
ξ2([n]) > ξ2([n]\ {1}), it follows that ξ1([n]) = ξ1([n]\ {2}). Then by in-
duction assumption, 1-BB, and (C.21),
C([n]\{2}) = ξ1([n]\{2}) +
n
X
i=3
ξi([n]\{2})
≤ξ1([n]\{2}) +
n
X
i=3
ξi([n]\{1,2})
=ξ1([n]) + C([n]\{1,2})
< C([n]) −C([n]\{1}) + C([n]\{1,2}),
contradicting submodularity of C. Thus, ξis cross-monotonic.
5. Fifth and last, consider Moulinξ= (Q0, ξ). Since ξis cross-monotonic by (4.),
Moulinξis GSP because of Theorem 4.1. As (1.)only utilized GSP of M, we can
conclude that for every bid vector b∈Rnnot only Q(b)but also Q0(b)is a strong
set for ξand b. Then by (2.),bi·ini(Q(b)) −ξi(Q(b)) = bi·ini(Q0(b)) −ξi(Q0(b))
for all i∈[n], i.e., Mand Moulinξare welfare equivalent.
Corollary C.4. [97] For any GSP mechanism Mthat employs cross-monotonic cost
shares specified by ξ,Moulinξis welfare equivalent to M.
Proof. The corollary simply follows from the proof of Theorem C.3 as submodular-
ity and 1-BB are only exploited for showing that cost-shares of a GSP and 1-BB
mechanism are cross-monotonic.
D
Further Optimization Problems
•SpanningTree:
–Input: set of players [n], complete undirected graph G= (V, E)with metric
edge weights we∈N, vector s∈Vnrepresenting a node sifor each player i,
subset of players S⊆[n].
–Feasible solution: subtree Tthat connects all nodes in {si}i∈Sand only con-
tains edges with both endpoints in {si}i∈S.
–Objective: minimize the weight of T.
•SteinerTree:
–Input: set of players [n], connected undirected graph G= (V, E)with metric
edge weights we∈N, vector s∈Vnrepresenting a node sifor each player i,
subset of players S⊆[n].
–Feasible solution: subtree Tthat connects all nodes in {si}i∈S.
–Objective: minimize the weight of T
•PriceCollectingSteinerForest:
–Input: set of players [n], undirected graph G= (V, E)with metric edge weights
we∈N, vectors s∈Vnand t∈Vnrepresening pairs (si, ti)of nodes for each
player i, penalty vector p∈Nn, subset of players S⊆[n].
–Feasible solution: forest Fand subset Q⊆[n]such that for all i∈S,siand
tiare either connected by For i∈Q.
–Objective: minimize the weight of Fplus the penalties of the players in Q, i.e.
of the not-connected pairs in S.
•ConnectedFacilityLocation:
–Input: set of players [n], set of facilities F, opening costs of∈Nfor each
facility, a metric d: ([n]∪F)×([n]∪F)→Nspecifying the distances between
all pairs of players and facilities, a parameter M≥1, subset of players S.
–Feasible solution: set of facilities Fand a Steiner tree Tconnecting all facilities
in F.
–Objective: minimize the sum of the opening costs of F, the connection costs
of players in Sto a closest open facility in F, and Mtimes the weight of T.
•TravelingSalesman
–Input: set of players [n], complete undirected graph G= (V, E)with metric
edge weights we∈N, vector s∈Vnrepresenting a node sifor each player i,
subset of players S⊆[n].
–Feasible solution: hamiltonian cycle Hfor the nodes in {si}i∈S, i.e., a cycle
that visits all nodes in {si}i∈Sexactly once, returns to the starting vertex, and
only contains edges with both endpoints in {si}i∈S.
–Objective: minimize the weight of H.
126 D Further Optimization Problems
•VertexCover:
–Input: set of players [n], undirected and unweighted graph G= (V, E), vector
s∈Enrepresenting an edge sifor each agent i, subset of players S⊆[n]
–Feasible solution: set of vertices Zsuch that each edge in {si}i∈Shas at least
one endpoint in Z.
–Objective: minimize the cardinality of Z.
•SetCover:
–Input: set of players [n], set of elements U, set {U1, . . . , Uk}of subsets of U
with SiUi=U, vector s∈Unrepresenting an element si∈Ufor each player
i, subset of players S⊆[n].
–Feasible solution: subset U0of {U1, . . . , Uk}such that SS∈U0S⊇ {si}i∈S.
–Objective: minimize the number of sets in U0.
•EdgeCover:
–Input: set of players [n], undirected and unweighted graph G= (V, E), vector
s∈Vnrepresenting an vertex sifor each agent i, subset of players S⊆[n].
–Feasible solution: set of edges Fsuch that each vertex in {si}i∈Sis adjacent
to an edge in F.
–Objective: minimize the cardinality of F.
•SingleSourceRentOrBuy:
–Input: set of players [n], connected undirected graph G= (V, E)with metric
edge weights we∈N, root vertex r, vector s∈Vnrepresenting a node sifor
each agent i, parameter M≥1, subset of players S⊆[n].
–Feasible solution: set EBof bought edges, set ERof rented edges, such that
for all i∈S, there is a path from ito rusing only edges in EB∪ER. Specify
these paths by P.
–Cost: The cost of e∈EBis M·we. An e∈ERcosts we·λ(P, e), where λ(P, e)
denotes the number of paths in Ptraversing e.
–Objective: minimize the cost.
•MulticommodityRentOrBuy:
–Input: set of players [n], connected undirected graph G= (V, E)with metric
edge weights we∈N, vectors s∈Vn,t∈Vnand f∈Nnrepresening pairs
(si, ti)of nodes and a flow requirement fifor each agent i, parameter M≥1,
subset of players S⊆[n].
–Feasible solution: set EBof bought edges, set ERof rented edges such that
for all i∈S, we can route flow fifrom sito tiusing only edges in EB∪ER.
Specify this flow by F.
–Cost: The cost of a bought edge e∈EBis M·we. A rented edges e∈ER
costs we·λ(F, e), where λ(F, e)denotes the total flow in Ftraversing e.
–Objective: minimize the cost.
•Multicast:
–Input: set of players [n], connected undirected graph G= (V, E)with metric
edge weights we∈N, root vertex r, vector s∈Vnrepresenting a node sifor
each agent i, subset of players S⊆[n].
–Feasible solution: Steiner tree Trooted at rconnecting all the nodes in {si}i∈S.
–Objective: minimize the weight of T.
–Note: It is often assumed that a fixed tree for all players [n]is given and
that the cost for serving S⊆[n]is simply the cost of the smallest subtree
containing S. Thus, there is essentially no optimization problem and the focus
lies on computing specific cost-shares (also in a distributed way).
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