Dissertation
Collusion-Resistant
Cost-Sharing Mechanisms:
Design Techniques, Analyses, Trade-Offs
Florian Schoppmann
Paderborn, 2. Juli 2009
Schriftliche Arbeit zur Erlangung des Grades
Doktor der Naturwissenschaften
an der Fakultät für Elektrotechnik, Informatik und Mathematik
der Universität Paderborn
Abstract
How can a system be designed so that autonomous self-interested players behave
in a “desirable” way? In this thesis, we study this question in the context of cost-
sharing problems, where finitely many players have an unknown valuation for some
non-rivalrous but excludable service (e.g., network connectivity). The challenge is to
design mechanisms that elicit truthful reports of the players’ valuations, determine which
set of players
Q
to serve, and decide how to distribute the incurred service cost
C
(
Q
). So
in particular, a cost-sharing mechanism has to give players an incentive to reveal truthful
information. Further constraints for cost-sharing problems include budget balance (i.e.,
recovery of the service cost with the prices charged) and economic efficiency (i.e., a
reasonable trade-off between the service cost and the excluded players’ valuations).
Practical applications moreover require that cost-sharing mechanisms are computable in
polynomial time.
Cost-sharing problems are fundamental in economics and have a broad area of ap-
plications; e.g., distributing volume discounts in electronic commerce, sharing the cost
of public infrastructure projects, allocating development costs of low-volume built-to-
order products, etc. Despite this fundamental nature, general techniques for solving
cost-sharing problems are rare. When requiring group-strategyproofness—i.e., collusion
resistance in a very strong sense—essentially only one technique has been known, the
so-called Moulin mechanisms. Unfortunately, there are several natural cost-sharing
problems for which any Moulin mechanism inevitably suffers poor budget balance and
economic efficiency.
In this thesis, we devise several alternative techniques for designing cost-sharing
mechanisms. We demonstrate the benefits of our novel techniques by applying them to
various natural cost-sharing problems where the costs
C
(
Q
)are induced by combinatorial
optimization problems. Moreover, we provide characterization results that contribute
towards understanding the inherent limitations of collusion resistance with respect to
the other desirable properties of cost-sharing mechanisms.
iii
Acknowledgments
This thesis would not have been possible without the continuous support of a number
of people to whom I wish to express my gratitude here. First and foremost, I want to
thank my advisor Burkhard Monien for his support, his steady encouragement, and his
insights that have been invaluable to my research. In fact, it also was a course lectured
by Burkhard almost four years ago that sparked my interest in algorithmic game theory
while still being an undergrad student.
I am furthermore indebted to Yvonne Bleischwitz who used to be the other Ph.D.
student in the “cost-sharing team”. I always greatly enjoyed working together with
Yvonne, and this feeling has only increased ever since she finished her dissertation and
left the group.
Special thanks also go to Martin Gairing and Karsten Tiemann for the fruitful collab-
oration on the topics of selfish routing. Besides the joint work, Martin has frequently
provided me with extremely helpful feedback on various matters. His support reaches
back to the time when he was the advisor of my diploma thesis. Together with Karsten I
attended more conferences and workshops than with anybody else. It not only was a
pleasure to work with him, but also to travel with him to various places in Europe.
During the past three years, I have been lucky to share my office with Tobias
Tscheuschner. Discussing research problems with Tobias, as well as the latest news
and everything in between, has contributed a lot to making my Ph.D. project delightful
and enjoyable.
I am very grateful to Johannes Berendt, Yvonne Bleischwitz, Rainer Feldmann, Martin
Gairing, Eva Neels, Jan Rieke, and Ulf-Peter Schroeder for carefully reading preliminary
parts of my thesis.
Crucial for my well-being throughout the past years have been my family and my
friends. I particularly thank my parents for teaching me to be open-minded and for
stimulating a passion for learning at an early age. Finally, I owe my sincerest thanks
to Rahel for her exceptional patience throughout the past years. She certainly had to
endure the most, and her support has been beyond words.
My research in the past three years was only possible due to a fellowship by the
International Graduate School Dynamic Intelligent Systems at the University of Paderborn.
I wish to thank everybody in the Graduate School team for the support and for organizing
the many extra-curricular activities. I enjoyed them a lot. Keep up the good work!
Paderborn, March 2009 Florian Schoppmann
v
Contents
1 Introduction 1
1.1 Algorithmic Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Cost Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Cost-Sharing Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Design Techniques for Cost-Sharing Mechanisms . . . . . . . . . . . . . . . . 4
1.4 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Lexicographic Maximization: Beyond Cross-Monotonicity . . . . . . 8
1.4.2 Cost Sharing Without Indifferences: To Be or Not to Be (Served) . 8
1.4.3 Does Coalition Size Matter? . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.4 Generalizing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Other Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.2 Characterizing Collusion-Resistant Cost Sharing . . . . . . . . . . . . 11
1.5.3 Outside the Realm of Cost Sharing . . . . . . . . . . . . . . . . . . . . 13
1.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Preliminaries 17
2.1 The Formal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 Cost-Sharing Problems and Mechanisms . . . . . . . . . . . . . . . . 17
2.1.3 Non-Manipulability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.4 Cost-Sharing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.5 Dealing with Indifferences . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.6 Budget Balance and Economic Efficiency . . . . . . . . . . . . . . . . 20
2.1.7 Special Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Previous Design Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Vickrey-Clarke-Groves Mechanisms . . . . . . . . . . . . . . . . . . . 22
2.2.2 Moulin Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Scheduling Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Bin Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Network Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
vii
Contents
3 Lexicographic Maximization: Beyond Cross-Monotonicity 27
3.1 Overview of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Symmetric Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Symmetric Cost-Sharing Methods and Mechanisms . . . . . . . . . . 29
3.2.2 Computing the Outcome of Symmetric Mechanisms . . . . . . . . . 32
3.3 Symmetric Mechanisms for Symmetric Subadditive Costs . . . . . . . . . . 35
3.3.1 Computing Two-Price Cost-Sharing Forms . . . . . . . . . . . . . . . 35
3.3.2 Lower Bound on the Performance of Symmetric Mechanisms . . . . 39
3.3.3 What Can Be Achieved with One Price? . . . . . . . . . . . . . . . . . 40
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Makespan Minimization with Identical Jobs . . . . . . . . . . . . . . 41
3.4.2 Makespan Minimization with Non-Identical Jobs . . . . . . . . . . . 43
3.5 Characterizing Symmetry and 1-BB . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.1 An Impossibility Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.2 GSP and 1-BB Cost-Sharing Mechanisms for Three Players . . . . . 47
3.6 Beyond Symmetric Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.1 Precedence-Monotonic Cost Shares . . . . . . . . . . . . . . . . . . . 49
3.6.2 The Missing Link to GSP . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Cost Sharing Without Indifferences: To Be or Not to Be (Served) 53
4.1 Overview of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Cost Sharing Without Indifferences . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.2 Equivalence of Models Without Indifferences . . . . . . . . . . . . . 56
4.2.3 WSGSP Implies Separability . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.4 WGSP Does Not Imply Separability . . . . . . . . . . . . . . . . . . . . 61
4.3 Egalitarian Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Efficiency of Egalitarian Mechanisms . . . . . . . . . . . . . . . . . . 63
4.3.2 Most Cost-Efficient Set Selection . . . . . . . . . . . . . . . . . . . . . 64
4.3.3 Submodular and Supermodular Costs . . . . . . . . . . . . . . . . . . 66
4.4 Acyclic Mechanisms and SGSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.1 Egalitarian Mechanisms Are Acyclic . . . . . . . . . . . . . . . . . . . 67
4.4.2 Sequential Stand-Alone Mechanisms . . . . . . . . . . . . . . . . . . . 70
4.5 A Framework for Polynomial-Time Computation . . . . . . . . . . . . . . . . 72
4.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6.1 Makespan Minimization and Bin Packing Cost-Sharing Problems . 76
4.6.2 Monotonic Approximation Algorithms . . . . . . . . . . . . . . . . . . 77
4.6.3
Non-Monotonic Approximation Algorithms with a Polynomial-
Time Computable Monotonic Bound . . . . . . . . . . . . . . . . . . . 80
4.6.4 Makespan Problems with Monotonic Optimal Costs . . . . . . . . . 82
4.6.5 Scheduling Problems with Supermodular Costs . . . . . . . . . . . . 83
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
viii
Contents
5 Does Coalition Size Matter? 85
5.1 Overview of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Notions of Non-Manipulability by Small Coalitions . . . . . . . . . . . . . . 85
5.2.1 Resistance Against Coalitions with Side-Payments . . . . . . . . . . . 86
5.2.2 Some Preliminary Implications by SP and WUNB . . . . . . . . . . . 87
5.2.3 k-GSP Is Strictly Weaker Than GSP . . . . . . . . . . . . . . . . . . . . 89
5.3 Group-Strategyproofness Against Only Two Players . . . . . . . . . . . . . . 89
5.3.1 Upper Continuity and 2-GSP Together Imply GSP . . . . . . . . . . . 89
5.3.2 Separability and 2-GSP Together Imply GSP . . . . . . . . . . . . . . 90
5.4 Weak Group-Strategyproofness and Non-Bossiness . . . . . . . . . . . . . . . 95
5.4.1 Separability and 2-WGSP Do Not Imply WGSP . . . . . . . . . . . . . 95
5.4.2 2-GSP Implies WGSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4.3
Relationship Between Collusion-Resistance and Non-Bossiness
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Generalizing the Model 101
6.1 Overview of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 General-Demand Cost Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.1 Generalized Moulin Mechanisms . . . . . . . . . . . . . . . . . . . . . 102
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Bibliography 107
Index 115
ix
Lists
List of Algorithms
2.1 Moulin mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Symmetric mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Two-price cost-sharing forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1 Egalitarian mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Acyclic mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Cost-sharing method and offer function of egalitarian mechanisms . . . . . 69
4.4 Sequential stand-alone mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Egalitarian mechanisms with β-relaxations . . . . . . . . . . . . . . . . . . . 75
4.6 ε-dual approximation algorithm for bin packing . . . . . . . . . . . . . . . . 81
4.7 Modified PTAS for the minimum makespan problem . . . . . . . . . . . . . . 82
5.1 3-Player mechanism that is 2-GSP but not GSP . . . . . . . . . . . . . . . . . 89
5.2 Separable mechanism that is 2-WGSP but not WGSP . . . . . . . . . . . . . 95
6.1 Generalized Moulin mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 104
List of Figures
4.1 Hierarchy of collusion-resistance properties . . . . . . . . . . . . . . . . . . . 58
4.2 Examples showing that collusion-resistance variants are not equivalent . . 59
4.3
For non-identical machines,
LPT
is not monotonic with regard to processing
requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 FFD is not monotonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 Two dimensions of coalition-forming capabilities . . . . . . . . . . . . . . . . 86
5.2 Overview of the various non-manipulability properties . . . . . . . . . . . . 99
List of Tables
1.1
Comparison of techniques for designing polynomial-time computable cost-
sharing mechanisms, using the example of minimum-makespan problems 10
xi
A note on the use of “we”
As a matter of style and to acknowledge that many of
the results presented in this thesis are based on collaborative work with others, “we” is
used throughout the thesis, except in places where the personal voice of the author is
expressed.
Chapter 1
Introduction
Present-day information and communication technologies crucially rely on large-scale
networks that are created and maintained by a huge number of autonomous players with
heterogeneous goals. Evidently, this includes the Internet as the most prominent example,
but no less also peer-to-peer, logistics, and social networks. Computer science and
research on distributed algorithms in particular have responded to the rapid emergence
of these new decentralized networks by a corresponding paradigm shift: Starting roughly
a decade ago, computer scientists have increasingly focused on optimization in networks
where the self-interested players’ behavior cannot be directly controlled [
56
,
43
,
60
].
The lack of central control that is encompassed by players’ selfishness is instead regarded
as an inevitable constraint, similar to the lack of computing power when devising
approximation algorithms or the lack of information about the future when designing
online algorithms.
Located at the intersection of computer science with mathematics and economics, this
rapidly growing field is now routinely called algorithmic game theory. Bringing together
algorithms and game theory has given rise to several new and important questions that
were not rigorously considered before: Can the loss due to selfish behavior be quantified,
when compared to a hypothetical optimum? In a more catchy term: What is the price of
anarchy? What is the complexity to compute stable states, i.e., game-theoretic equilibria,
or approximations of them? How should systems be designed in order to give players an
incentive to act in a particular fashion? This thesis is located in the subfield of algorithmic
game theory that deals with the last of the previous questions and that we will introduce
in the following.
1.1 Algorithmic Mechanism Design
When designing algorithms for traditional optimization problems, computer scientists can
often safely assume that their algorithms are run relatively isolated from other parts of
the system, sometimes even embedded in a black-box manner. Such an algorithm would
be easily interchangeable with any other algorithm that computes feasible solutions,
without changing any further aspects of the system beyond the confines of the black
box. Correspondingly, the quality of an algorithm has typically been measured only in
terms of its computational complexity (occasionally also its space or communication
complexity) and its performance according to either worst- or average-case analysis,
or—more practically—some reasonable benchmark.
1
Chapter 1 Introduction
In sharp contrast to devising algorithms as relatively isolated entities, decentralized
networks often necessitate algorithms and protocols that work on information received
from many different self-interested players. Clearly, these players only reveal what is
best for their own benefit. Hence, a system designer has to anticipate that the choice of
an algorithm or protocol crucially impacts what information the players will provide. In
more drastic words, an algorithm or protocol may have excellent performance—yet, if
the players do not adopt it, its implementation will not prove successful.
In economics, research on the implementation of system-wide desirable solutions in
the presence of self-interested players is a well-established field called mechanism design.
In the basic setting, one can imagine that each player has some private information that
is relevant to the problem at hand and that has to be elicited, e.g., by bids. Now the
goal of the system designer is to provide incentives (e.g., monetary transfers) so that
revealing truthful information is the best strategy players have available.
Auction design is the oldest and arguably the best-known motivation for mechanism
design. For instance, a naive way to conduct a sealed-bid auction is to sell the item at
stake to the highest bidder, for a price equal to his bid. However, this gives players an
incentive to bid less than their maximum willingness to pay—i.e., their true valuation for
the item. After all, winning only requires bidding strictly more than the second highest
bidder. As long as players cannot collude, there is an easy fix from the viewpoint of
incentives: In a “second-price” or “Vickrey” auction [
75
], the winner only has to pay the
value of the second highest bid. It is an easy observation that untruthful bidding could
not provide any player with a better “net benefit” (i.e., valuation for the item minus
payment if the player won the auction, and zero otherwise). Note, however, that Vickrey
auctions are not resistant against coordinated manipulation: In particular, the winner
would have an increased net benefit if he had convinced his competitors to bid less than
their true valuations.
The traditional application of game theory has been to predict outcomes of strategic
interaction, i.e., to identify the states of the game where every player is unlikely to change
his strategy. Arguably the most important solution concept here is the Nash equilibrium,
that is a state in which no player can unilaterally improve his utility by changing
his strategy, provided that all other players keep theirs unchanged. In mechanism
design, which is sometimes referred to as “inverse game theory” [
60
], the task is often
complementary: Design a game so that the state in which every player bids truthfully is
always a Nash equilibrium, for every possible combination of the true valuations. Hence,
since every state of the game could possibly be the truth, this requirement means that
revealing truthful information is even a dominant strategy for every player, i.e., the best
alternative regardless of what the other players are doing. Clearly—as long as one can
rule out collusion—such a dominant strategy equilibrium is a robust solution concept
because no player ever has to reason about his competitors.
Indeed, it contributes much to the importance and elegance of mechanism design
that suitable incentives provide the players with essentially effortless decision-making.
Obviously, this gives rise to many applications for the continuing increase in electronic
commerce and makes mechanism design an invaluable tool, e.g., to support automated
negotiations among groups of individuals and businesses.
2
1.2 Cost Sharing
Algorithmic mechanism design [
56
]constitutes a synthesis of mechanism design with
algorithmic methods for dealing with computationally intensive challenges, with the ulti-
mate goal of comprehending the necessary trade-off between incentives, computational
complexity, and approximation guarantees.
1.2 Cost Sharing
In this thesis, we work towards a more thorough understanding of a particular class of
algorithmic mechanism design problems, where the cost of a joint project is to be shared
between the participants. For illustration and motivation, we give two made-up and
arguably simplified examples.
1.2.1 Examples
Inclusion in a Schedule
Several logistics companies are interested in operating Sunday
flights at a cargo airport that has previously been used only six days a week. Specifically,
there are numerous possible flights with different requirements of ground handling,
yet only a limited number of terminals that allow processing at different speeds. How
can the overall cost for operating the airport infrastructure also on Sundays be shared?
This cost is assumed to be roughly proportional to the total time that there is at least
one aircraft needing ground handling. Every company has a maximum willingness to
contribute to the cost as there are several outside options: They could switch to another
day, to another airport, or they might use other means of transport altogether. So a
mechanism is sought that elicits truthful bids and then decides which flights are included
in the Sunday schedule and at what price (respecting the bids).
There are several challenges: Most naturally, the cost shares need to recover the total
cost. Moreover, the schedule should be “efficient” in an economic sense, meaning that for
each flight we assume an “external cost” when not including the flight in the schedule.
Therefore, a scheduling problem with rejection needs to be solved (see also Section 2.3.1):
Find a schedule that minimizes the overall schedule length (also called the makespan)
plus the external costs for the rejected flights. In particular, economic efficiency implies
that a trivial solution such as the empty schedule is typically not a good solution.
Network Connectivity
The residents of a remote village—long ignored by the telecom-
munication industry—wish to get out of the communication stone age by providing
high-speed Internet connectivity for every home that is interested. The willingness to
contribute to the construction and set-up costs is, however, different among the residents.
For instance, whereas some people have long been in desperate need of being able to
work from home over the Internet, some others do not even own a computer. Thus, the
local government is looking for a mechanism that elicits truthful bids and then decides
which homes participate and at what price.
Laying cables to provide the selected set of homes with high-speed Internet access
can be modeled as a “prize-collecting” Steiner tree problem (see also Section 2.3.3):
3
Chapter 1 Introduction
Find a minimum-cost network that connects all participating homes and the Internet
service provider in the next city. Here, “prize-collecting” is the usual term to indicate
that, similar as in the previous example, there is an external cost for not including homes
in the network.
1.2.2 Cost-Sharing Mechanisms
The above scenarios are examples of cost-sharing problems: A non-rivalrous but exclud-
able good (i.e., a service such as inclusion in a schedule or connectivity in a network) is
to be made available to
n∈N
players at non-negative prices. Each player
i∈{
1
,...,n}
is
completely characterized by his valuation
vi∈R
for receiving the service. A cost-sharing
mechanism is sought that elicits truthful reports of each player’s valuation and then
determines both the set of served players
Q⊆{
1
,...,n}
and a distribution of the service
cost C(Q)∈R≥0.
Cost-sharing problems are fundamental in economics and have a broad area of ap-
plications, similar in concept to the previous examples. This includes, e.g., distributing
volume discounts in electronic commerce, sharing the cost of public infrastructure
projects, allocating development costs of low-volume built-to-order products, etc.; see
also Moulin and Shenker
[52]
and their references. As mentioned before, the pivotal
constraint when designing cost-sharing mechanisms is incentive-compatibility (also called
truthfulness), meaning that each player has an incentive to act as desired by the provider
of the service. In the case of cost sharing, this is to submit truthful bids to the mechanism.
The cost-sharing literature typically requires a particular strong notion of truthfulness in
that even collusion (i.e., coordinated wrong-bidding) must never be profitable.
There are three essential goals for the design of truthful cost-sharing mechanisms: The
first and most natural constraint is, of course, recovery of the service cost. Together with
reasonable bounds on the generated surplus, this property is referred to as (approximate)
budget balance. As a second goal, a mechanism should satisfy (approximate) economic
efficiency, i.e., guarantee reasonable bounds on the social cost by appropriately trading off
the service cost and the excluded players’ valuations. Finally, for practical applications,
the computational complexity of the mechanism must be reasonable. Not least due to
this blend of optimization goals from different perspectives, cost sharing has attracted a
great deal of interest also in computer science.
1.3 Design Techniques for Cost-Sharing Mechanisms
Truthfulness in Non-Cooperative Settings
In the standard cost-sharing model, a
player who is served has a utility equal to his valuation minus his payment. If a player
is not served, then he will not be charged and his utility is zero. The basic notion of
truthfulness, called strategyproofness (SP), requires that no player can improve his utility
by false bidding when all other bids are kept fixed. Equivalently, it must hold for all
possible combinations of true valuations that the state in which every player
i
truthfully
bids viis a Nash equilibrium.
4
1.3 Design Techniques for Cost-Sharing Mechanisms
Truthfulness in Cooperative Settings
A form of manipulation that SP does not rule
out is manipulation by coalitions. Resistance against collusion is, however, especially
desirable in settings with a large number of players that the provider of the service
might not even know; e.g., in the Internet. Here, players often have the means to
coordinate deceit that is impossible to discover. Several concepts of collusion resistance
are known in the literature, of which we name only two here: A mechanism is called
group-strategyproof (GSP) if any defection of a coalition that increases some member’s
utility inevitably decreases the utility of one of its other members. A weaker notion
of collusion resistance is weak group-strategyproofness (WGSP) that is fulfilled if any
defecting coalition has at least one member whose utility does not strictly improve.
Equivalently, with a WGSP mechanism, it must hold for all possible combinations of
true valuations that the state in which every player
i
truthfully bids
vi
is a strong
equilibrium [4].
Efficient Solutions
The most universal technique for the design of truthful mechanisms
(not just for cost sharing) is the class of Vickrey-Clark-Groves (VCG) mechanisms [
75
,
14
,
30
]. By always picking a set of players so that the sum of the included players’ valuations
minus the service cost is maximized and by an appropriate payment scheme such that
each player’s payment does not directly depend on his own bid, these mechanisms are
truthful (SP) and satisfy optimal economic efficiency. In fact, Green and Laffont
[29]
revealed already in the 1970’s that under general assumptions
1
, the VCG mechanisms
are the only class of SP mechanisms with these properties.
Unfortunately, VCG mechanisms are not resistant against collusion and fail in general
to provide any guarantees for cost recovery, even when ignoring computational com-
plexity [
52
](see also Section 2.2.1 for details). Hence, there is an intrinsic conflict:
In general, truthful mechanisms cannot guarantee exact budget balance and optimal
economic efficiency at the same time. In fact, Feigenbaum et al.
[24]
gave simple
cost functions for which not even (relative) approximations of both budget balance
and economic efficiency can be achieved at the same time—presuming that economic
efficiency is measured in terms of the traditional social welfare (sum of the included
players’ valuations minus service cost). Roughgarden and Sundararajan
[64]
observed
that this impossibility is due to an incompatibility with the mixed-sign social welfare
and suggested an alternative measure of economic efficiency: Social cost, defined as the
sum of the excluded players’ valuations plus the service cost. This measure is clearly an
order-preserving transformation of the social welfare, and the absolute error is always
the same under both measures. Roughgarden and Sundararajan proved that measuring
economic efficiency in terms of the social cost indeed makes the desired bi-criteria (rela-
tive) approximation possible; i.e., there is a large class of mechanisms that guarantee
both budget balance and economic efficiency within constant approximation factors
β≥1 and α≥1, abbreviated as β-BB and α-EFF.
1
The set of possible bids is equal to the set of possible valuations, and utilities are quasi-linear (see
Section 2.1.2).
5
Chapter 1 Introduction
Cost-Sharing Methods
Due to the unsuitability of VCG mechanisms for cost-sharing
problems there is great need for other general design techniques. A straightforward idea
is to separate a mechanism into two parts: First, compute the set of served players
Q
depending on the bids. Then, compute the cost shares using a cost-sharing method that
only depends on
Q
, and not on the bids. In fact, it is known that all GSP mechanisms
could be separated this way [51].
Cooperative game theory has long dealt with the question of dividing costs among
coalitions. In particular, in the terminology of cooperative game theory, a cost-sharing
method is simply a value of the cooperative game defined by the costs
C
(see, e.g.,
Osborne and Rubinstein
[58]
). Consequently, many families of cost-sharing methods are
known in the literature. We consider three in the following:
A very simple cost-sharing method
ξ
is to let every player pay the marginal cost of
including him (according to a fixed order of the players). That is, if the players in
Q
are
numbered 1
,...,|Q|
, every player
i∈Q
pays
ξi
(
Q
) =
C
(
{
1
,
2
,...,i}
)
−C
(
{
1
,
2
,...,i−
1
}
).
Another well-known cost-sharing method is the Shapley value [
71
]. It assigns to each
player
i∈Q
his average marginal cost, over all orders of the players in
Q
. A third
method is the egalitarian solution by Dutta and Ray
[22]
, which is motivated by being
as egalitarian as possible. The details of the motivation are beyond the scope of this
introduction, so we refer to Dutta and Ray’s original article for a formal definition and
explanations. When the costs
C
are submodular (meaning that the marginal cost of
adding a player to a set
Q
can only decrease as
Q
gets larger), the egalitarian solution for
a set
Q
can be computed iteratively: Find the most cost-efficient subset
S
of 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.
GSP Mechanisms
Essentially the only technique for designing GSP cost-sharing mech-
anisms is due to Moulin
[51]
. Its main ingredient are cross-monotonic cost shares
ξi
(
Q
)
that never increase when the set of served players
Q
gets larger. Given such a cost-sharing
method
ξ
, a Moulin mechanism serves the maximal set of players who can afford their
corresponding price—due to cross-monotonicity, a unique maximal set always exists.
Algorithmically, this set can be found by simulating an iterative ascending auction: At
the beginning, all players are included in
Q
, and each player
i∈Q
is offered price
ξi
(
Q
).
If there is a player who cannot afford the price offered to him, he is dropped from
Q
and
a new iteration begins. The auction terminates once each of the remaining players can
afford the price offered to him.
The main benefit of Moulin mechanisms is that they reduce the design of GSP mecha-
nisms to finding cross-monotonic cost-sharing methods, which are solely responsible for
the mechanism’s performance. On the negative side, Immorlica et al.
[36]
and Rough-
garden and Sundararajan
[64
,
65]
showed that there are several natural cost-sharing
problems for which any Moulin mechanism inevitably suffers from poor budget balance
or/and poor economic efficiency.
6
1.4 Contribution
Immorlica et al.
[36]
and Penna and Ventre
[62]
gave another family of cost-sharing
mechanisms that are GSP only if voluntary non-participation is not an option players
have.
2
In this thesis, however, we very well assume that players may opt not to participate
in order to help others. Consequently, these mechanisms are not GSP according to the
definition used in this thesis.
The Role of Indifferent Players
An immediate implication of SP is that, by unilateral
deviation, each player can only influence whether he receives the service, but not his cost
share. In detail, for each player
i∈
[
n
]and every fixed combination of his competitors’
bids, there has to be some threshold value
θi
so that
i
is served for a price of
θi
when
bidding strictly more than
θi
, and
i
is not served when bidding strictly less. We call a
player indifferent if he bids exactly his threshold value. Note that SP does not imply a
particular rule for what to do with an indifferent player.
As a general rule of thumb, the major intricacy for the design of GSP mechanisms
with good performance—and thus for finding alternatives to Moulin mechanisms—is the
treatment of these indifferent players: Since the utility of an indifferent player is zero
regardless of whether he is served or not, his utility is completely unaffected by his own
bid. Consequently, this player is prone to manipulate and help others either by enforcing
his inclusion with a very large bid or by prompting his exclusion with a very low bid.
More Flexibility by Relaxing GSP
Replacing the fairly strong GSP axiom by demand-
ing only WGSP allows for greater flexibility when designing cost-sharing mechanisms. In
particular, manipulation by indifferent players is no longer problematic because their
utility does not strictly improve. A general technique for the design of WGSP mecha-
nisms, called acyclic mechanisms, is due to Mehta et al.
[50]
. Their mechanisms are
generalizations of Moulin mechanism and are likewise computed by simulating iterative
ascending auctions. However, for any set of remaining players, there is a specific order
in which prices are offered to the players. Now, whenever a player cannot afford this
offer, a new iteration is started prematurely. This way, lack of cross-monotonicity can be
“concealed” from the players and truthfulness be preserved, while the added versatility
of acyclic mechanisms allows for improved budget balance and economic efficiency.
1.4 Contribution
As discussed in the previous sections, cost-sharing problems involve a number of com-
peting goals, and it is known that not all goals are compatible with each other. As a
consequence, trade-offs are inevitable. In this thesis, we devise novel design techniques
for cost-sharing mechanisms, analyze their performance guarantees, and characterize the
compatibility of various desirable properties. In particular, we make intuitively “small”
modifications to the assumptions on players’ behavior or to certain design goals. It turns
out that some of these modifications allow for very improved performance guarantees,
2Technically, these mechanisms do not satisfy strong consumer sovereignty (see Section 2.1.2).
7
Chapter 1 Introduction
whereas others—somewhat counterintuitively—do not. For all new techniques, we
demonstrate their benefits by applying them to various natural cost-sharing problems
where the costs C(Q)are induced by combinatorial optimization problems.
The rest of this section provides a high-level overview of our results. We will give
more detailed and formal summaries at the beginning of the later chapters.
1.4.1 Lexicographic Maximization: Beyond Cross-Monotonicity
We propose a new technique for the design of GSP cost-sharing mechanisms that deliber-
ately treats players unequally by maintaining an order of precedence. The main idea is
as follows: Given a cost-sharing method
ξ
, call a set
Q
feasible if all players contained in
it can afford their cost share
ξi
(
Q
). Then, choose the set of players that lexicographically
maximizes the vector of all players’ utilities, over all feasible sets. In contrast, Moulin
mechanisms always choose the feasible set for which all players’ utilities attain their
maximum (note here that this maximum is only well-defined for cross-monotonic cost
shares). We show that our new technique allows for improved budget balance. However,
the trade-off to make is the unequal treatment of the players, which also entails a loss of
economic efficiency.
In detail: Costs are called symmetric if they only depend on the size of the served
players. They are called subadditive if the union of two sets of players is never more
costly than the sum of the two stand-alone costs. For costs that satisfy both properties,
we devise a novel family of cost-sharing mechanisms that we call symmetric mechanisms
and that achieve provably better budget balance than any Moulin mechanism could
guarantee.
We apply our findings to scheduling problems, where each player owns exactly one job
and the service is inclusion in a schedule, for processing the jobs on parallel machines.
The incurred service cost is the maximum completion time of all jobs—called the
makespan. Therefore, we give a polynomial-time algorithm that not only determines the
set of served players
Q
and a distribution of the service cost
C
(
Q
), but also a feasible
schedule for the jobs that belong to the players in Q.
Towards understanding the limitations imposed by GSP itself, we establish the follow-
ing impossibility result: For more than three players, cost-sharing mechanisms that are
both GSP and 1-BB do not exist in general, even if the cost function is symmetric. On
the other hand, for at most three players and symmetric costs, there is always a GSP and
1-BB mechanism.
1.4.2 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Our second technique is based on a modified assumption on players’ behavior: Specifi-
cally, it seems plausible that players prefer being served over not being served, also when
they have to pay a price equal to their valuation. In contrast, the “standard” model as-
sumes that players are indifferent between these two outcomes. We call mechanisms that
are collusion-resistant under our modified assumptions as “group-strategyproof against
service-aware players” (SGSP). It turns out that SGSP greatly increases the flexibility for
8
1.4 Contribution
designing collusion-resistant cost-sharing mechanisms. In particular, trading in GSP for
SGSP allows for improved budget balance and improved economic efficiency.
In detail: We give a novel family of SGSP mechanisms that we call egalitarian due to
being reminiscent to the algorithm for computing Dutta and Ray’s egalitarian solutions.
Our new mechanisms achieve 1-BB for arbitrary costs and additionally
O
(
logn
)-EFF for
the natural and large class of subadditive costs. Egalitarian mechanisms also fit into the
framework of acyclic mechanisms. Thus far, acyclic mechanisms were only known to be
WGSP; yet, we prove that they satisfy also the strictly stronger SGSP.
Our results show that for many cost functions defined by natural optimization prob-
lems, there are SGSP (and thus WGSP) egalitarian cost-sharing mechanisms that guar-
antee 1-BB and an economic efficiency known to be asymptotically optimal for truthful
and (approximately) budget-balanced cost-sharing mechanisms (see Section 1.5.2). Un-
fortunately, many of these egalitarian mechanisms are not computable in polynomial
time, unless
P
=
NP
. We therefore devise a framework for coping with the computational
complexity, and we identify suitable approximation algorithms. Using this framework,
we develop polynomial-time-computable egalitarian mechanisms for sharing the cost
induced by various scheduling problems. We show that these mechanisms achieve
provably better budget balance and economic efficiency than any Moulin mechanism. A
comparison of Moulin, symmetric, and egalitarian mechanisms is shown in Table 1.1.
1.4.3 Does Coalition Size Matter?
GSP implies that players have virtually unlimited means to communicate and make
binding agreements with all of their competitors. Since GSP is known to impose severe
limitations on the other goals in cost sharing, there is hence good reason to seek for a
weaker axiom: We study the following question: Does relaxing GSP to resistance only
against coalitions of bounded size yield a richer set of possible mechanisms? We show
that, surprisingly, the answer is essentially “no”.
In detail, we prove that if a mechanism has a cost-sharing method and is group-
strategyproof against coalitions of size only two (“2-GSP”), then it is also GSP. In other
words, 2-GSP and GSP are equivalent if we require that cost shares must only depend
on the set of served players, and not directly on the bids. Moreover, we show that
even without additional requirements, 2-GSP implies WGSP. Consequently, our results
give some justification that GSP may, after all, still be desirable in various scenarios.
As another benefit, we believe that our characterizations will facilitate devising and
understanding new GSP cost-sharing mechanisms. Finally, we also relate our findings to
other concepts of non-manipulability known in the literature.
1.4.4 Generalizing the Model
Up to this point, we have assumed a binary-demand cost-sharing model, where players
are either served or not served. However, this binary-demand model is not appropriate
in every context. This is particularly the case when fault tolerance is an issue: For
instance, when the service is connectivity to a network, players might have an increased
9
Chapter 1 Introduction
Table 1.1:
Comparison of techniques for designing polynomial-time computable cost-
sharing mechanisms, using the example of minimum-makespan problems
Makespan Problem
Technique
Collusion
Resistance BB EFF References and Remarks
arbitrary jobs on related machines (Q||Cmax)
Moulin mechanisms
GSP 2d2d·(1+Hn)[5, 6]
Symmetric mechanisms
GSP p17+1
4·dΩ(n)Theorem 3.4.8, Lemma 3.4.3
Egalitarian mechanisms
SGSP 2 4HnTheorem 4.6.6
arbitrary jobs on identical machines (P||Cmax)
Moulin mechanisms
GSP 2m
m+1Ω(n)[5]
GSP 2m−1
m
2m−1
m·(1+Hn)[10]
Egalitarian mechanisms
SGSP 1 +"2(1+")·Hn
Theorem 4.6.8, running time exponen-
tial in 1
"
SGSP 4
3−1
3m2(4
3−1
3m)·HnTheorem 4.6.4, practical mechanism
identical jobs on related machines (Q|pi=p|Cmax)
Egalitarian mechanisms
SGSP 1 2HnSection 4.6.4
Note: ddenotes the number of different processing requirements, Hndenotes the n-th harmonic number.
utility by having redundant connections—which corresponds to a higher reliability and
a higher quality of service. The general-demand model accounts for this by allowing
players to receive multiple levels of service. Put differently, a general-demand mechanism
must choose a multiset of served players. We generalize the binary-demand Moulin
mechanisms to the first general technique for designing GSP general-demand mechanism.
1.5 Other Related Work
1.5.1 Applications
Besides general design techniques as introduced in Section 1.3, most other work on
cost sharing has focused on devising “good” cross-monotonic cost-sharing methods and,
more recently, “good” acyclic mechanisms. In these works, costs stem from solutions
of combinatorial optimization problems, including the minimum spanning tree [
41
,
10
1.5 Other Related Work
37
,
38
], Steiner tree [
37
,
64
,
65
,
50
], fixed tree multicast [
23
,
24
,
3
], facility location
[
59
,
46
,
65
,
36
,
50
], rent-or-buy-network design [
59
,
65
,
32
], Steiner forest [
13
,
31
,
42
],
edge/vertex/set cover [
36
,
50
], and minimum makespan or minimum sum of completion
times when scheduling parallel machines [
5
,
10
,
9
,
6
,
8
,
11
]. In this list, three results
[
50
,
8
,
11
]fit into the framework of acyclic mechanisms, all other results develop Moulin
mechanisms with cross-monotonic cost shares.
Sharing Makespan Costs
In this thesis, the predominant sample application is the
minimum makespan problem, for which several results are known. For sharing the
makespan cost of either
n
identical jobs on
m
related machines or
n
arbitrary jobs on
m
identical machines, 2
m/
(
m
+1)-BB cross-monotonic cost-sharing methods are due to
Bleischwitz and Monien
[5]
. It is shown in the same paper that this is generally the best
that can be guaranteed under the constraint of cross-monotonicity. Brenner and Schäfer
[10]
later modified the cost-sharing methods for identical machines so that economic
efficiency is improved from
Ω
(
n
)-EFF to
O
(
logn
)-EFF. For arbitrary processing require-
ments and related machines, the best known cross-monotonic cost-sharing methods
achieve 2
d
-BB, where
d
is the number of different processing requirements [
5
]. This is
tight up to a factor of 2, since dis a lower bound [5].
1.5.2 Characterizing Collusion-Resistant Cost Sharing
Submodular Costs and Budget Balance
A complete characterization of the impact
of submodular costs on GSP and 1-BB was given by Moulin
[51]
: Any GSP mechanism
that is 1-BB with regard to submodular costs is a Moulin mechanism, or at least always
produces the same utilities as a Moulin mechanism. Conversely, for any submodular cost
function, a rich class of cross-monotonic 1-BB cost-sharing methods (and thus Moulin
mechanisms) always exists—including the marginal costs, the Shapley value [
74
], and
the egalitarian solution [
21
]. Interestingly, of all cross-monotonic cost-sharing methods,
the Shapley value is characterized by inducing the Moulin mechanism with the best
possible economic efficiency guarantee [
52
,
64
]. On the other hand, the egalitarian
solution is characterized by maximizing the probability that a given subset
S
of players
can afford the cost shares
ξ
(
S
); this result is obtained when assuming that the players’
valuations are independent random variables with a common distribution function
(under mild restrictions) [54].
Subadditive Costs and Economic Efficiency
Dobzinski et al.
[19]
studied the sce-
nario of a so-called excludable public good [
51
,
17
]where
C
(
S
) = 1 if
S6
=
;
and
C
(
;
) = 0.
They showed for any such cost-sharing problem with
n
players that no SP and
β
-BB
(
β≥
1) cost-sharing mechanism can guarantee social cost better than
Ω
(
logn
)times the
optimal social cost. The excludable-public-good case is a special instance of many natural
cost-sharing problems with subadditive optimal costs. This includes, e.g., makespan,
facility location, and rooted Steiner tree problems. Consequently,
Ω
(
logn
)-EFF is a lower
bound for all these cost-sharing problems.
11
Chapter 1 Introduction
Superadditive Costs and Singleton Mechanisms
Brenner and Schäfer
[11]
studied
several scheduling cost-sharing problems where the cost of a schedule is defined as the
sum of all players’ completion times. All problems they considered have superadditive
optimal costs (meaning that the union of disjoint sets is always more costly than sum of
the stand-alone costs), and the excludable-public-good case case therefore cannot occur.
In fact, Brenner and Schäfer
[11]
gave a simple subclass of acyclic mechanisms, called
singleton mechanisms, that guarantee 1-BB and constant-factor approximations of the
social cost (independent of the number of players), i.e., O(1)-EFF.
Intuitively, singleton mechanisms can be specified by a complete binary tree with
n
levels: Every node is labeled with a player, and every path from the source to a
leaf contains each player exactly once. Now applying the mechanism corresponds to
finding a path from the root to a leaf: Start at the root and initialize
Q
as the empty set.
Now proceed as follows: If the player at the current node can afford his marginal cost
C
(
Q∪{i}
)
−C
(
Q
), charge him this price, add him to
Q
, and go to the right successor
node. Otherwise, go to the left successor.
Indifference Rules and Cross-Monotonicity
Both Moulin and acyclic mechanisms
treat indifferent players in an extreme way, in that they are always served. This property
is referred to as upper continuity. For GSP mechanisms, an interesting characterization of
upper continuity is due to Immorlica et al.
[36]
: If a GSP mechanism is upper-continuous
then it has cross-monotonic cost shares. Consequently, it is a Moulin mechanism.
Non-Manipulability
Since in many mechanism-design scenarios it is unlikely that
players have unlimited means to communicate and make binding agreements with all
of their competitors, Serizawa
[70]
introduced and advocated relaxing GSP to effective
pairwise strategyproofness. This property is a little weaker than our “2-GSP” because it
means that a mechanism only needs to be resistant to pairs of defecting players if their
defection was stable (i.e., none of the two players could betray his partner to further
increase his utility). However, the models considered by Serizawa do not include the
cost-sharing scenario.
Besides the (coalitional) variants of strategyproofness, there are several other concepts
of non-manipulability. Satterthwaite and Sonnenschein
[66]
suggested a property called
(outcome) non-bossiness (ONB): If a single player changes his bid in a way so that his
own outcome does not change, then all other players should also get the same outcome
as before (hence, no player can “boss” others around). In an unpublished paper, Shenker
[72]
proved several results on the relationship between various forms of truthfulness,
non-bossiness, and other technical properties. His results are similar but different to
our work. With focus only on the cost-sharing model, several other relationships were
later studied by Mutuswami [55]. He introduced a variation of ONB called weak utility
non-bossiness (WUNB), meaning that if a single player changes his bid so that his utility
remains the same, then no other player may become better off. Mutuswami
[55]
showed
that SP and ONB together imply WGSP; moreover, SP, ONB, and WUNB together imply
GSP. Other variants of non-bossiness were also proposed by Deb and Razzolini
[17]
. For
12
1.6 Organization
scenarios when players are capable of side-payments, Schummer
[69]
studied bribe-
proof mechanisms, meaning that no player has an incentive to bribe another player into
submitting an untruthful bid. For the cost-sharing scenario, Schummer’s results imply
that collusion-resistance properties that include monetary transfers are too strong (see
also Section 5.2.1): They would rule out all but trivial mechanisms where each player’s
utility is completely independent of the other players’ actions.
General-Demand Cost Sharing
To the best of our knowledge, general-demand cost
sharing has previously only been considered by Moulin
[51]
, Devanur et al.
[18]
, and
Mehta et al.
[50]
. However, these works consider only SP and WGSP mechanisms,
respectively.
1.5.3 Outside the Realm of Cost Sharing
The Minimum-Makespan Problem
The discipline algorithmic mechanism design was
pioneered by Nisan and Ronen
[56]
roughly a decade ago. One of the optimization
problems the authors considered—based on the new idea that part of the input data
is held by selfish players—was minimizing the makespan when scheduling unrelated
parallel machines: In Nisan and Ronen’s work, the machines (and not the jobs as in
our cost-sharing applications) are controlled by selfish players and thus have to be
given monetary incentives to truthfully reveal the true time needed to process each
job. Until today, there is a huge gap between the best known lower and upper bounds
on the approximation ratio that “incentive-compatible” algorithms can achieve for this
problem. In fact, it is one of the central open problems in algorithmic mechanism design
whether “incentive-compatible” approximation is necessarily less powerful than “classical
approximation”. For a recent survey and further pointers into the literature we refer to
Roughgarden [63].
Small Coalitions and Solution Concepts
The idea of resistance to coalitions of only
bounded size, as described in Section 1.4.3, has been considered also from the perspective
of game-theoretic solution concepts: Andelman et al.
[2]
defined a
k
-strong equilibrium
so that it corresponds in our cost-sharing model to WGSP against coalitions of size at
most
k
. In the paper, the authors obtain results on the strong price of anarchy (i.e., the
worst-case loss in the system due to selfish behavior of the players) in job scheduling
and network creation games, depending on the maximum coalition size k.
1.6 Organization
Chapter 2 gives all technical preliminaries. We formally define cost-sharing problems,
the various notions of truthfulness, and all other desirable properties of cost-sharing
mechanisms. Afterwards, we give brief definitions of both VCG and Moulin mechanisms
and of the combinatorial optimization problems considered in this thesis.
13
Chapter 1 Introduction
Our main results are given in the remaining chapters of this thesis: In Chapter 3, we
introduce our new technique for designing GSP mechanisms, based on the idea of lexico-
graphic maximization of players’ utilities. In Chapter 4, we study cost-sharing models
without indifferences and develop our new egalitarian mechanisms. Afterwards, we
show in Chapter 5 that relaxing GSP to resistance only against coalitions of bounded size
does not allow for better mechanisms. Finally, we give generalized Moulin mechanisms
for the general-demand cost-sharing model in Chapter 6.
While Section 2.1 is a prerequisite for all what follows, readers with acquaintance
of mechanism design and combinatorial optimization may quickly flip through the
remaining two sections of Chapter 2. All further chapters may be read independently
from each other.
1.7 Prerequisites
Basic knowledge of game-theoretic concepts will help in comprehending explanations and
interpretations; however, all formal definitions necessary for our results are contained in
this thesis. For a general introduction to game theory from a (micro-)economic point of
view, we recommend Mas-Colell et al.
[47]
. Introductions to various aspects of game
theory with a bias towards computer science can be found in the book by Nisan et al.
[57]
. The essentials of mechanism design are presented in a concise and accessible
manner by Parkes [61].
While we likewise define all combinatorial optimization problems that we use in
our applications, we assume at least rudimentary acquaintance with the theory of
computational intractability and approximation algorithms. A standard reference for
NP
-
completeness is Garey and Johnson
[26]
. As a reference for approximation algorithms
we recommend the book by Hochbaum
[33]
as it discusses many of the optimization
problems studied in the application sections of this thesis.
1.8 Bibliographic Notes
Many of the results presented in this thesis are based on collaborative work with others,
and most have appeared in preliminary form in research papers. For this thesis, all joint
work has been revised, and all text and all proofs have been written only by me. This
thesis includes only results to which I contributed.
Chapter 3 is based on joint work with Yvonne Bleischwitz, Burkhard Monien, and
Karsten Tiemann. It has been published in the Proceedings of the 32nd International
Symposium on Mathematical Foundations of Computer Science (MFCS’07) [
9
]. Chapter 4 is
based on joint work with Yvonne Bleischwitz and Burkhard Monien. It has been published
in the Proceedings of the 3rd International Workshop on Internet and Network Economics
(WINE’07) [
8
]. Chapter 5 has appeared in the Proceedings of the 4th International
Workshop on Internet and Network Economics (WINE’08) [
67
]. A small part of Chapter 3
and all of Chapter 6 are joint work with Yvonne Bleischwitz and have been published
14
1.8 Bibliographic Notes
in Information Processing Letters 107(2), 2008 [
6
]and in the Proceedings of the 1st
International Symposium on Algorithmic Game Theory (SAGT’08) [7].
Since the focus of this thesis is on cost-sharing mechanisms, I did not include results
in other research areas that I also published during my PhD project. These publications
deal with the loss due to selfish behavior (the price of anarchy) in selfish routing and
competitive location scenarios. They appeared, as joint work, in the Proceedings of the
23rd International Symposium on Theoretical Aspects of Computer Science (STACS’06) [
1
],
the Proceedings of the 2nd International Workshop on Internet and Network Economics
(WINE’06) [
49
], the Proceedings of the 3rd International Workshop on Internet and Network
Economics (WINE’07) [
25
], and the Proceedings of the 33rd International Symposium on
Mathematical Foundations of Computer Science (MFCS’08) [48].
15
Chapter 2
Preliminaries
2.1 The Formal Model
2.1.1 Notation
Sets
The set of positive integers
{
1
,
2
,
3
,...}
is denoted by
N
, the set of non-negative
integers by
N0
. For
n,m∈Z
, we write
{n...m}
:=
{n,n
+1
,...,m}
, which is the
empty set if
m<n
. Moreover, [
n
]:=
{
1
...n}
. Given a finite set
S⊂N
and
i∈N
,
we let
rank
(
i,S
):=
|{j∈S|j≤i}|
be the rank of
i
in
S
. Moreover,
MINkS
:=
{i∈
S|rank
(
i,S
)
≤k}
is defined as the set of the
k
smallest elements in
S
, and
MAXkS
is
likewise defined as the set of the klargest elements in S.
Vectors
Given an arbitrary set
X
and a non-empty index set
I⊂N
, a vector
x∈XI
is a family of elements of
X
indexed by
I
. It is denoted
x
= (
xi
)
i∈I
. Given a subset
K⊆I
, we define the vector
xK
:= (
xi
)
i∈K
as the subfamily of
x
indexed by
K
. Similarly,
x−K
:= (
xi
)
i∈I\K
. Given two vectors
x∈XI
and
z∈XK
, we use the usual game-theoretic
notation (
x−K,z
)to denote the vector
y∈XI
with
yi
=
zi
for
i∈K
and
zi
=
xi
for
i∈I\K. We say two vectors x,y∈XIare K-variants if x−K=y−K.
To simplify notation, we will often omit curly brackets around singleton sets when it
is unambiguous to do so. E.g., we will write
i
-variants instead of
{i}
-variants or
K∪i
instead of
K∪{i}
. Moreover, we identify
Xn
with
X[n]
and apply the previous notation
also to vector-valued functions. E.g., if
f
:
A→Xn
is a function,
a∈A
,
f
(
a
) =
x
, and
I⊆[n], then fI(a) = xI.
Binary Relations
If
xi≤yi
for all
i∈I
, we write
x≤y
. If
x≤y
and
x6
=
y
, we write
x<y
. Moreover, if
xi<yi
for all
i∈I
, we write
xy
. If
x
is lexicographically no
larger than
y
, we write
xy
. Correspondingly, if
xy
but
x6
=
y
, we write
x≺y
.
The maximum element of on Xis denoted lexmaxX.
Others We denote the n-th harmonic number by Hn:=Pn
i=11
i.
2.1.2 Cost-Sharing Problems and Mechanisms
A (binary-demand) cost-sharing problem with
n∈N
players is specified by a cost function
C
: 2
[n]→R≥0
that associates all possible sets of served players to the incurred service
17
Chapter 2 Preliminaries
cost. A set of served players
Q∗⊆
[
n
]together with a cost distribution
x∗∈Rn
is called
an outcome. We denote player
i
’s true valuation for being served by
vi∈R
. Unless
otherwise stated, we assume quasi-linear utilities, i.e., player
i
’s utility for outcome
(Q∗,x∗)is vi·q∗
i−xiwhere q∗
i∈{0,1},q∗
i=1 :⇔i∈Q∗.
Definition 2.1.1.
Acost-sharing mechanism
M
= (
Q,x
)consists of a pair of functions
Q
:
Rn→
2
[n]
and
x
:
Rn→Rn
that associate any bid vector
b
to an outcome (
Q
(
b
)
,x
(
b
)).
Sometimes the set notation will be inconvenient, and we therefore implicitly define
q
:
Rn→{
0
,
1
}n
by
qi
(
b
) = 1 :
⇔i∈Q
(
b
). Given a cost-sharing mechanism
M
= (
Q,x
),
we write
Mi
(
b
):= (
qi(b),xi
(
b
)) and define
ui(b|vi)
:=
vi·qi(b)−xi
(
b
). When utilities
are quasi-linear,
ui(b|vi)
is hence player
i
’s utility for outcome (
Q
(
b
)
,x
(
b
)). Provided
that there is no confusion about the true valuation
vi
, we simply write
ui(b)
instead of
ui(b|vi).1We let M(b):= (M1(b),..., Mn(b)) and u(b):= (u1(b),...,un(b)).
Unless otherwise noted, we will always require three standard axiomatic properties in
this thesis:
•No positive transfers (NPT): Players never get paid, i.e., xi(b)≥0.
•
Voluntary participation (VP): When served, players never pay more than they
bid; otherwise, they are charged nothing, i.e., if
i∈Q
(
b
)then
xi
(
b
)
≤bi
, else
xi(b) = 0.
•
Consumer sovereignty (CS): Each player can bid in a way so that he is served,
regardless of the other players’ bids; i.e., there is a
b∞∈R≥0
such that if
bi≥b∞
then i∈Q(b).
VP and NPT imply that players may opt not to participate. Technically, these players
submit a negative bid. This property in conjunction with CS is sometimes referred to as
strong CS. It strengthens the collusion-resistance requirements and rules out otherwise
implausible and undesirable mechanisms [36].
2.1.3 Non-Manipulability
The basic notion of truthfulness is strategyproofness (SP). It requires a mechanism
M
to guarantee that for all possible valuation vectors
v∈Rn
, all players
i∈
[
n
], and all
i
-variants
b
of
v
it holds that
ui(b)≤ui(v)
. In this thesis, as well as in many related
works on cost sharing, a stronger notion is required that also ensures resistance against
coordinated manipulation.
Definition 2.1.2.
A cost-sharing mechanism
M
is group-strategyproof (GSP) if for all true
valuations
v∈Rn
and all non-empty coalitions
K⊆
[
n
]there is no
K
-variant
b
of
v
with
uK(b)>uK(v).
1Since the true valuation viis an “optional” argument, we separate it by “|” for better readability.
18
2.1 The Formal Model
We say a non-empty coalition
K⊆
[
n
]is GSP-successful at
v
(or simply a successful
coalition) if there is some
K
-variant
b
of
v
so that the coalition improves by deviating,
i.e.,
uK(b)>uK(v)
. With the corresponding modifications, we will use this terminology
also for other kinds of collusion resistance.
Aweaker notion of collusion resistance is obtained by strengthening the requirements
for successful coalitions:
Definition 2.1.3.
A cost-sharing mechanism
M
is weakly group-strategyproof (WGSP) if
for all true valuations
v∈Rn
and all non-empty coalitions
K⊆
[
n
]there is no
K
-variant
b
of vwith uK(b)uK(v).
Besides the coalitional variants of strategyproofness, there are several other concepts
of non-manipulability. In this thesis, we consider a property introduced by Satterthwaite
and Sonnenschein
[66]
: If a single player changes his bid in a way so that his own
outcome does not change, then all other players should also get the same outcome as
before.
Definition 2.1.4 (Satterthwaite and Sonnenschein [66]).
A cost-sharing mechanism
M
is (outcome) non-bossy (ONB) if for all players
i∈
[
n
]and all
i
-variants
b,b0∈Rn
it
holds that Mi(b)6=Mi(b0)or M(b) = M(b0).
Another notion of non-bossiness was later introduced by Mutuswami [55].
Definition 2.1.5 (Mutuswami [55]).
A cost-sharing mechanism
M
is weakly utility non-
bossy (WUNB) if for all true valuations
v∈Rn
, all players
i∈
[
n
], and all
i
-variants
b
of
vit holds that ui(b)6=ui(v)or u−i(b)≤u−i(v).
2.1.4 Cost-Sharing Methods
A straightforward idea for devising cost-sharing mechanisms is to separate a mechanism
into two parts: First, compute the set of served players depending on the bids. Then,
compute the cost shares only depending on the set of served players. In fact, it is known
that all GSP mechanisms have to work this way. This has been originally observed by
Moulin [51]. Later on, we will give a more general statement in Theorem 4.2.9.
Definition 2.1.6.
Acost-sharing method is a function
ξ
: 2
[n]→Rn
≥0
that associates
each set of players to a cost distribution, where for all
S⊆
[
n
]and all
i/∈S
it holds that
ξi(S) = 0.
Definition 2.1.7.
A cost-sharing mechanism
M
= (
Q,x
)is separable if there exists a
cost-sharing method ξso that x =ξ◦Q, i.e., for all b∈Rn:x(b) = ξ(Q(b)).
Given a cost-sharing method
ξ
and a bid vector
b
, we say a set of players
S⊆
[
n
]is
b
-feasible with regard to
ξ
if for all
i∈S
it holds that
ξi
(
S
)
≤bi
. Trivially, the empty set
is always b-feasible.
19
Chapter 2 Preliminaries
2.1.5 Dealing with Indifferences
A general rule of thumb for the design of truthful mechanisms is that a player’s payment
must not depend directly on his own bid. In particular, the following simple proposition
is well-known and a standard fact (see, e.g., Deb and Razzolini [16]).
Proposition 2.1.8 (Threshold Property).
A cost-sharing mechanism
M
= (
Q,x
)is SP if
and only if the following holds: For all
i∈
[
n
]and all
b−i∈R[n]\i
, there is a non-negative
threshold value
θi
(
b−i
)so that if
bi> θi
(
b−i
)then
i∈Q
(
b
), if
bi< θi
(
b−i
)then
i/∈Q
(
b
),
and if i ∈Q(b)then xi(b) = θi(b−i).
We call a player
i
indifferent at
b
if
bi
=
θi
(
b−i
), i.e., player
i
bids exactly his threshold
value. Clearly, the threshold property leaves open how to handle indifferent player. Two
extreme options are to always serve players who bid their respective threshold value
or to always reject them. Formally, these two extremes are captured by the notions
upper- and lower-continuity.
Definition 2.1.9.
A cost-sharing mechanism
M
= (
Q,x
)is called upper-continuous if for
all players
i
and all bid vectors
b
the following holds: If
i∈Q
(
b−i,z
)for all
z>bi
then also
i∈Q
(
b
). Likewise,
M
is called lower-continuous if the following holds: If
i/∈Q
(
b−i,z
)for
all z <bithen also i /∈Q(b).
We remark that an alternative property called upper semi-continuity is sometimes defined
in the literature (e.g., by Deb and Razzolini
[16]
), meaning that
qi
(
b−i,·
)is always upper
semi-continuous. That is, if
qi
(
b
) = 0 then
∃" >
0 :
∀z∈
(
bi−",bi
+
"
):
q
(
b−i,z
) = 0.
For mechanisms that fulfill the threshold property, upper semi-continuity and upper
continuity are clearly equivalent.
2.1.6 Budget Balance and Economic Efficiency
In typical applications, cost functions are implicitly defined by combinatorial optimization
problems, i.e.,
C
(
S
)is the value of a minimum-cost solution for the problem instance
that corresponds to the set of served players
S
. Due to the
NP
-hardness of many natural
problems, usually only approximations with cost
C0
(
S
)
≥C
(
S
)can be computed in
polynomial time, unless
P
=
NP
. Still, the budget of the mechanism should be reasonably
balanced:
Definition 2.1.10.
A mechanism
M
= (
Q,x
)is
β
-budget-balanced (
β
-BB) with regard
to actual costs C0and optimal costs C if for all bid vectors bit holds that
C0(Q(b)) ≤
n
X
i=1
xi(b)≤β·C(Q(b)),
where β≥1is a constant (independent of b).
20
2.1 The Formal Model
We define
β
-BB in the corresponding way also for cost-sharing methods. Since the
definition of budget balance is meaningless otherwise, we will always assume and only
consider problems with C(;) = 0.
For economic efficiency, the service cost and the rejected players’ valuations should
be traded off as good as possible. A measure for this trade-off is the social cost function
SC
: 2
[n]→R≥0
. Given actual costs
C0
and true valuations
v
, social costs are defined by
SC(S):=C0(S) + Pi/∈Smax{vi,0}.
Definition 2.1.11.
A mechanism
M
= (
Q,x
)is
α
-efficient (
α
-EFF) with regard to actual
costs C0and optimal costs C if for all true valuations vit holds that
SC(Q(v)) ≤α·min
P⊆[n](C(P) + X
i/∈P
max{vi,0}),
where α≥1is a constant (independent of v).
Note here that there are two potential sources for loss of economic efficiency: First, the
selected set Smay be suboptimal; and second, the actual cost C0(S)may be too high.
In the sections where polynomial-time computability is not an issue, we implicitly
assume that the actual costs
C0
coincide with the optimal costs
C
. When there is no
confusion, we will usually only write β-BB and α-EFF (and omit the “with regard to”).
2.1.7 Special Cost Functions
In many cases, costs exhibit a special structure that can be exploited when designing
cost-sharing mechanisms. In this thesis, we discuss costs with the following properties:
•
Symmetric costs: Costs depend only on the number of served players. That is, for
any two sets S,T⊆[n]with |S|=|T|:C(S) = C(T).
•
Subadditive costs: The cost of the union of two sets is never more than the sum of
the stand-alone costs. That is, for any two sets
S,T⊆
[
n
]:
C
(
S∪T
)
≤C
(
S
)+
C
(
T
).
•
Superadditive costs: The cost of the union of two disjoint sets is never less than
the sum of the stand-alone costs. That is, for any two sets
S,T⊆
[
n
]
,S∩T
=
;
:
C(S∪T)≥C(S) + C(T).
•
Submodular costs: The marginal costs of adding players to some set
S
are non-
increasing in the size of
S
. That is, for all players
i∈
[
n
]and any two sets
S⊆T⊆
[
n
]:
C
(
T∪{i}
)
−C
(
T
)
≤C
(
S∪{i}
)
−C
(
S
). It can be shown that this
condition is equivalent to that for all
S,T⊆
[
n
]:
C
(
S∪T
)+
C
(
S∩T
)
≤C
(
S
)+
C
(
T
).
•
Supermodular costs: Marginal costs are non-decreasing, i.e., for any two sets
S⊆T
:
C
(
T∪{i}
)
−C
(
T
)
≥C
(
S∪{i}
)
−C
(
S
). Equivalently, for all
S,T⊆
[
n
]:
C(S∪T) + C(S∩T)≥C(S) + C(T).
21
Chapter 2 Preliminaries
Note that subadditivity seems very natural in cost-sharing scenarios as it conveys the
idea of synergies between players. On the other hand, superadditivity may be seen as
the result of congestion. Clearly, sub- and supermodularity are special cases of sub- and
superadditivity, respectively.
2.2 Previous Design Techniques
For completeness, we give formal definitions of Vickrey-Clarke-Groves (VCG) mech-
anisms [
75
,
14
,
30
]and Moulin mechanisms [
51
,
52
]in this section. The acyclic-
mechanism framework by Mehta et al.
[50]
will be needed only in Chapter 4. We
therefore postpone its formal introduction to Section 4.4, where we give a slight general-
ization of the original algorithm for computing the outcome of these mechanisms.
2.2.1 Vickrey-Clarke-Groves Mechanisms
Recall that VCG mechanisms are the unique class of 1-EFF mechanisms [
29
]. They
are, of course, a general technique in mechanism design and in no way restricted to
cost-sharing problems. We give a definition using our cost-sharing notation here.
Definition 2.2.1.
A cost-sharing mechanism
M
= (
Q,x
)is a VCG mechanism with respect
to non-decreasing costs C if for all bid vectors band players i ∈[n]it holds that
Q(b)∈arg max
T⊆[n]¨X
j∈T
bj−C(T)«
xi(b) = C(Q(b)) −X
j∈Q(b)\{i}
bj+hi(b−i),
where hi:R[n]\i→Ris a function independent of bi.
By definition, if all players bid truthfully, a VCG mechanism selects a set of players with
maximum social welfare (sum of the included players’ valuations minus service cost).
Equivalently, it picks a set with minimum social cost.
We give a short explanation why VCG mechanisms are SP: Consider an arbitrary player
i∈
[
n
]. Note that his cost share
xi
(
b
)and thus also his utility do not directly depend on
his own bid
bi
, but only on the set of served players
Q
(
b
)and the other players’ bids
b−i
. Moreover, the function
hi
is irrelevant for incentive considerations, so we might as
well assume
hi≡
0. Let now
v
contain the true valuations. Player
i
’s cost share
xi
(
v
)
is defined in a way such that his utility
ui(v)
=
vi·qi(v)
+
xi
(
v
)is exactly equal to the
maximum social welfare. Hence, it holds for any i-variant bof vthat
ui(b) = vi·qi(b) + X
j∈Q(b)\{i}
bj−C(Q(b)) = X
j∈Q(b)
vj−C(Q(b)) ≤ui(v),
where the last inequality holds because, by Definition 2.2.1, the set of players
Q
(
v
)that
the mechanisms chooses for input vhas optimal social welfare. Consequently:
22
2.2 Previous Design Techniques
Proposition 2.2.2. VCG mechanisms are SP and 1-EFF.
The functions
hi
can be chosen so that the respective VCG mechanism satisfies NPT, VP,
and CS [
52
]. Unfortunately, however, VCG mechanisms do not provide any guarantees
for cost recovery: Consider the following simple example in the context of cost sharing:
Let there be two players, and consider the excludable-public-good case, i.e.,
C
(
S
) = 1 if
S6
=
;
and
C
(
;
) = 0. If
b1
=0 and
b2
=1 it holds that
x1
(
b
) = 0 due to NPT and VP, and
thus
h1
(
b2
) = 0. Similarly,
h2
(
b1
) = 0 if
b1
=1. Consequently, if
b1
=
b2
=1 we have
Q(b) = {1,2}but no cost is recovered.
Besides not being budget-balanced, one can also find simple examples where VCG
mechanisms are not resistant against collusion (see, e.g., Moulin and Shenker [52]).
2.2.2 Moulin Mechanisms
In the following, we formally introduce Moulin mechanisms, which are the most universal
technique for the design of GSP cost-sharing mechanisms. Their main characteristic is
having cross-monotonic cost shares.
Definition 2.2.3.
A cost-sharing method
ξ
is cross-monotonic if for all sets
A,B
with
A⊆B and all players i ∈A it holds that ξi(A)≥ξi(B).
Given cross-monotonic cost shares, Moulin mechanisms are characterized by serving
the largest feasible set of players. We remark here that the original definition by Moulin
[51]
and by Moulin and Shenker
[52]
was different and essentially algorithmic
2
(see
Algorithm 2.1). Yet we will see very soon that both definitions are equivalent.
Definition 2.2.4.
A cost-sharing mechanism
M
= (
Q,x
)is called a Moulin mechanism
if it has a cross-monotonic cost-sharing method
ξ
and for every bid vector
b
it holds that
Q(b)is the largest b-feasible set with regard to ξ.
Recall here that there is a unique largest
b
-feasible set because
ξ
is cross-monotonic:
Suppose both
A
and
B
were largest
b
-feasible sets, then also
A∪B
is
b
-feasible due to
cross-monotonicity. By assumption, it must then hold that
A
=
B
. Consequently, Moulin
mechanisms are well-defined. Equivalently to Definition 2.2.4, a mechanism
M
= (
Q,x
)
is a Moulin mechanism if and only if there is a cost-sharing method
ξ
so that for all
b∈Rn
it holds that
Q
(
b
) =
max{S⊆
[
n
]
|∀i∈S
:
bi≥ξi
(
S
)
}
and
x
(
b
) =
ξ
(
Q
(
b
)). It
turns out that this non-constructive definition allows for a much simpler proof of GSP
than the one given by Moulin and Shenker [52]:
Theorem 2.2.5. Moulin mechanisms are GSP.
Proof.
Let
v
contain the true valuations, let
K
be a non-empty coalition, and let
b
be a
K
-variant of
v
with
uK(b)≥uK(v)
. By assumption and due to VP, it then holds that
Q
(
b
)
2
To be exact, Moulin and Shenker defined
Q
(
b
)as the limit of the decreasing set sequence
Q0
:= [
n
],
Qj+1:={i∈Qj|bi≥ξi(Qj)}.
23
Chapter 2 Preliminaries
is
v
-feasible. Consequently, due to cross-monotonicity, also
Q
(
v
)
∪Q
(
b
)is
v
-feasible.
Since
Q
(
v
)is the largest
v
-feasible set by definition of Moulin mechanisms, this implies
Q
(
b
)
⊆Q
(
v
). Therefore, again by cross-monotonicity,
ui(b)≤ui(v)
for all
i∈Q
(
v
)and
thus uK(b) = uK(v).ut
The outcome of Moulin mechanisms can be efficiently computed in a straightforward
manner:
Input: cross-monotonic cost-sharing method ξ, bid vector b∈Rn
Output: set of players Q∈2[n], cost distribution x∈Rn
≥0
1: Q:= [n]
2: while ∃i∈Q:bi< ξi(Q)do
3: Q:={i∈Q|bi≥ξi(Q)}
4: x:=ξ(Q)
Algorithm 2.1: Moulin mechanisms
Lemma 2.2.6.
Let
ξ
be a cross-monotonic cost-sharing method. Then, for all bid vectors
b
,
Algorithm 2.1 computes the outcome of the respective Moulin mechanism.
Proof.
Denote by
S
the set chosen by the Moulin mechanism, and by (
Q∗,x∗
)the outcome
returned by Algorithm 2.1.
We show that no player dropped by Algorithm 2.1 can be contained in
S
. In detail,
we verify by induction that the invariant
S⊆Q
holds throughout the algorithm. Clearly,
this holds before the first iteration of the while-loop. Therefore, consider an arbitrary
iteration and fix all variable values immediately before line 3. Suppose
S⊆Q
holds.
Now assume, by way of contradiction, that there is a player
i∈S
who will be dropped in
the current iteration, i.e.,
bi< ξi
(
Q
). Then
bi< ξi
(
Q
)
≤ξi
(
S
), where the last inequality
holds due to cross-monotonicity. This is a contradiction to the fact that
S
is
b
-feasible.
Hence, any player removed in the current iteration cannot be contained in
S
, and the
invariant continues to hold also after line 3.
Now, since the exit condition of the while-loop was fulfilled in the last iteration,
Q∗
is
clearly
b
-feasible. Moreover,
S⊆Q∗
, and
S
is the largest
b
-feasible set. Consequently,
S=Q∗.ut
2.3 Optimization Problems
2.3.1 Scheduling Problems
Problem Definition
Machine scheduling is one of the fundamental areas in combina-
torial optimization, with a huge number of specific problem types (see, e.g., Brucker
[12]
). We therefore define in detail only the problem of minimizing the maximum
24
2.3 Optimization Problems
completion time when scheduling
n
jobs on
m
parallel related machines. This problem is
the predominant application considered in this thesis.
Each job
i
is characterized by its processing requirement
pi∈N
. Similarly, each machine
j
is characterized by its speed
sj∈N
, which is the amount of processing that machine
j
can finish within one unit of time. A schedule maps, in a non-overlapping way, each
job
i
to some machine
j
and a starting time
t
. The completion time of job
i
is then
Ci
:=
t
+
pi/sj
. The objective is to minimize the maximum completion time of any job,
i.e., to minimize
Cmax
:=
maxi∈[n]Ci
. A different term for
Cmax
is the makespan. In
makespan problems, the precise schedule is often not needed and we consider only an
allocation
a∈
[
m
]
n
of the jobs to machines. Then, assuming that jobs are processed
without gaps,
Cmax
=
maxj∈[m]Pi∈[n]|ai=jpi/sj
. Special cases we also consider are
identical jobs (all piare equal) and identical machines (all sjare equal).
In the context of cost sharing, each player corresponds to a job. The (optimal) service
cost
C
(
S
)is defined as the optimal value of the objective function for scheduling only
the jobs in set S. For instance, if the objective is to minimize the makespan, then
C(S):=min
a∈[m]S(max
j∈[m]Pi∈S|ai=jpi
sj).
We remark that the social cost
SC
(
S
) =
C0
(
S
) +
Pi/∈Smax{vi,
0
}
is the objective function
of the respective scheduling problem with rejection where
max{vi,
0
}
is the external cost
(typically called “penalty” in the optimization literature) for not serving player i∈[n].
The Three-Field Classification Scheme
In order to distinguish between the various
variants of scheduling problems, we make use of the three-field notation
α|β|γ
intro-
duced by Graham et al.
[28]
: The field
α
represents the machine environment: E.g., “1”
denotes a single machine, “
P
” identical parallel machines, “
Q
” related parallel machines,
and an optional
m∈N≥2
after
P
or
Q
denotes that the number of machines is
m
and
thus a constant. The field
β
defines job characteristics:
pi
=
p
means that all processing
requirements are identical,
ri
means that the jobs may have release times (earliest
starting time), and “
pmtn
” indicates that preemption is allowed. Finally, the field
γ
refers
to the objective function. E.g,
Cmax
is the makespan,
PCi
is the sum of completion times,
and PwiCiis the weighted sum of completion times.
Algorithms
Even
P
2
||Cmax
is an
NP
-hard problem [
45
]. A simple approximation
algorithm for
Q||Cmax
is the longest processing time first (LPT) algorithm [
27
], which
assigns jobs in the order of decreasing processing requirements. Every job is allocated
to the machine where its completion time is minimal (taking only into account the
jobs have been assigned already). The approximation ratio of LPT is upper bounded by
1+
p3/
3
≈
1
.
58 in the general case [
44
]and is exactly
4
3−1
m
in the special case of
identical machines (i.e.,
P||Cmax
)[
27
]. If jobs are identical (
Q|pi
=
p|Cmax
), LPT produces
optimal schedules [
12
]. Both for
P||Cmax
and
Q||Cmax
,polynomial-time approximation
schemes (PTAS) have been devised by Hochbaum and Shmoys [34, 35].
25
Chapter 2 Preliminaries
2.3.2 Bin Packing
Problem Definition
In a bin packing problem, we are given
n∈N
items with rational-
valued item sizes
ς1,...,ςn∈
(0
,
1]. The task is to pack them into a minimum number
of unit-sized bins. That is, we seek to partition [
n
]into a minimum number of subsets
B1,...,Bm
so that for each
j∈
[
m
]it holds that
Pi∈Bjςi≤
1. In the cost-sharing variant,
each player corresponds to an item and the (optimal) service cost
C
(
S
)is determined by
the minimum number of bins needed for packing the items in S.
Algorithms
With an easy reduction from 2-Partition [
26
], it is straightforward to see
that bin packing is
NP
-hard to approximate within a factor of less than
3
2
. A simple
approximation algorithm for bin packing is next fit decreasing (NFD), which assigns items
in the order of decreasing size. Every item is put into the same bin as the preceding one
if it fits; otherwise it is put into a new bin. Clearly,
NFD
can be implemented to run in
time
O
(
n·logn
). Its approximation guarantee is 2
·opt−
1; see, e.g., Hochbaum
[33]
. A
better approximation algorithm is first fit decreasing (FFD), which also assigns items in
the order of decreasing sizes but puts every item into the first bin where it fits. Using a
balanced tree as data structure, it can be implemented to run in time
O
(
n·logn
). The
tight bound of FFD is
11
9·opt
+
6
9
,[
20
]. It will prove useful later on that in the special
case where all item sizes are a power of 2, FFD produces optimal solutions [15].
2.3.3 Network Problems
Most other problems considered in the cost-sharing literature are network problems
(compare Section 1.5.1). While we occasionally refer to such problems, we will not
elaborate on them in our application sections. We only give two examples:
Steiner Tree
We are given a connected undirected graph
G
= (
V,E
)and a distinct root
node
r∈V
. Associated with each edge
e∈E
is a non-negative edge cost. The task is to
find a subgraph of
G
with minimum total edge cost so that for each
v∈V
there is a path
from
v
to
r
. In the cost-sharing variant, each player corresponds to a node and
C
(
S
)is
the cost of an optimal Steiner tree for the nodes in S(and r).
Metric Facility Location
We are given a bipartite graph
G
= (
V,E
). The partitioning
of
V
is given by
V
=
N∪F
, where
N
is the set of consumers and
F
is the set of facilities.
Each facility
v∈F
has opening costs
fv
, and each edge
e∈E
is associated with a
connection cost
ce
. The triangle inequality is fulfilled. The task is to find a set of facilities
to be opened and a mapping of all consumers to an open facility, under the objective to
minimize the sum of opening costs plus connection costs. In the cost-sharing variant,
each player corresponds to a consumer and
C
(
S
)is the cost of an optimal solution
including only the consumers from S.
26
Chapter 3
Lexicographic Maximization:
Beyond Cross-Monotonicity
3.1 Overview of Contribution
We devise a novel technique for the design of GSP mechanisms based on the following
idea: Given a cost-sharing method
ξ
, a set
S
is called feasible if every player
i∈S
can afford his cost share
ξi
(
S
). Now choose the set of players that lexicographically
maximizes the vector of all players’ utilities, over all feasible sets. In contrast, Moulin
mechanisms always choose the feasible set for which all players’ utilities attain their
maximum—which is only well-defined for cross-monotonic cost shares.
As an integral part of our new technique, we identify a property of cost-sharing
methods that is sufficient to induce GSP mechanisms under lexicographic maximization.
Cost-sharing methods that satisfy this property are called valid. Moreover, the resulting
cost-sharing mechanisms are called symmetric mechanisms. For cost functions that
are both symmetric and subadditive, we give a family of valid cost-sharing methods—
and hence GSP symmetric mechanisms—that guarantee (
p17
+1)
/
4-BB. Interestingly,
these cost-sharing methods assign at most two different cost shares, for any set of served
players. Nevertheless, the aforementioned budget-balance factor is the best that any valid
cost-sharing method can generally guarantee. Our result is a significant improvement
over cross-monotonic methods (and thus Moulin mechanisms), which can only guarantee
2-BB (see Section 1.5.1). All computation needed for our new technique can be carried
out efficiently: For the case of subadditive symmetric costs, we give algorithms both for
computing the cost-sharing method and for computing the outcome of the mechanism
itself. The total running time is O(n2).
As an application of our findings, we look at sharing the makespan cost when schedul-
ing jobs on related machines (
Q||Cmax
). Since the makespan cost function is not symmet-
ric if jobs are non-identical, we group all jobs by their processing requirements and then
use symmetric mechanisms separately for each group. Altogether, this yields a GSP and
(
d·
(
p17
+1)
/
4)-BB mechanism, where
d
is the number of different processing require-
ments. Note that this result beats the previously best-known budget balance of 2
d
(see
again Section 1.5.1). Together with a solution for the scheduling instance, the outcome
of the mechanism can be computed in time
O
(
n2
+
n·logm
). Hence, while symmetric
costs might seem to be of limited practical interest, our technique can still be used in
settings without symmetric costs. Unfortunately, though, the better budget balance com-
27
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
pared to Moulin mechanisms comes at a price: We give an example with identical jobs
on identical machines where our mechanisms achieve only Ω(
n
)-EFF, whereas Brenner
and Schäfer
[10]
gave Moulin mechanisms with
O
(
logn
)-EFF. Nevertheless, we regard
symmetric mechanisms as an important systematic first step for finding GSP mechanisms
that perform better than Moulin mechanisms.
Unfortunately, we have to leave open an exact characterization of when cost-sharing
methods induce GSP mechanisms under lexicographic maximization. Yet, we discuss a
possible direction towards finding properties that are not based on symmetric costs.
Towards understanding the limitations imposed by GSP itself, we study the impact of
symmetry of costs on 1-BB and GSP. An impossibility result can be shown already for this
restricted class of cost functions: Even for just 4 players, we prove that symmetry of costs
is not sufficient for the existence of a GSP and 1-BB mechanism. This is an exact bound,
because for the case of only 3 players we do give a family of 1-BB and GSP mechanisms.
3.2 Symmetric Mechanisms
In this section, we develop our novel technique for designing GSP cost-sharing mech-
anisms. As an intermediate step, we define precedence mechanisms. The main idea
(formalized below) is to always choose the set of players that lexicographically maxi-
mizes the utility vector. In contrast, Moulin mechanisms always perform a maximization
of all components in the utility vector, which is only possible in the case of cross-
monotonic cost shares. We remark that the name precedence mechanisms refers to the
following fact: As a necessity for lexicographic maximization, the mechanism serves the
players according to an order of precedence that is determined by the players’ indices.
Given a fixed cost-sharing method
ξ
, we define the benefit distribution with respect
to
ξ
as the function
Pξ
:
Rn×
2
[n]→Rn
, where
Pξ
(
v,S
):= (
vi·si−ξi
(
S
))
i∈[n]
. Here,
s∈ {
0
,
1
}n
is the service-allocation vector corresponding to
S
. Obviously, if the true
valuation vector is
v
and a mechanism with cost-sharing method
ξ
serves the player set
S
,
then
Pξ
(
v,S
)is the vector of all players’ utilities. By definition,
Pξ
(
b,S
)is non-negative
if and only if
S
is
b
-feasible. We denote the set of all
b
-feasible player sets by
F
(
b
). To
simplify notation, we omit the superscript ξwhen there is no confusion about ξ.
Now recall that the main ingredient of a Moulin mechanism
M
= (
Q,x
)is a cross-
monotonic cost-sharing method
ξ
, and
M
serves the maximal set
S
of players
i
who
can afford their corresponding
ξi
(
S
)—due to cross-monotonicity, a unique maximal set
always exists. Formally,
Q
(
b
) =
maxF
(
b
), i.e.,
Q
(
b
)is always the greatest element of
F
(
b
)ordered by
⊆
. Due to cross-monotonicity, the largest feasible set is also a utility
maximizer. Thus, an equivalent formalization of Moulin mechanisms using the above
definition of
P
is
Q
(
b
) =
max
(
argmaxS∈F(b){P
(
b,S
)
}
). The crucial idea is now the
following generalization.
Definition 3.2.1.
A mechanism
M
= (
Q,x
)with cost-sharing method
ξ
is called a prece-
dence mechanism if, for all bid vectors
b
, it chooses a set of players
S
so that the benefit
distribution
P
(
b,S
)is lexicographically maximal over all
b
-feasible sets
S
. Equivalently,
Q(b)∈arglexmaxS∈F(b){P(b,S)}.
28
3.2 Symmetric Mechanisms
For historic reasons [
9
], we take the lexicographic order with reversed significance in
Section 3.2: For instance, (1,2)(2,1). Likewise for sets, {2,4}{1,2,3}.
We remark that in this generality, precedence mechanisms only satisfy NPT and VP,
but not necessarily CS: As an example, let
n
=2,
ξ2
(
{
2
}
) = 1, and
ξ2
(
{
1
,
2
}
)
>
1. Now,
if player 2 bids more than 1, player 1 will not be served, regardless of his bid. Also note
that a precedence mechanism is not uniquely defined by its cost-sharing method because,
in general, arglexmaxS∈F(b){P(b,S)}contains more than one feasible set.
3.2.1 Symmetric Cost-Sharing Methods and Mechanisms
We define in this section a condition on cost-sharing methods that will turn out to be
sufficient to induce precedence mechanisms that satisfy also CS and that are GSP. We
first need:
Definition 3.2.2.
A cost-sharing method
ξ
is symmetric if for all
S,T⊆
[
n
]with
|S|
=
|T|
and all i ∈S, j ∈T with rank(i,S) = rank(j,T)it holds that ξi(S) = ξj(T).
Clearly, a symmetric cost-sharing method
ξ
is completely defined by
ξ
([1])
,...,ξ
([
n
]).
In order to increase readability, we will therefore slightly abuse notation and simply
write ξλ(p)to denote ξλ([p]). Hence, for any arbitrary S⊆[n]and i∈S, it holds that
ξi(S) = ξrank(i,S)(|S|).
Given a fixed
ξ
, we always implicitly define two vectors
l∈Rn
≥0
and
d∈
[
n
]
n
as
follows. For each cardinality
p∈
[
n
], we say
lp
:=
ξp
(
p
)is the
l
ow cost share and
dp
:=
|{λ∈
[
p
]
|ξλ
(
p
)
>lp}|
is the number of
d
isadvantaged players who pay more than
the low cost share. Every cost share ξλ(p)>lpis called a high cost share.
We call a contiguous range
{y...z}⊆
[
n
]of cardinalities with
dy
=0,
dz+1
=0 (or
z
=
n
), and
dλ>
0 for
λ∈ {y
+1
...z}
asegment. Note that
d1
=0 by definition. In
order to improve readability, we stick to the convention of denoting ranks (in sets) by
λ,µ,ν, players by i,j,k, and cardinalities by p,r,s,....
Definition 3.2.3.
A symmetric cost-sharing method
ξ
is valid if for every segment
{y...z}
and every cardinality p ∈{y...z}the following holds:
V1) Cost shares are always non-increasing in the rank, i.e., ξ1(p)≥···≥ξp(p) = lp.
V2)
Within a segment, low cost shares are constant. In general, they are non-increasing
in the cardinality, i.e., ∀r∈{y...z}:lr=lpand ∀r∈{z+1...n}:lr≤lp.
V3)
Within a segment, high cost shares are non-increasing in the cardinality. That is,
∀r∈{p...z}:ξ[dp](r)≤ξ[dp](p).
V4)
The number of players paying a low cost share is non-decreasing in the cardinality,
i.e.,
dp≤dp−1
+1. Moreover, if a high cost share has strictly decreased, all higher-
precedence ranks are set to the low cost share; i.e., ξλ(p)< ξλ(p−1) =⇒dp≤λ.
29
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
Example 3.2.4. We give two valid symmetric cost-sharing methods for illustration:
p lpdpξ1(p),...,ξp(p)
1 1 0 1
2 1 1 5,1
3 1 2 5,4,1
4 1 3 5,4,2,1
5 1 2 5,3,1,1,1
p lpdpξ1(p),...,ξp(p)
1 1 0 2
2 1 1 4,2
3 1 1 3,2,2
4 1 2 3,3,2,2
5 1 0 1,1,1,1,1
The symmetric cost-sharing method on the left side consists only of the single segment
{
1
...
5
}
, whereas the right method has the two segments
{
1
...
4
}
,
{
5
}
. If
S
=
{
2
,
4
}
, then
ξ(S) = (0,5,0,1,0)for the left method and ξ(S) = (0,4,0,2,0)for the right method.
Definition 3.2.5.
Let
ξ
be a valid symmetric cost-sharing method. The precedence mecha-
nism uniquely defined by
Q
(
b
):=
lexmax
(
arglexmaxS∈F(b){P
(
b,S
)
}
)is called a symmet-
ric mechanism.
Roughly speaking, valid symmetric cost-sharing methods satisfy a property reminiscent
to a “half-sided” version of cross-monotonicity:
Lemma 3.2.6.
Let
ξ
be a valid symmetric cost-sharing method. Inclusion of a lower-
precedence player never makes a higher-precedence player worse off, i.e., for every cardinality
p∈[n−1]and every rank λ∈[p]it holds that ξλ(p)≥ξλ+1(p+1).
Proof.
Let
p∈
[
n−
1]and
λ∈
[
p
]. If
λ≤dp
, then
ξλ+1
(
p
+1)
≤ξλ
(
p
+1)
≤ξλ
(
p
)due
to (V3) and (V1). If
λ > dp
, then
λ
+1
>dp+1
due to (V4) and
ξλ+1
(
p
+1) =
lp+1≤
lp=ξλ(p)due to (V2). ut
The previous property guarantees that symmetric mechanisms indeed fulfill our require-
ments for cost-sharing mechanisms:
Lemma 3.2.7. Symmetric mechanisms satisfy NPT, VP, and CS.
Proof.
CS is the only non-obvious property to show: Let
b
be a bid vector,
i∈
[
n
]be a
player, and
S⊆
[
n
]
\i
be an arbitrary set of players that does not contain
i
. Suppose
bi>maxp∈[n]ξ1
(
p
). We show that a symmetric mechanism would not choose
S
. There
are two cases:
•Case S=;or i<minS:
Define
S0
:=
S∪i
. Then
|S0|
=
|S|
+1 and for all
j∈S0\i
it holds that
rank
(
j,S0
) =
rank(j,S) + 1. Therefore, ξj(S0)≤ξj(S)due to Lemma 3.2.6.
•Otherwise:
Define
k
:=
max
(
S∩
[
i
]) and
S0
:= (
S\k
)
∪i
. Then
|S0|
=
|S|
and for all
j∈S0\i
it holds that
rank
(
j,S0
) =
rank
(
S
). Therefore,
ξj
(
S0
) =
ξj
(
S
)because
ξ
is a
symmetric cost-sharing method.
30
3.2 Symmetric Mechanisms
In both cases, we have ξi(S0)<biand for all j∈S0\ithat ξj(S0)≤ξj(S)≤bj. Hence,
S0is b-feasible and P(b,S0)P(b,S). This completes the proof. ut
Theorem 3.2.8. Symmetric mechanisms are GSP.
Proof.
Let
M
= (
Q,x
)be a symmetric mechanism. The proof of GSP is by way of
contradiction: Let
v
contain the true valuations, let
K
be a non-empty coalition, and
let
b
be a
K
-variant of
v
with
uK(b)≥uK(v)
. Suppose there is a player
i∈K
with
ui(b)>ui(v)
. We will show that the mechanism would not have chosen
Q
(
v
)for input
vthen.
Denote
S
:=
Q
(
v
),
l
:=
l|S|
, and
T
:=
Q
(
b
). Note that
T
is also
v
-feasible. Moreover,
let
{y...z}3|S|
be the segment that
|S|
is in. We partition
S
into
S0
:=
S∩
[
i
]and
S00
:=
S∩{i
+1
...n}
. Likewise, we partition
T
into
T0
:=
T∩
[
i
]and
T00
:=
T∩{i
+1
...n}
.
Finally, let s0:=|S0|,s00 :=|S00|,t0:=|T0|,t00 :=|T00|, and define
p:=minr∈{t0+s00 ...|T∪S00|}ξ[t0](r)≤ξ[t0](|T|). (3.2.9)
For any cardinality r∈{t0+s00 ...|T∪S00|}, define
Rr:=T0∪S00 ∪MINr−t0−s00 (T00 \S00).
Note that
T0
,
S00
, and
MINr−t0−s00 (T00 \S00)
are pairwise disjoint. Moreover, the definition
ensures that i∈Rrand |Rr|=r.
We show in the rest of the proof that for input
v
a precedence mechanisms would
rather have chosen
Rp
instead of
S
. We start by verifying that
p
is well-defined and
p∈{y...z}:
i) |T|≤z,vi> ξi(T)≥l, and |T∪S00|≤z
Let
{y0...z0}3|T|
be the segment that
|T|
is in. By way of contradiction, assume
that
|T|>z
. Then, all players in
S
and
T
have a true valuation of at least
ly0
, so
Ry0
is
v
-feasible and
P
(
v,Ry0
)
P
(
v,S
). A contradiction. Consequently,
|T|≤ z
and we also have vi> ξi(T)≥l.
Now, all players in
T∪S
have a true valuation of at least
l≥lz+1
. So if
|T∪S00|>z
then Rz+1would be v-feasible and P(v,Rz+1)P(v,S), a contradiction.
ii) y≤s00 and y≤|T|
Define
S00
+
:=
{j∈S00 |vj>l}
. By way of contradiction, assume
y>|S00
+|
. Define
R0
:=
S00
+∪i∪MAXy−|S00
+|−1
(
S\
(
S00
+∪i
)). Then
|R0|
=
y
,
i∈R0
, and
R0
is
v
-feasible,
but P(v,R0)P(v,S). This is a contradiction. Hence, y≤|S00
+|≤s00.
By way of contradiction, assume that
y>|T|
. Note that for all
j∈S00
+\T
it holds
that
uj(b)
=0
<uj(v)
, so
j/∈K
and
bi
=
vi
. Define
R00
:=
T∪MAXy−|T|
(
S00
+\T
).
Then
|R00|
=
y
because
y−|T|≤|S00
+\T|
. Moreover,
R00
is
b
-feasible, but
P
(
b,R00
)
P(b,T). A contradiction.
31
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
iii) pis well-defined and p∈{y...z}
Since
y≤ |T| ≤ |T∪S00| ≤ z
and due to (V3), we have that
ξ[t0]
(
|T∪S00|
)
≤
ξ[t0]
(
|T|
). Moreover, also
y≤s00 ≤t0
+
s00 ≤ |T∪S00|
, so
p
is well-defined and
p∈{y...z}.
Finally, we show that
Rp
is
v
-feasible. By
(3.2.9)
, it holds for all players
k∈T0
that
ξk
(
Rp
)
≤ξk
(
T
)
≤vk
. Therefore, fix an arbitrary player
k∈Rp\T0
. There are three
cases:
•Case p<|S|:
Since t0+s00 ≤p, this implies 1 ≤t0≤p−s00 <|S|−s00 =s0.
If
i/∈S
and
vi> ξs0
(
|S|
)then
R0
:= (
S\maxS0
)
∪i
is
v
-feasible but
P
(
v,R0
)
P
(
v,S
); hence,
i/∈S
implies
ξi
(
T
)
<vi≤ξs0
(
|S|
). On the other hand, if
i∈S
,
then ξi(T)< ξi(S) = ξs0(|S|).
In both cases, we get
ξt0(p)≤ξt0(|T|)due to (3.2.9)
=ξi(T)< ξs0(|S|)as explained
≤ξt0(|S|)due to (V1).
Now due to (V3), this is only possible if t0>dp. Consequently, ξk(Rp) = l.
•Case p≥|S|and p=t0+s00:
Then Rp\T0=S00; so ξk(Rp)≤ξk(S)because of p≥|S|and Lemma 3.2.6.
•Case p≥|S|and p>t0+s00:
Then ξ[t0](p)< ξ[t0](p−1)by (3.2.9) and therefore ξk(Rp) = ldue to (V4).
Hence,
Rp
is
v
-feasible. Now,
P
(
v,Rp
)
P
(
v,S
)because
ξk
(
Rp
)
≤ξk
(
S
)for all
k∈S00
and ξi(Rp)≤ξi(T). This is a contradiction and completes the proof. ut
3.2.2 Computing the Outcome of Symmetric Mechanisms
In the rest of this section, we show that the outcome of symmetric mechanisms can be
efficiently computed by Algorithm 3.1. We start by giving some intuition. Afterwards,
we provide a formal verification of correctness.
Algorithm 3.1 takes an optimistic approach in that it starts with the full player set
Q
= [
n
]and then removes players as long as
Q
is not
b
-feasible. We divide the algorithm
into three phases. The first phase consists of lines 1–2, and finds the largest set
Q
so
that all players in
Q
can afford
l|Q|
, i.e., the low cost share corresponding to cardinality
|Q|
. This phase is reminiscent to a Moulin mechanism—in particular, if high cost shares
would never be used (i.e., all
dr
’s were 0), then the symmetric cost-sharing method is
cross-monotonic, and phase 1 works exactly as the corresponding Moulin mechanism.
32
3.2 Symmetric Mechanisms
Input: valid symmetric cost-sharing method ξ, bid vector b∈Rn
Output: set of players Q∈2[n], cost distribution x∈Rn
≥0
1: Q:= [n]
2: while ∃i∈Q:bi<l|Q|do Q:={i∈Q|bi≥l|Q|}
3: I:={i∈Q|bi=l|Q|};H:=;;p:=max{r∈[|Q|]|dr=0}∪{0}
4: while p<|Q\I|do
5: j:=min(Q\H);I:=I\j
6: if bj≥ξj(Q)then H:=H∪j;p:=max{r∈[|Q|]|dr=|H|}
7: else Q:=Q\j
8: Q:=Q\MIN|Q|−pI;x:=ξ(Q)
Algorithm 3.1: Symmetric mechanisms
In the second phase, from line 3–7, least-precedence players are offered a high cost
share if including them for the low share
l|Q|
would require dropping higher-precedence
players who are not contained in
I
. Note here that the set
I
contains all remaining
players who are indifferent to being served for l|Q|, and Hcontains all players ialready
assigned a high cost share
ξi
(
Q
)
>l|Q|
. Moreover,
p
contains the maximum number of
players in
Q
that could be served when only the players in
H
pay a high cost share. Player
j
is of least precedence among all remaining players that have not yet been assigned a
high cost share.
The third phase consists only of line 8: Here, indifferent players are dropped for the
benefit of players with a lower precedence.
Lemma 3.2.10.
Suppose Algorithm 3.1 is given a valid symmetric cost-sharing method
and an arbitrary bid vector as input. The following holds:
i)
In every iteration of the while loop in the second phase, either
p
increases or
|Q|
decreases; but not both within the same iteration.
ii) The algorithm can be implemented to terminate after at most O(n2)steps.
iii)
Let
{y...z}
be the segment that
|Q|
is in after the first phase. Then,
|Q|
stays within
this segment until the algorithm terminates.
iv)
In line 6, if
bj≥ξj
(
Q
)is fulfilled, then
ξj
(
Q
)is the price that player
j
will be charged
at the end of the algorithm. In particular, in the third phase, all players in
H
are
charged a high cost share, and all players in
Q\H
are charged the low cost share
l|Q|
.
Proof. i)
We show the claim together with the following invariant, which holds
during and immediately after the second phase: “
∀r∈{p
+1
...|Q|}
:
|H|<dr
”.
Clearly, the invariant holds immediately after line 3, so consider an arbitrary
iteration and assume that the invariant holds immediately before line 5. Let the
updated variable values at the end of the same iteration (i.e., immediately after
line 7) be indicated by a star (∗). One of the following two cases happens:
33
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
•
In line 6, a player is added to
H
, i.e.,
|H∗|
=
|H|
+1. Since
p<|Q|
,
d|Q|>|H|
,
and due to (V4), there must be a cardinality
r∈{p
+1
...|Q|}
with
dr
=
|H∗|
.
It follows that p∗>pand ∀r∈{p∗+1...|Q|}:|H∗|<dr.
•In line 7, a player is removed from Q, i.e., |Q∗|<|Q|.
Clearly, the invariant continues to hold in both cases.
ii)
Every iteration of the while loop in the first phase decreases
|Q|
by at least 1.
Similarly, every iteration of the while loop in the second phase decreases
|Q\H|
by 1. Hence, the algorithm terminates. By sorting players’ bids, the first phase can
be implemented to take at most
O
(
n·logn
)steps. The second phase takes
O
(
n2
)
steps and the third phase O(n)steps.
iii)
In the second phase, in line 3,
p
is initialized with
p
:=
y
. During the rest of the
algorithm,
p
is non-decreasing,
|Q|
is non-increasing, and
p≤|Q|
is an invariant
throughout the algorithm. Together with the invariant from (i), this implies that
always |Q|∈{y...z}.
iv)
Consider an arbitrary iteration, immediately after line 7. By the invariant from (i),
it holds for all
r∈{p
+1
...|Q|}
that
|H|<dr
, and thus by (V3) and (V4) also that
ξ[|H|]
(
r
) =
ξ[|H|]
(
|Q|
). Moreover, after line 8, it holds that
|Q|
=
p
. By the previous
assignment in line 3 or 6, dp=|H|. This completes the proof. ut
Theorem 3.2.11.
Let
ξ
be a valid symmetric cost-sharing method. Then, for all bid
vectors b, Algorithm 3.1 computes the outcome of the respective symmetric mechanism.
Proof.
Consider an arbitrary input
ξ
and
b
. Denote by
S
the set chosen by the respective
symmetric mechanism, and by (
Q∗,x∗
)the outcome returned by Algorithm 3.1. Note
first that no player dropped in the first phase can be contained in
S
as otherwise
S
would
not be
b
-feasible. Hence, the loop invariant
S⊆Q
holds immediately before line 3.
Define l:=l|Q|.
Now consider the second phase. We show that, again, no player dropped here can be
contained in
S
. Therefore, consider an iteration of the while loop and fix all variable
values immediately before line 5. Assume that the loop invariant
S⊆Q
holds. Note that
|S|
and
|Q|
are in the same segment, due to Lemma 3.2.10 (iii). Let
j
:=
min
(
Q\H
)be the
player who is either dropped or added to
H
in this iteration. It holds that
l≤bj< ξj
(
Q
)
because
rank
(
j,Q
) =
|H|
+1=
dp
+1
≤d|Q|
. We only need to consider the case that
j
is
dropped in line 7, i.e.,
j/∈Q∗
; otherwise,
Q
will not be changed in this iteration and the
loop invariant S⊆Qcontinues to hold.
By way of contradiction, assume
j∈S
and thus
ξj
(
S
)
< ξj
(
Q
). This implies
ξj
(
S
) =
l
because otherwise
rank
(
j,S
)
≤d|S|
and therefore
ξj
(
S
) =
ξrank(j,S)
(
|S|
)
≥
ξrank(j,S)
(
|Q|
)
≥ξrank(j,Q)
(
|Q|
) =
ξj
(
Q
)
.
Here the inequalities are due to (V3) and
(V1). Since
S⊆Q
and
ξj
(
S
) =
l< ξj
(
Q
), it must hold that
d|S|≤ |H|
. There-
fore, we have
|S| ≤ p
=
max{r∈
[
|Q|
]
|dr
=
|H|}
. Since
p<|Q\I|
, there are
r>p−|H|
players
k∈Q∩{j
+1
...n}
with
bk>l
. Note here that (V4) further implies
34
3.3 Symmetric Mechanisms for Symmetric Subadditive Costs
r>p−|H|≥|S|−d|S|≥|S∩{j...n}|
. Among these
r
players, at least one player
k
is not
included in
S
. Now,
R
:= (
S\j
)
∪k
is
b
-feasible and
P
(
b,R
)
P
(
b,S
). A contradiction.
Finally, consider phase 3 and fix the variable values immediately before line 8. Suf-
ficiently many players
k
with
bk
=
l
are removed so that all players
k∈Q\H
with
bk>l
receive the service for
l
. Since
S⊆Q
, this implies
P
(
b,Q∗
)
P
(
b,S
). It remains
to be shown that
Q∗
is lexicographically maximal. This holds because only the least
necessary number of indifferent players
|Q|− p
are removed and these players are of
least precedence within I.ut
3.3 Symmetric Mechanisms for Symmetric Subadditive Costs
In this section, we devise an efficient algorithm for computing
p17+1
4≈
1
.
28-BB valid
symmetric cost-sharing methods for arbitrary symmetric subadditive costs
C
: 2
[n]→
R≥0
. For simplicity of notation, we will specify the costs
C
by a function (or array)
c:[n]→R≥0. That is, for any non-empty set Swe have C(S) = c(|S|).
The valid symmetric cost-sharing methods we propose are of a particularly simple
structure in that for any set of served players, at most two different cost shares are
assigned. Therefore, the cost-sharing methods can be specified in a succinct way.
Definition 3.3.1.
Atwo-price cost-sharing form (2P-CSF) is a 4-tuple (
n,d,h,l
)with
n∈N
,
d∈
[
n
]
n
, and
h,l∈Rn
≥0
that specifies a symmetric cost-sharing method
ξ
as follows:
For every cardinality
p∈
[
n
]and every rank
λ∈
[
p
], define
ξλ
(
p
):=
hp
if
λ≤dp
and
ξλ(p):=lpotherwise.
Corresponding to the previous section, we denote by
lp
the low cost share and by
hp
the high cost share used for cardinality
p∈
[
n
]. Moreover,
dp
specifies the number of
disadvantaged players who pay the high cost share. Note that only the least-precedence
players pay the high cost share. We say a 2P-CSF is valid if it specifies a valid symmetric
cost-sharing method.
3.3.1 Computing Two-Price Cost-Sharing Forms
We compute 2P-CSFs with Algorithm 3.2. The intuition is as follows: The minimum
average per-player cost up to the respective cardinality is used as the low cost share.
More precisely, we also multiply with the budget-balance factor to leave some flexibility
when the average cost increases for a larger cardinality. Now, if assigning all players the
low cost share recovers the total cost, no player has to pay a high cost share.
Otherwise, the general idea is to let the least-precedence player pay the rest. However,
the high cost share cannot increase within a segment (V3), and therefore it might have
to be assigned to two players. Now, as long as there is more than one disadvantaged
player, the high cost share cannot change (V4). Hence, special care is needed to ensure
that the high cost share is neither too large nor too small.
Note that “
∞
” is used as a placeholder for any “sufficiently large” value in order to
simplify the presentation (a value strictly larger than β·c(s)is sufficient).
35
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
Input: function c:[n]→R≥0that specifies symmetric costs C,
BB parameter β≥p17+1
4
Output: 2P-CSF (n,d,h,l)
1: d1:=0; h1:=∞;l1:=β·c(1);s:=1
2: for p:=2,...,ndo
3: if β·c(p)
p≤lsthen dp:=0; hp:=∞;lp:=β·c(p)
p;s:=p
4: else
5: dp:=1; lp:=lp−1;hp:=min{β·c(p)−(p−1)·lp,hp−1}
6: if p·lp≥c(p)then
7: dp:=0; hp:=∞
8: else if hp+ (p−1)·lp<c(p)then
9: dp:=2
10: else if 2·c(s)>hp+ (p−1)·lp>(β2−β)·c(s)+(p−1)·lp≥c(p)then
11: hp:= (β2−β)·c(s)
Algorithm 3.2: Two-price cost-sharing forms
Lemma 3.3.2.
Suppose Algorithm 3.2 is given symmetric subadditive costs
c
:[
n
]
→R≥0
as input. Then, the output 2P-CSF (
n,d,h,l
)is valid and the algorithm terminates after
O(n)steps.
Proof. Consider an arbitrary cardinality p∈[n]. Clearly, dp∈{0,1,2}and d1=0.
(V1) lp<hp
This can be verified by induction. The base case
p
=1 holds as
h1
=
∞> β ·c
(1) =
l1
. For the induction step
p−
1
→p
, assume that
lp−1<hp−1
. Since there
is nothing to show if
hp
=
∞
, assume
hp<∞
. Then the condition in line 3
evaluated to false for cardinality
p
. Let
s
:=
max{r∈
[
p−
1]
|β·c(r)
r
=
lp}
be
the last cardinality previous to
p
for which the lower cost share was set in line 1
or line 3. Now,
c(p)
p>c(s)
s
=
ls
β
and
lp
=
lp−1
=
···
=
ls
. In line 5,
hp
was set
for the first time, either to
hp−1>lp−1
=
lp
or to
β·c
(
p
)
−
(
p−
1)
·lp>lp
. If
hp
was set again in line 11, then
p·lp<c
(
p
)because line 6 evaluated to false
and (
β2−β
)
·c
(
s
)+(
p−
1)
·lp≥c
(
p
)because line 10 evaluated to true. Hence,
hp= (β2−β)·c(s)≥c(p)−(p−1)·lp>lp.
(V2) lp≤lp−1and (lp<lp−1=⇒dp=0)
This holds since lp6=lp−1only if lpwas set in line 3.
(V3) hp>hp−1=⇒dp=0
This holds since line 5 ensures that from
dp>
0 it follows that
hp≤hp−1
. Note
here that in line 11, the value of
hp
was not increased due to the condition in
line 10.
36
3.3 Symmetric Mechanisms for Symmetric Subadditive Costs
(V4) dp≤dp−1+1 and (hp<hp−1=⇒dp≤1)
This is fulfilled trivially if
dp≤
1. So assume
dp
=2. Then
hp<c
(
p
)
−
(
p−
1)
·lk
because line 8 evaluated to true in iteration
p
and thus
hp
=
min{β·c
(
p
)
−
(
p−
1)
·
lp,hp−1}
=
hp−1
. Now assume
dp>dp−1
+1, i.e.,
dp−1
=0. Then,
hp
=
hp−1
=
∞
,
a contradiction to hp<c(p)−(p−1)·lp.
It is trivial to see that the running time of the algorithm is O(n).ut
Theorem 3.3.3.
Given symmetric, subadditive, and non-decreasing costs
c
:[
n
]
→R≥0
and an arbitrary parameter
β≥p17+1
4
, Algorithm 3.2 efficiently computes a valid and
β-BB 2P-CSF (n,d,h,l).
Proof.
Due to Lemma 3.3.2, it only remains to be shown that (
n,d,h,l
)is
β
-BB. Let
γ
:[
n
]
→R≥0
,
γ
(
p
):=
dp·hp
+ (
p−dp
)
·lp
, be the recovered cost. Consider now an
arbitrary cardinality
p∈
[
n
]. Like in iteration
p
of the algorithm, let
s
:=
max{r∈
[
p
]
|
β·c(r)
r
=
lp}
be the last cardinality previous or equal to
p
for which the lower cost share
was set in lines 1 or 3.
•Case dp=0:
For the upper bound, note that
γ
(
p
) =
p·ls≤p·β·c(p)
p
=
β·c
(
p
). The lower
bound γ(p)≥c(p)holds by line 6.
•Case dp=1:
Since the value of
hp
is never increased in line 11, it holds due to line 5 that
hp≤β·c
(
p
)
−
(
p−
1)
·lp
and thus
γ
(
p
) =
hp
+ (
p−
1)
·lp≤β·c
(
p
). Furthermore,
since the condition in line 8 evaluated to false and since the value of
hp
is only
decreased in line 11 to (
β2−β
)
·c
(
s
)if line 10 evaluated to true, it holds that
γ(p)≥c(p).
Before considering the case dp=2, we need two general observations: First, we show
hp≥(β2−β)·c(s). (3.3.4)
Obviously,
(3.3.4)
holds when
hp
=
∞
. Therefore assume
hp<∞
, i.e.,
dp>
0 and
s<p
.
By line 6 we then also have
c
(
p
)
>p·lp> β ·c
(
s
). Consequently,
β·c
(
p
)
−
(
p−
1)
·lp>
β·c
(
p
)
−c
(
p
)+
lp>
(
β−
1)
·c
(
p
)
>
(
β2−β
)
·c
(
s
)
.
Hence, by line 5, if
hp−1≥
(
β2−β
)
·c
(
s
),
then also hp≥(β2−β)·c(s). Since pwas chosen arbitrarily, this proves (3.3.4).
Next, we show
c(p)≤2·c(s). (3.3.5)
Let
t
:=
min
(
{r∈{p
+1
...n}|β·c(r)
r≤lp}∪{n
+1
}
)be the next cardinality after
p
for
which the lower cost share was set in lines 1 or 3 (or
t
=
n
+1 if
s
is the largest such
cardinality). Then
s≤p<t≤
2
s
. Otherwise, if
t>
2
s
, then
t
would not be minimal
due to
c(2s)
2s≤2·c(s)
2s
=
c(s)
s
because of subadditivity. Since
c
is also non-decreasing,
c(p)≤c(t)≤c(s) + c(t−s)≤2·c(s).
37
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
•Case dp=2:
Define
p0
:=
max{r∈
[
p
]
|dr
=1
}
, and let
y
:=
max{r∈
[
p
]
|dp
=0
}
be the
start of the segment that
p
is in. Then,
s≤y<p0<p
,
dy+1
=1,
dp0+1
=2, and
hy+1≥hp=hp0. We show
2·c(s)>hy+1+y·ly+1. (3.3.6)
By way of contradiction, assume 2
·c
(
s
)
≤hy+1
+
y·ly+1
. Then,
c
(
p0
+1)
≤c
(
p
)
≤
2
·c
(
s
)
≤hy+1
+
y·ly+1
due to
(3.3.5)
. Observe now that the value
hr
+(
r−
1)
·lr
is increasing in
r∈ {y
+1
... p0
+1
}
, because lines 3, 6, 8, and 10 evaluated to
false for iterations
y
+1
,..., p0
. This is a contradiction to
dp0+1
=2 and to the
fact that line 8 must have evaluated to true in iteration
p0
+1. Therefore,
(3.3.6)
holds.
As a next step, we show
hp= (β2−β)·c(s). (3.3.7)
Recall
(3.3.4)
and assume, by way of contradiction, that
hp>
(
β2−β
)
·c
(
s
). Since
hy+1≥hp
, line 10 must have evaluated to false for cardinality (
y
+1). Since
the other two inequalities of line 10 are fulfilled according to
(3.3.6)
and by our
assumption
hp>
(
β2−β
)
·c
(
s
), it must hold that (
β2−β
)
·c
(
s
)+
y·ly+1<c
(
y
+1).
This implies
β2·c
(
s
) =
β2·c
(
s
)
−β·c
(
s
)+
s·ly+1≤
(
β2−β
)
·c
(
s
)+
y·ly+1<c
(
y
+1)
.
Moreover, by definition of
y
and due to line 5, we have
hy+1
=
β·c
(
y
+1)
−y·ly+1
.
Then, however, 2
·c
(
s
)
< β3·c
(
s
)
< β ·c
(
y
+1) =
hy+1
+
y·ly+1
. This is a
contradiction to (3.3.6), which then proves (3.3.7)
Now, we get as lower bound
γ(p) = 2hp+ (p−2)·lp
=2hp+ (p−2)·β·c(s)
sdue to the definition of s
≥(2β2−β)·c(s)due to (3.3.7) and p−2≥s
≥2·c(s)due to β≥p17+1
4
≥c(p)due to (3.3.5).
For the upper bound, note that β·c(s)<p·lp<c(p)due to line 6. Hence,
γ(p) = 2hp+ (p−2)·lp
=hp−lp+hp+ (p−1)·lp
<hp+c(p)due to line 8
= (β−1)·β·c(s) + c(p)due to (3.3.7)
<(β−1)·c(p) + c(p)as explained
=β·c(p).ut
38
3.3 Symmetric Mechanisms for Symmetric Subadditive Costs
3.3.2 Lower Bound on the Performance of Symmetric Mechanisms
We show that
p17+1
4
-BB is in general the best that can be achieved by valid symmetric
cost-sharing methods.
Theorem 3.3.8.
For all
" >
0, there is a symmetric, subadditive, and non-decreasing cost
function c for which no valid p17+1
4−"-BB symmetric cost-sharing method exists.
Proof.
Fix
β
:=
p17+1
4
, let 0
< " ≤β−
1, and set
α
:=
β−"
. Additionally, let
p,l∈N
with
l>logβ
β−1
"
and
p>(l+1)·α
"
=
(l+1)·β
"−
(
l
+1). Set
r
:=
p
+
l
+1 and
n
:=
r
+1
and consider function cdefined below:
k1··· p p +1p+2··· p+l r n
c(k)1··· 1β−β−1
β1β−β−1
β2··· β−β−1
βlβ2
Clearly,
c
is subadditive and non-decreasing. By way of contradiction, assume there
is a valid symmetric cost-sharing method
ξ
which is
α
-BB. For all
s∈
[
n
], we define
γ
(
s
):=
ds·hs
+ (
s−ds
)
·ls
. Moreover, let
hs
:=
ξ1
(
s
)be the largest cost share used for
cardinality s.
The idea is the following: It can be shown that
dr≥
1 and
dn
=
dr
+1. Let
y
:=
max{s∈
[
r−
1]
|ds
=0
}
be the start of the segment that cardinality
n
is in. Then
y<r
and
hn≤hy+1
due to (V3). By case analysis,
hy+1≤α·c
(
y
+1)
−c
(
y
)
< β2−β
. Thus
γ
(
n
)
≤hn
+
α·β <
2
·β2−β
=2=
c
(
n
), a contradiction to
α
-BB. In detail, we can show
the following:
•dr≥1: Otherwise, dr=0 and we would obtain a contradiction to α-BB:
γ(r) = r·lr≤r·lp≤r·α
p=α·1+l+1
p< α ·1+"
α=β=c(r).
•dn=dr+1: Otherwise, dn≤drand we again obtain a contradiction to α-BB:
γ(n)≤γ(r) + ln≤α·β+ln< β2+α
p
< β2+"
l+1≤β2+"
2≤β2+β−1
2<2.
•
Bounds on
hy+1
: Due to
α·c
(
y
+1)
≥γ
(
y
+1) =
hy+1
+
γ
(
y
)
≥hy+1
+
c
(
y
), it
holds that hy+1≤α·c(y+1)−c(y). There are three cases:
–Case y∈[p−1]: Then
hy+1≤α·c(y+1)−c(y) = α−1< β2−β.
39
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
–Case y∈{p... p+l−1}: Let s:=y+1−p. Then
hy+1≤α·c(y+1)−c(y)
=α·c(p+s)−c(p+s−1)
=α·β−β−1
βs−β−β−1
βs−1
< β2−β−1
βs−1−β−β−1
βs−1
=β2−β.
–Case y=p+l=r−1: Then
hy+1≤α·c(r)−c(r−1) = α·β−β−β−1
βl
< α ·β−(β−") = α·(β−1)< β2−β.ut
3.3.3 What Can Be Achieved with One Price?
The previous result on two-price cost-sharing forms gives rise to the question what can
be achieved using only one price. In the following, we say a symmetric cost-sharing
method
ξ
is
β
-uniform with regard to costs
c
if for every cardinality
p∈
[
n
]and every
rank
λ∈
[
p
]it holds that
ξλ
(
p
):=
β·minr∈[p]{c(r)
r}
. Clearly, the definition ensures that
ξ
is cross-monotonic. Note that if
c
is submodular, then
c(r)
r
is non-increasing in
r
, and
the 1-uniform cost-sharing method is also 1-BB. Moreover, it coincides with the Shapley
value [71]and the egalitarian solution [22].
Lemma 3.3.9.
For every symmetric, subadditive, and non-decreasing cost function
c
, the
2-uniform cost-sharing method ξis cross-monotonic and 2-BB.
Proof.
Let
µ
:[
n
]
→R≥0
so that
µ
(
p
)is the unique cost share for cardinality
p
. Fix now
an arbitrary
p∈
[
n
]. The upper bound
p·µ
(
p
)
≤
2
·c
(
p
)holds by definition. Now, since
for all cardinalities
r≤n
2
we have
c(2r)
2r≤2·c(r)
2r
=
c(r)
r
due to subadditivity, we get that
µ
(
p
) = 2
·c(r)
r
for some
r∈ {dp
2e... p}
. Furthermore, since 2
·c
(
dp
2e
)
≥c
(
p
), we have
that p·µ(p) = p·2·c(r)
r≥2·c(dp
2e)≥c(p), which proves the lower bound. ut
In the following, we show that 2-BB is in fact a lower bound for all GSP “one-price”
cost-sharing mechanisms.
Lemma 3.3.10.
For any
" >
0, there is a symmetric, subadditive, and non-decreasing cost
function for which no GSP and (2
−"
)-BB mechanism exists that always assigns all served
players the same price.
40
3.4 Applications
Proof.
Let
"∈
(0
,
1]and
n∈N
with
n>2
"
. Consider the cost function
c
(
p
):=1 for all
p∈
[
n−
1]and
c
(
n
):=2. By way of contradiction, assume there is a GSP mechanism
M
= (
Q,x
)that is (2
−"
)-BB and charges all players equally. Since
M
is GSP, it is
separable (see Moulin
[51]
or Theorem 4.2.9). Denote the equal cost share used for
player set Sby µ(S).
We first observe that for each
i∈
[
n
]:
µ
([
n
]
\i
)
< µ
([
n
]). Otherwise, there is a player
i
so that
µ
([
n
]
\i
)
≥c(n)
n
=
2
n
and then (
n−
1)
·µ
([
n
]
\i
)
≥
2
−2
n>
2
−"
, which is a
contradiction to (2−")-BB.
Now assume the true valuations
v
are given by
v1
=
v2
=
µ
([
n
]) and
vi
:=
b∞
for all
other players
i
. It has to hold that
Q
(
v−1,b∞
)=[
n
]
\
2. Otherwise, if
Q
(
v−1,b∞
)=[
n
],
player 2 could help all other players by bidding
−
1. Correspondingly,
Q
(
v−2,b∞
) =
[n]\1.
Now there are four possibilities for
Q
(
v
): If
Q
(
v
) = [
n
],
Q
(
v
) = [
n
]
\
1, or
Q
(
v
) =
[
n
]
\{
1
,
2
}
, then player 1 could improve by bidding
b∞
. Correspondingly, if
Q
(
v
)=[
n
]
\
2,
then player 2 could improve by bidding b∞. A contradiction to GSP. ut
3.4 Applications
In this section, we consider the problem of sharing the makespan cost when scheduling
n
jobs on
m
parallel machines, where there is a one-to-one correspondence between
players and jobs. Recall that the cost
C
(
S
)is defined as the maximum completion time
in an optimal schedule for the jobs in S(see Section 2.3.1).
It is a simple observation that
C
is a subadditive function. Moreover, if jobs are
identical, then
C
is symmetric and
C
(
S
)can be computed in time
O
(
n·logm
)using the
LPT (longest processing time first) algorithm [27].
If jobs are not identical,
C
is not symmetric anymore and an optimal schedule is
in general
NP
-hard to compute. However, to keep finding a solution computationally
tractable, we want algorithms to be polynomial-time computable in the size of the
scheduling instance plus the players’ bids. We therefore need to resort to approximation
algorithms. For any algorithm ALG that computes feasible schedules, we define
CALG
(
S
)as
the makespan of the schedule that ALG computes for the jobs in
S
. The objective is now
to find mechanisms that are
β
-BB with regard to the actual cost
CALG
and the optimal
cost C.
3.4.1 Makespan Minimization with Identical Jobs
Theorem 3.4.1.
For sharing the makespan cost when scheduling
n
identical jobs on
m
related machines (
Q|pi
=
p|Cmax
), there is always a valid
p17+1
4
-BB 2P-CSF. The outcome
of the corresponding symmetric mechanism, together with a schedule for the served players,
can be computed in time O(n2+n·logm).
Proof.
Since the optimal costs are symmetric, subadditive, and non-decreasing, it is
sufficient to apply Theorem 3.3.3. Consider now the running time:
c
(1)
,...,c
(
n
)can be
41
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
computed by a single run of LPT, which takes time
O
(
n·logm
). Afterwards, computing the
corresponding 2P-CSF with Algorithm 3.2 can be done in time
O
(
n
). Finally, computing
the outcome of the symmetric mechanism with Algorithm 3.1 takes time
O
(
n2
). Summing
up yields the desired result. ut
Lemma 3.4.2.
For sharing the makespan cost when scheduling
n
identical jobs on
m
identical machines (
P|pi
=
p|Cmax
), there is always a 1-BB 2P-CSF that can be computed in
time O(n).
Proof.
Define the 2P-CSF (
n,d,h,l
)as follows: For cardinality
p∈
[
n
], define
σ∈N,τ∈
[m−1]0by p=σ·m+τ.
•If p∈[m], let dp:=0, hp:=∞, and lp:=1
p.
•If p>mand τ=0, let dp:=0, hp:=∞, and lp:=1
m.
•Otherwise, let dp:=1, hp:=1−τ
m, and lp:=1
m.
It can easily be verified that this 2P-CSF is valid and 1-BB. We remark that the same
2P-CSF can be obtained by applying Algorithm 3.2 and dividing all cost shares by
p17+1
4
.
ut
Lemma 3.4.3.
There is a family of scheduling instances with
n
identical jobs and
m
identical machines, so that any symmetric mechanism driven by the 2P-CSF computed by
Algorithm 3.2 (or as in Lemma 3.4.2) cannot guarantee an economic efficiency better than
Θ(n).
Proof.
Let
n
=2
m
. Hence, the cost function
C
is specified by
c
(
p
) = 1 for cardinalities
p∈
[
m
]and
c
(
p
) = 2 for
p∈{m
+1
...
2
m}
. Let (
n,d,h,l
)as in Lemma 3.4.2 (as noted
before, this is the same 2P-CSF as the one computed by Algorithm 3.2 but divided by
p17+1
4). We have:
p1 2 ... m m +1m+2 ... 2m
c(p)1 1 ... 1 2 2 ... 2
ξ(p)11
2,1
2... 1
m,... 1, 1
m,... m−1
m,1
m,... ... 1
m,...
Suppose the true valuations are
v
= (
1
m−",2
m−",...,
1
−",1
m
+
",..., 1
m
+
"
)for some fixed
" >
0. Let
M
= (
Q,x
)be the symmetric mechanism defined by (
n,d,h,l
). Then
Q
(
v
) =
{m
+1
...m}
,
C
(
Q
(
v
)) = 1, and
Pm
i=1vi
=
m+1
2−m"
, hence
C
(
Q
(
v
))+
Pi/∈Q(v)vi
= Θ(
n
).
However C([n]) = 2. This completes the proof. ut
It is turns out that there are “one-price” Moulin mechanisms that offer a better
guarantee for economic efficiency. As a corollary of Lemma 3.3.9 and the subsequent
Lemma 3.4.7 we get:
42
3.4 Applications
Lemma 3.4.4.
Let
c
be a symmetric, subadditive, and non-decreasing cost function. Then,
the Moulin mechanism induced by the 2-uniform cost-sharing method is 2-BB and 2
Hn
-EFF.
For the proof, we need a result by Roughgarden and Sundararajan
[64]
that relates the
economic efficiency of Moulin mechanisms to a property of their cost-sharing methods.
Definition 3.4.5 (Roughgarden and Sundararajan [64]).
A cost-sharing method
ξ
is
α
-summable with regard to costs
C
if for all sets
A⊆
[
n
]and all orders
a1,...,a|A|
of
A
it
holds that P|A|
p=1ξap({a1,...,ap})≤α·C(A).
Proposition 3.4.6 (Roughgarden and Sundararajan [64]).
Let
ξ
be a cross-monotonic
cost-sharing method. Suppose
ξ
is
β
-BB with regard to approximate costs
C0
and optimal
costs
C
, and moreover
α
-summable with regard to
C
. Then, the Moulin mechanism induced
by ξis α-EFF.
We remark here that in the conference version by Roughgarden and Sundararajan
[64]
,
the definition of budget balance is slightly different to our Definition 2.1.10. This leads
to a different economic-efficiency guarantee in Proposition 3.4.6. We therefore refer to
the extended version in which the same definition as in this thesis is used.
Lemma 3.4.7. Let c be a symmetric, subadditive, and non-decreasing cost function. Then
the 2-uniform cost-sharing method is O(logn)-summable.
Proof. Let A⊆[n]be a set of players and a1,...,a|A|an arbitrary order of A. Then
|A|
X
p=1
ξap({a1,...,ap})≤2·|A|
X
p=1
c(p)
p≤2H|A|·c(|A|)≤2Hn·c(|A|).ut
3.4.2 Makespan Minimization with Non-Identical Jobs
Theorem 3.4.8.
For sharing the makespan cost when scheduling
n
(not necessarily identi-
cal) jobs on
m
related machines (
Q||Cmax
), there is always a (
d·p17+1
4
)-BB cost-sharing
mechanism. Here,
d
is the number of different processing requirements. The outcome of the
cost-sharing mechanism can be computed in time O(n2+n·logm).
Proof.
Let
p∈Nn
be the vector with the processing requirements and let
s∈Nm
contain
the machine speeds. Let
P
:=
Si∈[n]{pi}
be the set of different processing requirements
and define
d
:=
|P|
. Moreover, for
φ∈P
, define
Nφ
:=
{i∈
[
n
]
|pi
=
φ}
as the set of
all players whose job has processing requirement φ.
i)
We denote by
c
:[
n
]
→R≥0
the optimal-makespan cost function if all jobs were
identical with processing requirement 1. Compute
c
(1)
,...,c
(
n
)with a single run
of LPT. This takes time O(n·logm).
43
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
ii)
Use Algorithm 3.2 to compute the 2P-CSF (
n,d,h,l
)that corresponds to costs
c
.
This can be done in time O(n).
iii)
For each processing requirement
φ∈P
, do the following: Use (
|Nφ|,d,φ·h,φ·l
)
and only the bids of the players in
Nφ
as input for Algorithm 3.1. This yields a
set of served players
Qφ⊆Nφ
and a cost distribution
xφ
. Moreover, use LPT to
compute a schedule
aφ
for the players in
Qφ
. The time needed for each processing
requirement
φ
is
O
(
|Nφ|2
)for Algorithm 3.1 and
O
(
|Nφ|·logm
)for LPT. Altogether,
this step hence can be computed in time O(n2+n·logm).
iv)
Let the set of served player be
Q
:=
Sφ∈PQφ
. Assign each player
i
the cost share
xi
:=
xpi
i
that was computed for processing requirement
pi
. Finally, merge all
schedules aφ. This takes time O(n).
Hence, the total running time is
O
(
n2
+
n·logm
). It remains to be shown that this
mechanism is (
d·p17+1
4
)-BB with respect to the actual cost
C0
(
Q
), which is the makespan
resulting from merging all schedules aφ, and the optimal cost C(Q):
C0(Q)≤X
φ∈P
φ·c(|Qφ|)because merging is subadditive
≤X
φ∈PX
i∈Qφ
xφ
iby Theorem 3.4.1
≤X
φ∈P
p17 +1
4·C(Qφ)by Theorem 3.4.1
≤d·p17 +1
4·C(Q).ut
3.5 Characterizing Symmetry and 1-BB
3.5.1 An Impossibility Result
Theorem 3.5.1.
There is no GSP mechanism that is 1-BB with regard to the symmetric
4-player cost function C specified as follows:
p1 2 3 4
c(p)1 3 6 7
For the proof of Theorem 3.5.1, we need the subsequent Propositions 3.5.2, 3.5.4, and
3.5.5, together with Lemma 3.5.6.
Proposition 3.5.2 (Moulin [51]).
Let
M
= (
Q,x
)be a GSP cost-sharing mechanism and
ξ
its cost-sharing method. Then, for all
S⊆
[
n
],
i,j/∈
[
n
]
\S
,
i6
=
j
, at least one of the
following three conditions holds:
44
3.5 Characterizing Symmetry and 1-BB
i) ξi(S∪{i,j})< ξi(S∪{i})and ξj(S∪{i,j})< ξj(S∪{j}),
ii) ξi(S∪{i,j}) = ξi(S∪{i}),
iii) ξj(S∪{i,j}) = ξj(S∪{j}).
Note here that Proposition 3.5.2 implies the following simple observation: Suppose
ξ
is
the cost-sharing method of a GSP cost-sharing mechanism,
S(
[
n
],
j,k∈
[
n
]
\S
, and
j6=k. Then the following implication holds:
ξj(S∪{j,k})> ξj(S∪{j}) =⇒ξk(S∪{k}) = ξk(S∪{j,k}). (3.5.3)
Proposition 3.5.4 (Immorlica et al. [36]).
Let
M
= (
Q,x
)be a GSP cost-sharing mech-
anism and
ξ
its cost-sharing method. Then, if every player
i
bids
bi> ξi
([
n
]) it holds that
Q(b) = [n].
Proposition 3.5.5 (Immorlica et al. [36]).
Let
M
= (
Q,x
)be a GSP cost-sharing mech-
anism and
ξ
its cost-sharing method. Then, for all
S⊆
[
n
]and
i∈
[
n
], it holds that
∀j∈S:ξj(S)≤ξj(S∪{i})or ∀j∈S:ξj(S)≥ξj(S∪{i}).
Lemma 3.5.6. Let the symmetric 3-player cost function C be specified as follows:
p1 2 3
c(p)1 3 6
Then, every 1-BB and GSP cost-sharing mechanism has one of the following cost-sharing
methods µ,ν(up to renumbering of the players).
U; {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}
µ(U)(0,0,0) (1,0,0) (0,1,0) (0,0,1) (2,1,0) (2,0,1) (0,2,1)(3,2,1)
ν(U) (2,3,1)
Proof.
Let
M
be a 1-BB GSP cost-sharing mechanism and
ξ
its cost-sharing method. We
will show that
ξ∈{µ,ν}
. Note that in order to increase readability, we omit parentheses
in this proof when it is unambiguous to do so.
Since
C{j,k}>C{j}
+
C{k}
for any
j,k∈
[3],
j6
=
k
, it follows that
ξj{j,k}>
ξj{j}
=
C{j}
or
ξk{j,k}> ξk{k}
=
C{k}
. Then, by
(3.5.3)
, we have
ξj{j,k}
=
C{j}
or
ξk{j,k}=C{k}.
Let
/
denote the binary relation
/
:=
{
(
a,b
)
∈
[3]
2|ξa{a,b} ≥ ξb{a,b}}
. W.l.o.g.,
there are two cases.
i) Case ξ3{2,3}=1, ξ2{1,2}=1, and ξ1{1,3}=1 (/is cyclic order):
Again, w.l.o.g., we may assume that
ξ1{
1
,
2
,
3
}>
1=
ξ1{
1
,
3
}
. It follows by
(3.5.3)
that
ξ2{
1
,
2
,
3
}
=
ξ2{
2
,
3
}
=3
−
1=2. Then,
ξ2{
1
,
2
,
3
}>
1=
ξ2{
1
,
2
}
45
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
and hence, by the same argument,
ξ3{
1
,
2
,
3
}
=
ξ3{
1
,
3
}
=3
−
1=2. Now 1-BB
implies ξ1{1,2,3}=6−2−2=2, i.e., ξ{1,2,3}= (2,2,2).
Now assume that the true valuation vector is
v
= (2
,
2
,
2). Since
Q
(
v
) =
{
1
,
2
,
3
}
would make all players indifferent, a GSP mechanism would, w.l.o.g., select player
1 as the only player having strictly positive utility, i.e.,
Q
(
v
)
∈{{
1
},{
1
,
2
}}
. Then
players 2 and 3 could cooperate to help player 3 because with the same argument
as before it has to hold that Q(2, b∞,b∞) = {2,3}. A contradiction to GSP.
ii) Case ξ3{2,3}=1, ξ3{1,3}=1, and ξ2{1,2}=1 (/is linear order):
Assume first that
ξ3{
1
,
2
,
3
}>
1. By
(3.5.3)
we get that
ξ2{
1
,
2
,
3
}
=
ξ2{
1
,
2
}
=1
and
ξ1{
1
,
2
,
3
}
=
ξ1{
1
,
2
}
=3
−
1=2, hence
ξ3{
1
,
2
,
3
}
=6
−
2
−
1=3. Suppose
now the true valuations are
v
= (2
,b∞,b∞
). Then player 1 is indifferent to getting
the service and he could help player 3 by bidding
−
1 or help player 2 by bidding
b∞, a contradiction.
Assume now that
ξ3{
1
,
2
,
3
}<
1. Since
ξ2{
1
,
2
,
3
}>
1=
ξ2{
1
,
2
}
would im-
ply
ξ3{
1
,
2
,
3
}
=1 (again, by
(3.5.3)
), we have that
ξ2{
1
,
2
,
3
} ≤
1 in this
case. Now consider the true valuation vector
v
= (
b∞,ξ2{
1
,
2
,
3
},v3
)where
v3∈
(
ξ3{
1
,
2
,
3
},
1). Here, player 2 is obviously indifferent about getting the
service and as a result he could either help player 3 by bidding
b∞
(using Proposi-
tion 3.5.4) or player 1 by bidding −1 (since ξ1{1,2,3}≥6−1−1>1=ξ1{1}).
Hence,
ξ3{
1
,
2
,
3
}
=1. Assume next that
ξ1{
1
,
2
,
3
}>
3. Then,
ξ2{
1
,
2
,
3
}<
2.
Moreover, when
v
= (
b∞,v2,b∞
)with
v2∈
(
ξ2{
1
,
2
,
3
},
2), all players are served
for
v
(Proposition 3.5.4). However, player 1 would be better off by bidding some
b1∈
(2
,ξ1{
1
,
2
,
3
}
)instead of
b∞
, in which case only players 1 and 3 would be
served. A contradiction to SP.
Now if
ξ1{
1
,
2
,
3
}<
3, then
ξ2{
1
,
2
,
3
}>
2=
ξ2{
2
,
3
}
and therefore, by
(3.5.3)
,
ξ1{1,2,3}=ξ1{1,3}=2, i.e., ξ{1,2,3}= (2,3,1).
Hence, we have shown that ξ{1,2,3}∈{(3,2,1),(2,3,1)}, which completes the proof.
Proof (of Theorem 3.5.1).
The proof is by contradiction and consists of several steps.
Assume that there is a 1-BB and GSP cost-sharing mechanism. Let
ξ
be its cost-sharing
method. Note first that since players may opt to not participate (by submitting a negative
bid), the results of Lemma 3.5.6 hold for all subset U⊂[4]with |U|=3.
i) ξinduces a unique order on the set of players.
Let
/
denote the binary relation
/
:=
{
(
a,b
)
|ξa{a,b} ≥ ξb{a,b}} ⊆
[4]
2
. By
Lemma 3.5.6, for any 3-element-subset
U⊂
[4], the restriction of
/
to
U
is a linear
order on U.
Now assume
/
is not a linear order on the whole of [4]. Since reflexivity and
antisymmetry are obviously fulfilled, this means that there are
a,b,c∈
[4]with
a6
=
b
,
a6
=
c
,
b6
=
c
,
a/b
,
b/c
, and
c/a
. Clearly, this is a contradiction for the
3-element-set U={a,b,c}.
46
3.5 Characterizing Symmetry and 1-BB
Hence,
/
is a linear order, and we may, w.l.o.g., assume 1
/
2
/
3
/
4 in the following.
ii) Finally, we will show that GSP and 1-BB lead to a contradiction.
By Lemma 3.5.6, for each 3-element subset
U⊂
[4], there are only two possible
vectors of cost shares, e.g., ξ{1,2,3}∈{(3,2,1,0),(2,3,1,0)}.
Assume first
ξ4{
1
,
2
,
3
,
4
}>
1. Then, since 1 =
ξ4
(
{
1
,
2
,
3
,
4
} \ i
)for all
i∈
{
1
,
2
,
3
}
it follows by
(3.5.3)
that
ξi{
1
,
2
,
3
,
4
}
=
ξi{
1
,
2
,
3
}
, a contradiction to
1-BB. Similarly, if
ξ4{
1
,
2
,
3
,
4
}<
1, then
(3.5.3)
implies for all
i∈ {
1
,
2
,
3
}
that
ξi{
1
,
2
,
3
,
4
} ≤ ξi{
1
,
2
,
3
}
. Again a contradiction to 1-BB. Consequently,
ξ4{1,2,3,4}=1.
Because of Proposition 3.5.5 it holds that either
ξ[3]{
1
,
2
,
3
,
4
} ≥ ξ[3]{
1
,
2
,
3
}
or
ξ[3]{
1
,
2
,
3
,
4
} ≤ ξ[3]{
1
,
2
,
3
}
. Consequently, 1-BB implies
ξ[3]{
1
,
2
,
3
,
4
}
=
ξ[3]{1,2,3}.
Hence, we only need to consider the following two cases:
a) Case ξ{1,2,3}= (3,2,1,0), i.e., ξ{1,2,3,4}= (3,2,1,1):
If
ξ{
1
,
3
,
4
}
= (3
,
0
,
2
,
1)then player 2 can help either player 3 or 1 in case
that the true valuation vector is
v
= (
b∞,
2
,3
2,b∞
): He could bid
b∞
or
−
1
because Q(v−2,b∞) = {1,2,3,4}and Q(v−2,−1) = {1,4}.
Also, if
ξ{
1
,
3
,
4
}
= (2
,
0
,
3
,
1), player 2 can again help either player 3 or 1 in
case that the true valuation vector is
v
= (
b∞,
2
,b∞,b∞
), by bidding
b∞
or
−1.
b) Case ξ{1,2,3}= (2,3,1,0), i.e., ξ{1,2,3,4}= (2,3,1,1):
We can use essentially the same arguments as in the previous case: If
ξ{
2
,
3
,
4
}
= (0
,
3
,
2
,
1)then player 1 can help either player 3 or 2 in case
that the true valuation vector is
v
= (2
,b∞,3
2,b∞
): He could bid
b∞
or
−
1
because Q(v−1,b∞) = {1,2,3,4}and Q(v−1,−1) = {2,4}.
Also, if
ξ{
2
,
3
,
4
}
= (0
,
2
,
3
,
1), player 1 can again help either player 3 or 2 in
case that the true valuation vector is
v
= (2
,b∞,b∞,b∞
), by bidding
b∞
or
−1.
Hence, all cases are in contradiction to GSP. Consequently, there is no mechanism that
satisfies GSP and 1-BB. ut
3.5.2 GSP and 1-BB Cost-Sharing Mechanisms for Three Players
Theorem 3.5.7.
If the number of players is at most 3, then for every symmetric costs
C
there is a 1-BB and GSP mechanism.
47
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
Proof. Define the symmetric cost-sharing method ξas follows:
pξ1(p),...,ξp(p)condition
1c(1)
2c(2)
2,c(2)
2if c(2)
2≤ξ1(1) = c(1)
c(2)−c(1),ξ1(1)otherwise
3c(3)
3,c(3)
3,c(3)
3if c(3)
3≤ξ2(2)
c(3)−2·ξ2(2),ξ2(2),ξ2(2)otherwise, if c(3)−c(2)< ξ2(2)
ξ1(2),c(3)−c(2),ξ2(2)otherwise, if c(3)−c(2)< ξ1(2)
c(3)−c(2),ξ1(2),ξ2(2)otherwise
If it does not holds that
c
(3)
−c
(2)
>c
(2)
−c
(1)
>c
(1), it can easily be verified that
ξ
is a valid symmetric cost-sharing method. Hence, due to Theorem 3.2.8, the symmetric
mechanism defined by ξis GSP.
Otherwise, if
c
(3)
−c
(2)
>c
(2)
−c
(1)
>c
(1), the cost function
c
is supermodular.
The following mechanism that shares cost incrementally is GSP: Start with the empty
player set
Q
and do the following iteratively, for every player
i
=3
,
2
,
1: If
i
bids strictly
more than his marginal cost
C
(
Q∪{i}
)
−C
(
Q
), then charge him his marginal cost and
add him to Q. Otherwise, he will not be served.
Now a player can only improve if the number of players added before him decreases.
However, this would require a coalition where some member loses utility. ut
We close by remarking that the mechanism proposed for the case
c
(3)
−c
(2)
>c
(2)
−
c
(1)
>c
(1)is called a sequential stand-alone mechanism (see also Section 4.4.2). Some-
what counterintuitively, if costs are supermodular but not necessarily symmetric, then
sequential stand-alone mechanisms are in general not GSP [
51
, p. 297, second remark
after Proposition 1].
3.6 Beyond Symmetric Costs
In this section, we outline a possible direction for generalizing precedence mechanisms
to arbitrary costs. In detail, we formulate conditions for non-symmetric cost-sharing
methods so that the resulting precedence mechanisms are WGSP. Afterwards, we show
what is the “missing link” to GSP. Note that in this section, we take the lexicographic
order with the usual significance. For instance, (1,2)≺(2,1).
We start with a simple observation that may be of independent interest:
Lemma 3.6.1.
Consider an arbitrary precedence mechanism. Let
v
contain the true
valuations, let
K
be a non-empty coalition, and let
b
be a
K
-variant of
v
with
uK(b)≥uK(v)
.
Suppose
ui(b)>ui(v)
for some
i∈
[
n
]. Then, there is a player
j<i
with
uj(b)<uj(v)
.
Consequently, P(v,Q(b)) ≺P(v,Q(v)).
48
3.6 Beyond Symmetric Costs
Proof.
W.l.o.g.,
i
is the first improving player. By way of contradiction, assume
uj(b)
=
uj(v)
for all
j<i
. By definition of precedence mechanisms, it must hold that there
is a player
k>i
with
xk
(
b
)
>vk
. Otherwise,
P
(
v,Q
(
b
)) would be non-negative and
P
(
v,Q
(
b
))
P
(
v,Q
(
v
)). Due to VP, it now follows that also
bk>vk
and hence
k∈K
.
This contradicts the assumption that Kis successful. ut
3.6.1 Precedence-Monotonic Cost Shares
Definition 3.6.2.
A cost-sharing method
ξ
is precedence monotonic if for all
S,T⊆
[
n
]
and
i∈S
with
ξi
(
T
)
> ξi
(
S
)or
i/∈T
there is a set
R
with (
S∩T∩
[
i
])
∪i⊆R⊆S∪T
so
that ∀k∈S:ξk(R)≤ξk(S)and ∀k∈T:ξk(R)≤ξk(T).
Theorem 3.6.3.
Precedence mechanisms with a precedence monotonic cost-sharing method
satisfy NPT, VP, and CS.
Proof.
NPT is fulfilled because cost-sharing methods are non-negative. VP is fulfilled
since the mechanism only chooses feasible sets. In order to show CS, consider arbitrary
valuations
v
and a player
i
with
i/∈Q
(
v
) =:
S
. Let
T
:=
S∪i
,
bi>maxA⊆[n]|i∈Aξi
(
A
),
and b:= (v−i,bi).
By Definition 3.6.2, there is a set
R
(for
S
,
T
, and
i
) with (
S∩
[
i
])
∪i⊆R⊆S∪i
and
∀k∈S
:
ξk
(
R
)
≤ξk
(
S
)and
∀k∈T
:
ξk
(
R
)
≤ξk
(
T
). Hence,
P
(
b,R
)
P
(
b,S
)
and
R
is
b
-feasible. Now note that
S∈lexmaxA⊆[n]|i/∈A{P
(
b,A
)
}
. Consequently, since
P(b,Q(b)) P(b,R)P(b,S), it must hold that i∈Q(b).ut
Theorem 3.6.4.
Let
M
be a precedence mechanism with precedence monotonic cost shares.
Then M is WGSP.
Proof.
The proof is by way of contradiction: Let
v
contain the true valuations,
K⊆
[
n
]be
a non-empty coalition, and
b
be a
K
-variant of
v
with
uK(b)uK(v)
. Define
S
:=
Q
(
v
)
and
T
:=
Q
(
b
). Clearly, this implies
T⊇K
. By Lemma 3.6.1, there is a player
i<minK
with
ui(b)<ui(v)
, so
i∈S
and
vi>xi
(
v
) =
ξi
(
S
). W.l.o.g., assume that
i
is the first
such player. Note that, again by Lemma 3.6.1, this implies
uj(b)
=
uj(v)
for all
j<i
. By
Definition 3.6.2, there is a set R(for S,T, and i) with (S∩T∩[i]) ∪i⊆R⊆S∪Tand
∀k∈S:ξk(R)≤ξk(S)and ∀k∈T:ξk(R)≤ξk(T).
For every
k∈
(
T\S
)
∩
[
i
], we have that
uk(b)
=
uk(v)
=0 and
bk
=
vk
, so
uk(b|bk)
=
0. Moreover, for all
k∈
(
T∩S
)
∩
[
i
], it holds that
k∈R
and
ξk
(
R
)
≤ξk
(
T
). Since
ξi
(
R
)
≤ξi
(
S
)and either
ξi
(
S
)
< ξj
(
T
)or
i/∈T
, it holds that
P
(
b,R
)
P
(
b,T
). Finally,
R
is
b
-feasible because for all
k∈R∩T
, it holds that
ξk
(
R
)
≤ξk
(
T
)
≤bk
, and for
all
k∈R\T
, it holds that
k∈S
and
k/∈K
, so
ξk
(
R
)
≤ξk
(
S
)
≤vk
=
bk
. This is a
contradiction. ut
Example 3.6.5.
Any cross-monotonic cost-sharing method is trivially precedence mono-
tonic, because for all
S,T⊆
[
n
], the set
R
:=
S∪T
satisfies the desired properties of
Definition 3.6.2.
49
Chapter 3 Lexicographic Maximization: Beyond Cross-Monotonicity
Example 3.6.6.
Suppose
C
is a supermodular cost function. Then, a cost-sharing method
that charges every player his marginal cost is precedence monotonic because for all
S,T⊆
[n], the set R := (S∩T∩[i]) ∪i satisfies the desired properties of Definition 3.6.2.
3.6.2 The Missing Link to GSP
We show that the missing link to GSP is only a very weak version of WUNB: It must be
guaranteed that if an indifferent player
i
changes his bid to either
−
1 or
b∞
, then no
other player’s utility must improve. We first need several simple properties of precedence
mechanisms.
Lemma 3.6.7.
Consider an arbitrary precedence mechanism. Let
v
contain the true
valuations, let i be an arbitrary player, and let bbe an i-variant of v.
i) Suppose i /∈Q(v)and bi<vi. Then i /∈Q(b)and u(v) = u(b).
ii) Suppose i /∈Q(v)and i ∈Q(b). Then xi(b)≥vi.
iii)
Suppose
i∈Q
(
v
)and
bi>xi
(
v
). Then
i∈Q
(
b
)and
ui(b)≤ui(v)
. If also
bi≤vi
,
then ui(b) = ui(v).
iv) Suppose Mi(v) = Mi(b). Then u(b) = u(v).
Proof. i)
Since VP implies
xi
(
b
)
≤bi
, it holds that
ui(b)≥ui(v)
. Hence, if
u(b)6
=
u(v)
, then
P
(
b,Q
(
v
))
P
(
b,Q
(
b
)) due to Lemma 3.6.1. A contradiction because
Q(v)is b-feasible. Finally, ui(b) = ui(v)and bi<viimply i/∈Q(b).
ii)
By way of contradiction, assume
xi
(
b
)
<vi
. Then
ui(b)>ui(v)
and
P
(
b,Q
(
v
))
P(b,Q(b)) due to Lemma 3.6.1. A contradiction because Q(v)is b-feasible.
iii)
By (ii) (and changing the roles of
v
and
b
), we have
i∈Q
(
b
). Now if
ui(b)>ui(v)
,
then
P
(
b,Q
(
b
))
≺P
(
b,Q
(
v
)) by Lemma 3.6.1. This is a contradiction, since
Q
(
v
)
is b-feasible. Hence, if bi≤vi, then xi(b)≤xi(v)≤xi(b).
iv)
If
i/∈Q
(
v
)and
i/∈Q
(
b
), then the claim follows by (i). If
i∈Q
(
v
)and
i∈Q
(
b
)and
u(b)6
=
u(v)
, then
P
(
b,Q
(
v
))
P
(
b,Q
(
b
)) due to Lemma 3.6.1. A contradiction
because Q(v)is b-feasible. ut
In the following, we will need Theorem 5.3.8, which is a result from Section 5.3.2.
Lemma 3.6.8.
Let
M
be a WGSP precedence mechanism. Suppose that for all true valua-
tions
v
, all players
i
, and all
i
-variants
b
of
v
with
bi∈{−
1
,b∞}
and
ui(v)
=
ui(b)
=0
it holds that ∀j∈[n]:uj(b)≤uj(v). Then M is GSP.
Proof.
By way of contradiction, assume that
M
is not GSP. Then, due to Theorem 5.3.8,
it is not even 2-GSP. This means there are true valuations
v
, a coalition
K
=
{i,i0}
, and a
K
-variant
b
of
v
so that, w.l.o.g.,
ui0(b)>ui0(v)
and
ui(b)
=
ui(v)
. Denote
S
:=
Q
(
v
)
50
3.7 Conclusion
and
T
:=
Q
(
b
)and
bi
:= (
v−i,bi
). By Lemma 3.6.1, there is a player
j<i0
with
uj(b)<uj(v), so j∈Sand vj>xj(v) = ξj(S).
By way of contradiction, assume that
ui(v)>
0. Then
i∈S
,
i∈T
, and
bi≥
xi
(
b
) =
xi
(
v
). Since
M
is SP, it must hold that
ui0(bi)
=
ui0(b)>ui(v)
. Otherwise, if
ui0(bi)<ui0(b)
, player
i0
could improve and manipulate at
bi
by bidding
bi0
. Similarly,
if ui0(bi)>ui0(b), then player i0could improve at bby bidding vi0.
It holds that
i/∈Q
(
bi
). Otherwise,
xi
(
bi
) =
xi
(
v
)due to SP and then
u(bi)
=
u(v)
due to Lemma 3.6.7 (iv). Hence,
bi≤xi
(
v
)
<vi
due to Lemma 3.6.7 (iii). Now, if
bi<xi(v), then xi(b)≤bi<xi(v), a contradiction. Therefore, bi=xi(v).
For all
k<i0
,
k6
=
i
, it must hold that
uk(bi)≥uk(v)
. Otherwise,
P
(
bi,S
)
P
(
bi,Q
(
bi
)), which is a contradiction because
S
is
bi
-feasible. In particular, we have now
uj(b)
=
uj(bi)≥uj(v)
where the equality is due to
Mi0
(
bi
) =
Mi0
(
b
)and Lemma 3.6.7
(iv). A contradiction. Hence, it holds that ui(v) = ui(b) = 0.
Now, if
i∈Q
(
b
), then
u(b−i,b∞)
=
u(b)
because of the threshold property and due
to Lemma 3.6.7 (iv). Similarly, if
i/∈Q
(
b
), then
u(b−i,−1)
=
u(b)
due to Lemma 3.6.7
(i). Hence, we may assume w.l.o.g., that bi∈{−1, b∞}. This completes the proof. ut
3.7 Conclusion
In this chapter, we made a systematic first step for finding GSP mechanisms that perform
better than Moulin mechanisms. Furthermore, we continued the line of characterization
efforts by specifically looking at symmetric costs. It came as a surprise that despite
their simplicity, these costs do not necessarily allow for GSP and 1-BB mechanisms.
While symmetric costs are arguably of limited practical interest, we yet transferred our
techniques to the minimum makespan problem as an application and also to a setting
with non-symmetric costs. Clearly, our work leaves open many issues:
•
For symmetric and/or subadditive costs, we still need an exact characterization
with respect to the best possible budget balance that GSP mechanisms can achieve.
•
Section 3.6 gave directions in which to generalize precedence mechanisms. How
can these ideas be used to develop polynomial-time mechanisms for combinatorial
optimization problem that do not necessarily induce symmetric costs?
•
Finally: Is it possible at all to design GSP mechanisms that improve both on the
budget balance and on the economic efficiency of Moulin mechanisms?
51
Chapter 4
Cost Sharing Without Indifferences:
To Be or Not to Be (Served)
4.1 Overview of Contribution
Models Without Indifferences
In this chapter, we study cost sharing where indif-
ferences do not occur. In detail, we discuss three alternative—yet in a precise sense
related—models, together with explanations why we regard them as equally reasonable
as the “standard model” where indifferences may exist:
i)
First, we find it plausible that players’ utilities might not be strictly quasi-linear;
instead, when players have the choice between being served for their valuation
and not being served at all, they would still prefer the service. For an illustrative
example, one might think of auctions here: A collector would probably prefer
receiving an item also if this requires spending the maximum amount of money
he could afford. We call mechanisms that are GSP with respect to these modified
utilities as group-strategyproof against service-aware players1(SGSP).
ii)
Second, even when utilities are quasi-linear as in the standard model, we find it a
credible assumption on human behavior that a player would not join a coalition
that prevents further service to himself (let alone a coalition that decreases his
utility). Under this behavioral assumption, the GSP requirement is unnecessarily
strong and could be relaxed so to not imply indifference about losing the service. In
detail, we say that a mechanism is weakly group-strategyproof against service-aware
players (WSGSP) if any defection by a coalition that increases some member’s
utility inevitably either decreases the utility of one of its other members or prevents
one of its other members from further service.
iii)
Third and last, one could also assert that the case where a player’s valuation equals
one of the—only finitely many—prices used by the cost-sharing mechanism is a
rare and negligible event (see also Juarez
[39]
). Hence, the argument is that a
slightly changed model where indifferences cannot occur by definition is sufficient
for practical applications. Specifically, we are (only) interested here whether a
mechanism is group-strategyproof on the restricted domain of valuations that results
from excluding all possible cost shares used by the mechanism.
1
In our WINE’07 paper [
8
], we called this property “group-strategyproof against collectors” due to the
connotations explained before. In this thesis, I chose a more general term because I think this behavior
is in no way limited to collectors and auctions.
53
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Immediately from the definition, it follows that SGSP implies WSGSP, which in turn
implies GSP on the restricted domain of valuations that does not contain the cost shares
used by the mechanism. Conversely, we show a canonic way to extend any mechanism
that is GSP on such a restricted domain to be SGSP on the usual domain Rn.
The main benefit of the three models without indifferences is that they allow for
cost-sharing mechanisms with much improved budget balance and economic efficiency
compared to mechanisms that are required to be GSP on the usual domain of valuations
Rn
. Yet, only a small amount of collusion resistance is sacrificed: In particular, even
WSGSP is a strictly more robust collusion-resistance property than WGSP.
Techniques for Designing SGSP Mechanisms
We introduce a novel family of SGSP
mechanisms by devising an iterative algorithm based on set selection and price functions
σ
and
ρ
: In each iteration, a set of players
S
not yet assigned a cost share is selected
by
σ
and offered a price specified by
ρ
. If there are players in
S
who cannot afford
this price, they are rejected. Otherwise, each player in
S
is assigned the price. We call
mechanisms induced by our new algorithm egalitarian because the algorithmic idea is
reminiscent of Dutta and Ray’s algorithm for computing egalitarian solutions [22].
If
σ
always selects the most cost-efficient set and
ρ
the respective price, we prove that
egalitarian mechanisms guarantee 1-BB for arbitrary costs and additionally 2
Hn
-EFF for
the large class of subadditive costs. In particular, our result implies for many natural
optimization problems that there are SGSP (and thus WGSP) cost-sharing mechanisms
that provide exact budget balance and an economic efficiency that is asymptotically
optimal for truthful and (approximately) budget-balanced cost-sharing mechanisms (i.e.,
O
(
logn
)-EFF, recall Section 1.5.2). This great advantage over Moulin mechanism is, e.g.,
illustrated by the rooted Steiner tree problem: Here, (optimal) costs are subadditive,
which ensures existence of a 1-BB and
O
(
logn
)-EFF egalitarian mechanism. In contrast,
no Moulin mechanism can be better than 2-BB [
42
, Theorem 7.1]and
Ω
(
log2n
)-EFF
[65, Theorem 4.2].
Afterwards, we show that the family of acyclic mechanisms by Mehta et al.
[50]
is in
fact also SGSP and thus notably more collusion-resistant than just WGSP. Moreover, we
prove that our egalitarian mechanisms constitute a subclass of acyclic mechanisms.
Finally, we observe that SGSP and 1-BB alone is not hard to achieve. Even trivial
sequential stand-alone mechanisms that charge all players marginal costs are SGSP
and 1-BB. However, for many natural cost-sharing problems with subadditive costs,
the economic efficiency of sequential stand-alone mechanisms is poor. Hence, more
advanced mechanisms like our egalitarian mechanisms should be used here. On the
other hand, when costs are supermodular, sequential stand-alone mechanisms perform
very well: Using a result by Brenner and Schäfer
[11]
(recall Section 1.5.2), we show
that they even achieve O(1)-EFF.
Efficient Computation
We develop a framework and techniques for coping with the
computational complexity of egalitarian mechanisms, especially if the underlying opti-
mization problems are hard. Besides the use of approximation algorithms, the key idea
54
4.2 Cost Sharing Without Indifferences
here are “monotonic” costs
C
(
S
)that must not increase when replacing a player
i∈S
by some other player
j/∈S
with index
j<i
. In this case, finding the most cost-efficient
set only requires iterating through all possible cardinalities (and not all possible subsets
any more). The main issue here is how to pair good (but possibly non-monotonic)
approximation algorithms with a monotonic cost function.
Finally, we give applications that underline the power of our new approach. For
sharing the makespan cost of scheduling
n
jobs on
m
parallel machines, we achieve
better budget balance than all known GSP mechanisms, while maintaining
O
(
logn
)-EFF
(see Table 1.1). Moreover, we achieve 1-BB and
O
(
logn
)-EFF for essentially all makespan
scheduling problems that are optimally solvable in polynomial time. Lastly, we also
obtain results for the bin packing problem and for problems with supermodular optimal
cost functions. The latter includes, e.g., several scheduling problems when the objective
is to minimize the sum of completion times.
4.2 Cost Sharing Without Indifferences
In this section, we formally define the three new collusion-resistance properties without
indifferences. Afterwards, we will show how they are related and that, in a precise
sense, there are some natural constraints under which all three models are equivalent.
Recall that relaxing the collusion-resistance requirements implies weaker coordination
capabilities (or simply less willingness to cooperate), and is done by strengthening the
assumptions on successful coalitions.
4.2.1 Definitions
Service-Aware Players
It seems plausible that utilities are not strictly quasi-linear:
When players have the choice between being served for their valuation and not being
served at all, they might still prefer the service. Formally, we say that players are service-
aware if a player
i
prefers outcome (
Q∗,x∗
)over (
Q0,x0
)if and only if
q∗
i·vi−x∗
i>
q0
i·vi−x0
i
or (
q∗
i·vi−xi
=
q0
i·vi−x0
i
and
q∗
i>q0
i
). In order to more easily compare our
results, we do not define a new utility function but instead incorporate service-awareness
into a new variant of collusion resistance.
Definition 4.2.1.
A mechanism
M
is group-strategyproof against service-aware players
(SGSP) if for all true valuations
v∈Rn
and all non-empty coalitions
K⊆
[
n
]there is
no
K
-variant
b
of
v
with (
uK(b)>uK(v)
and
qK(b)≥qK(v)
)or (
uK(b)≥uK(v)
and
qK(b)>qK(v)).
Limited Coalition Formation
Alternatively, we propose a new notion of coalition-
resistance in between GSP and WGSP. The main motivation is the behavioral assumption
that players are not willing to sacrifice being served for no personal reward, i.e., players
are not indifferent to losing the service. We believe that this behavior is very plausible
for human beings.
55
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Definition 4.2.2.
A mechanism
M
is weakly group-strategyproof against service-aware
players (WSGSP) if for all true valuations
v∈Rn
and all non-empty coalitions
K⊆
[
n
]
there is no K-variant bof vwith uK(b)>uK(v)and qK(b)≥qK(v).
Restricted Valuation Domain
A third model without indifferences was proposed by
Juarez
[39]
. Here, the domain of valuations is restricted so that players’ valuations are
always different to the cost shares used by the mechanism. Formally, this can be done
as follows: Let
M
be a cost-sharing mechanism with cost-sharing method
ξ
. For each
player
i
, let
Di
:=
R\SS⊆[n]ξi
(
S
). That is,
Di
contains all reals except player
i
’s possible
payments. Note that
Di
is still dense in
R
. Define
D
:=
D1×···×Dn
. When valuations
equal to some payment are a rare and in practice a negligible event, the argument is
that one could just as plausibly focus only on the restriction
M|D
= (
Q|D,x|D
), where
Q|D:D→2[n]and x|D:D→Rn
≥0.
Definition 4.2.3.
A mechanism
M
is group-strategyproof on the restricted domain of
valuations
D⊆Rn
if for all true valuations
v∈D
and all non-empty coalitions
K⊆
[
n
]
there is no K-variant b∈D of vwith uK(b)>uK(v).
4.2.2 Equivalence of Models Without Indifferences
We show that for any mechanism that is GSP on the restricted domain
D
there is a
canonical SGSP mechanism (on
Rn
). Conversely, even WSGSP (on
Rn
) implies GSP on
the restricted domain
D
. Specifically, we show that all three models are equivalent once
we require upper continuity, non-bossiness, and the threshold property.
Lemma 4.2.4. Let M be a SGSP cost-sharing mechanism. Then M is upper-continuous.
Proof.
By way of contradiction, assume that
M
= (
Q,x
)is not upper-continuous. Hence,
there is a true valuation vector
v
and a player
i
so that
i/∈Q
(
v
)whereas for all
z>vi
it
holds that
i∈Q
(
v−i,z
). Due to the threshold property, it holds that
x
(
v−i,z
) =
vi
for all
z>vi. Hence, {i}is a successful SGSP-coalition; a contradiction. ut
Lemma 4.2.5.
Let
M
be a WSGSP and upper-continuous cost-sharing mechanism. Then
M is (outcome) non-bossy.
Proof.
Denote
M
= (
Q,x
). Let
v∈Rn
contain the true valuations, let
i∈
[
n
], and let
b
be an i-variant. As intermediate steps, we prove the following implications:
i) qi(b) = qi(v) =⇒u(b) = u(v)
ii) qi(b) = qi(v) = 1=⇒Q(b) = Q(v)
iii) qi(b) = qi(v) = 0 and bi>vi=⇒Q(b) = Q(v)
iv) qi(b) = qi(v) = 0 and bi<vi=⇒Q(b) = Q(v)
56
4.2 Cost Sharing Without Indifferences
Note that the threshold property implies that
Mi
(
b
) =
Mi
(
v
)is equivalent to
qi(b)
=
qi(v). Implication (i) is clearly fulfilled because of WSGSP.
To see the other implications, suppose that
qi(b)
=
qi(v)
indeed holds. By way of
contradiction, assume there is a player
j6
=
i
such that, w.l.o.g.,
j/∈Q
(
v
)and
j∈Q
(
b
),
i.e.,
xj
(
b
) =
vj
. Let
v∗
be a
j
-variant of
v
with
vj<v∗
j< θj
(
v−j
). Such a
v∗
j
exists due
to upper continuity. The threshold property implies
Mj
(
v∗
) =
Mj
(
v
). We also define
b∗
:= (
b−j,v∗
j
). Hence,
b∗
j
=
v∗
j>vj
=
bj
and again the threshold property implies
Mj(b∗) = Mj(b). We can now verify the remaining three implications:
ii) Case qi(b) = qi(v) = 1:
Then
ui(b)
=
ui(v)
=
ui(v∗)
where the last equality is due to (i). Hence, the
coalition {i,j}can help player jat v∗by bidding as in b. This is a contradiction.
iii) Case qi(b) = qi(v) = 0 and bi>vi:
Since
Mj
(
b∗
) =
Mj
(
b
)and
j∈Q
(
b
), we have
M
(
b∗
) =
M
(
b
)due to implica-
tion (ii); so, in particular,
i/∈Q
(
b∗
). Since
b∗
i
=
bi>vi
=
v∗
i
, the threshold
property implies also
i/∈Q
(
v∗
). Consequently, player
i
can help player
j
at
v∗
by
bidding bi; a contradiction.
iv) Case qi(b) = qi(v) = 0 and bi<vi
As in the previous case, we have
i/∈Q
(
b∗
)due to implication (ii). Moreover,
Mj
(
v∗
) =
Mj
(
v
)and
v∗
j>vj
, so
qi(v∗)
=
qi(v)
=0 due to (iii). Hence, player
i
can again help player jat v∗by bidding bi; a contradiction.
Consequently,
qi(b)
=
qi(v)
implies
q(b)
=
q(v)
. Together with implication (i), this
completes the proof. ut
Lemma 4.2.6.
Let
M
be a cost-sharing mechanism with cost-sharing method
ξ
, and let
D
be the restricted domain of valuations as in Section 4.2.1. Then,
M
is SGSP if and only if
M|D
is GSP,
M
is upper-continuous,
M
is (outcome) non-bossy, and
M
fulfills the threshold
property.
Proof.
Sufficiency (“
⇒
”) is straightforward: Upper continuity and outcome non-bossiness
follow from Lemmata 4.2.4 and 4.2.5. Moreover, SGSP clearly implies SP. It also implies
GSP on the restricted domain of valuations
D
: By way of contradiction, suppose there
are true valuations
v∈D
, a non-empty coalition
K
, and a
K
-variant
b∈D
of
v
with
uK(b)>uK(v)
. Then, it must hold that also
q(b)≥q(v)
. Consequently,
K
is SGSP-
successful, a contradiction.
In the rest of the proof, we verify necessity (“
⇐
”). Let
v∈Rn
contain the true
valuations, let
K
be a non-empty coalition, and let
b∈Rn
be a
K
-variant of
v
. By way of
contradiction, assume that (
uK(b)>uK(b)
and
qK(b)≥qK(v)
)or (
uK(b)≥uK(b)
and
qK(b)>qK(v)
, i.e.,
K
is a SGSP-successful coalition. We show that both
v
and
b
can be
transformed into
K
-variants
v0,b0∈D
so that
ui(b|v0
i)≥ui(v|v0
i)
for all
i∈K
, with
at least one strict inequality. We give an algorithmic argument: First, initialize
v0
:=
v
57
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
and
b0
:=
b
. Then, for every player
i
=1
,...n
, increase both
v0
i
and
b0
i
by the same
amount
" >
0 so that still
v0
i,b0
i∈D
and neither
Mi
(
v0
)nor
Mi
(
b0
)changes. Such an
"
exists because
Di
is dense in
R
and because
M
is upper-continuous. Due to (outcome)
non-bossiness, M−i(v0)and M−i(b0)do no change, either.
In the end, both
v0∈D
and
b0∈D
. Moreover,
M
(
v
) =
M
(
v0
),
M
(
b0
) =
M
(
b
), and
v0
and
b0
are
K
-variants. It holds for all players
i∈K
that
ui(b0|v0
i)≥ui(v0|v0
i)
. As
a last step, we show that there is at least one player
i∈K
with
ui(b0|v0
i)>ui(v0|v0
i)
.
This is obvious if there is a player
i∈K
with
ui(b|vi)>ui(v|vi)
. If this is not the case,
then there is a player
i∈Q
(
b
)
\Q
(
v
)with
xi
(
b
) =
vi
. Now, since
v0
i>vi
, it follows that
ui(b0|v0
i)
=
v0
i−xi
(
b0
) =
v0
i−xi
(
b
)
>
0=
ui(v|v0
i)
. This completes the contradiction.
ut
Corollary 4.2.7.
Let
M
be a cost-sharing mechanism. Then
M
is SGSP if and only if
M
is
WSGSP and upper-continuous.
Proof. This follows from Lemma 4.2.5 and Lemma 4.2.6. ut
Figure 4.1 gives an overview of the previous implications. In the following, we give
examples showing that the converse directions do not hold true.
GSP
WGSP SP
+
upper-
continuous
+
SGSP
WSGSP
treshold
property
outcome
non-bossy
GSP on D
Figure 4.1: Hierarchy of collusion-resistance properties
Lemma 4.2.8. There are cost-sharing mechanisms that are
i) GSP but not SGSP,
ii) SGSP but not GSP, or
iii)
WGSP and upper-continuous but not GSP on
D
(where
D⊂Rn
is the restricted
domain of valuations as in Section 4.2.1).
Proof. i) Any lower-continuous GSP cost-sharing mechanism is not SGSP.
58
4.2 Cost Sharing Without Indifferences
ii)
Let there be
n
=2 players and define
M
= (
Q,x
)as follows: The cost-sharing
method is ξ1({1}) = ξ2({2}) = 1 and ξ({1,2}) = (2,1). Moreover,
Q(b):=
{1,2}if b1≥2 and b2≥1
{1}if b1≥1 and b2<1
{2}if b1<2 and b2≥1
;otherwise.
For an illustration, see Figure 4.2a. Note first that
M
is upper-continuous and
satisfies the threshold property (Proposition 2.1.8). Moreover, for player 2, this
threshold value is constantly
θ2
(
b1
) = 1. The threshold value for player 1 is
θ1
(
b2
) = 2 if player 2 gets the service (i.e.,
b2≥
1) and 1 otherwise; i.e., the
threshold value only changes if player 2 loses the service. This proves SGSP.
M
is not GSP: Assume
v
= (2
,
1). Then player 2 could help player 1 by bidding
strictly less than 1.
iii)
Let there be
n
=2 players and define
M0
= (
Q0,x0
)as follows: The cost-sharing
method is ξ1({1}) = ξ2({2}) = 2 and ξ({1,2}) = (3,1). Moreover,
Q0(b):=
{1,2}if b1≥3 and b2≥1
{1}if b1≥2 and b2<1
{2}if b1<3 and b2≥2
;otherwise.
For an illustration, see Figure 4.2b. Note first that
M0
is SP as the threshold
property is fulfilled. Moreover,
M0
is upper-continuous. Now,
M0
always chooses a
b
-feasible set that maximizes player 2’s utility. Hence, both players can never form
a successful WGSP-coalition together. This proves WGSP.
Suppose now
v
= (2
.
5
,
1
.
5). Then, player 2 could help player 1 by bidding strictly
less than 1, e.g., let
b
= (2
.
5
,
0
.
5). It holds that
v,b∈D
, so
M0
is not GSP on the
restricted domain D.ut
{1,2}
{2}
{1}
b1
b2
123
1
2
(a) Graphical illustration
of Q
{1,2}
{2}
{1}
b1
b2
123
1
2
(b) Graphical illustration
of Q0
Figure 4.2: Examples showing that collusion-resistance variants are not equivalent
59
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
4.2.3 WSGSP Implies Separability
We show in the following that already WSGSP is sufficient for separability; hence,
collusion resistance according to any of the three models without indifference implies
existence of a cost-sharing method. A corresponding result for GSP mechanisms is due
to Moulin
[51]
. However, our Theorem 4.2.9 is stronger, since GSP is relaxed to WSGSP,
and strong CS is relaxed to CS. In particular, our result continues to hold if the domain
of valuations is restricted to non-negative bids.
Theorem 4.2.9.
Let
M
be a WSGSP mechanism. Then,
M
is separable. This result holds
even if the domain of valuations is restricted to Rn
≥0(i.e., even if M does not satisfy strong
CS).
Proof.
Let
D
denote the domain of valuations. We establish the result both for
D
=
Rn
and
D
=
Rn
≥0
. Define a cost-sharing method
ξ
: 2
[n]→Rn
≥0
as follows: Denote
b−
:=
−
1
if
D
=
Rn
and
b−
:=0 if
D
=
Rn
≥0
. Now, define
bb
: 2
[n]→D
by
bbi
(
S
):=
b∞
if
i∈S
and
bbi
(
S
):=
b−
otherwise. Then, let
ξi
(
S
):=
x
(
bb
(
S
)). In order to prove the theorem, it is
sufficient to show that for any true valuations
v
it holds that
x
(
v
) =
ξ
(
Q
(
v
)). We do
this by induction over m∈[n]:
Claim (Induction Hypothesis).
Suppose
; 6
=
S⊆
[
n
]with
|S| ≤ m
, and
b∈Rn
is an
S
-variant of
v
such that
bi
=
b∞
if
i∈S∩Q
(
v
),
bi
=
b−
if
i∈S\Q
(
v
), and
bi
=
vi
otherwise. Then
u(v)
=
u(b)
, and for all
i∈S
with
vi>
0or
i∈Q
(
v
)it holds that
Mi(v) = Mi(b).
Proof (of claim).
The base case is
m
=1: Suppose
S
=
{i}
and
b
is as in the induction
hypothesis. Due to the threshold property,
Mi
(
b
) =
Mi
(
v
). Hence, also
u
(
b
) =
u
(
v
)due
to WSGSP. Otherwise, player
i
could help some other player either by bidding
vi
if the
true valuation vector was bor or biif it was v.
For the induction step “(
m−
1)
→m
”, assume the induction hypothesis holds up to
m−
1. Suppose
S⊆
[
n
]with
|S|
=
m
and
b
is as in the induction hypothesis. Define
S0
:=
{j∈S|vj>
0 or
j∈Q
(
v
)
}
. If
D
=
Rn
then consider the coalition
K
:=
S
, otherwise
if
D
=
Rn
≥0
, let
K
:=
S0
. In the latter case it holds for all
j∈S\K
that
vj
=
bj
=0. The
proof of the claim proceeds in several steps:
i) uS(b)≤uS(v)
This holds because otherwise there is a player
i∈S
with
ui(b)>ui(v)
=
ui(b−i,vi)
. Here the equality is due to the induction hypothesis. This contra-
dicts SP.
ii) For all i∈K:(i∈Q(b)⇐⇒ i∈Q(v))
Let
i∈K
. Obviously, if
i∈Q
(
v
), then
i∈Q
(
b
)because
bi
=
b∞
. If
D
=
Rn
then
i/∈Q
(
v
)similarly implies
i/∈Q
(
b
)because
bi
=
b−<
0. On the other hand,
if
D
=
Rn
≥0
then
i/∈Q
(
v
)implies
vi>
0 by definition of
S0
and
bi
=
b−
=0.
Consequently,
i/∈Q
(
b
)because otherwise
ui(b)
=
vi>
0=
ui(v)
. Here, the first
equality is due to NPT and VP. A contradiction to (i).
60
4.2 Cost Sharing Without Indifferences
iii) uS(b)≥uS(v)
This holds because otherwise there is a player
i∈S
with
ui(b)<ui(v)
. Note that
this implies
vi>
0, so
i∈K
. Then, due to (i) and (ii), the coalition
K
can help
i
at
bby bidding as in v. This contradicts WSGSP.
iv) MK(b) = MK(b)and u(b) = u(v)
Since
K⊆S
, steps (i), (ii), and (iii) clearly imply
MK
(
b
) =
MK
(
v
). Now if there is
a player
j∈
[
n
]
\S
with
uj(b)6
=
uj(v)
, then the coalition
K
can help
j
either at
v
or at bby bidding bKor vK, respectively.
We remark that if
D
=
Rn
≥0
then it may happen that
Q
(
bb
(
Q
(
v
)))
)Q
(
v
). This is not a
problem because all players
j∈Q
(
bb
(
Q
(
v
)))
\Q
(
v
)satisfy
bj
=
vj
=0, so
xj
(
b
) = 0=
xj(v).ut
4.2.4 WGSP Does Not Imply Separability
We close this section by observing that WGSP, in contrast to WSGSP, does not imply
separability, i.e., existence of a cost-sharing method. This observation gives some
intuition why WSGSP is notably stronger than only WGSP.
As a simple corollary of the threshold property, the outcome of any SP mechanism
could be computed as follows: For every player
i
, compute the threshold value
θi
,
together with a rule what to do in case of indifference,
φi∈{serve,reject}
. Then, serve
all players
i
with
bi> θi
or (
bi
=
θi
and
φi
=
serve
)for price
θi
; reject all others. Next,
we give a simple idea to transform this observation into a WGSP mechanism.
Definition 4.2.10.
Suppose the outcome of a mechanism
M
can be computed as follows:
For each
j
=1
,...,n
, compute
σj∈
[
n
],
θσj∈R≥0
, and
φσj∈{serve,reject}
as functions
of
bσ1,..., bσj−1
so that
σ1,...,σn
becomes a permutation of [
n
]. Then
M
is called a
sequential mechanism.
Note that, for every sequential mechanism, σ1is a constant.
Lemma 4.2.11. Every sequential mechanism is WGSP.
Proof.
Let
v
contain the true valuations, let
K
be a non-empty coalition, and let
b
be
a
K
-variant of
v
. Consider the first iteration
i
in which a player from
K
is considered
for input
b
, i.e.,
i
:=
min{j∈
[
n
]
|σj
(
b
)
∈K}
. Obviously we have for all
j∈
[
i
]that
σj
(
b
) =
σj
(
v
)and
θσj(b)
=
θσj(v)
. Hence,
uσi(b)(b)≤uσi(b)(v)
because the threshold
value for player
σi
(
b
)
∈K
has not changed. Therefore,
K
cannot be a WGSP-successful
coalition. ut
Corollary 4.2.12. There is a WGSP and 1-BB mechanism that is not separable.
Proof.
Let there be
n
=3 players, and let the cost function be defined by
C
(
S
):=1
if
S6
=
;
and
C
(
;
) = 0. Define a sequential mechanism
M
as follows: If
b1≥1
2
, let
61
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
σ
:= (1
,
2
,
3)and otherwise
σ
= (1
,
3
,
2). Now find the first player (according to the
order
σ
) who can pay for himself and also for all remaining players with a non-negative
bid. Let Mserve this set.
Formally: The threshold value is
θσi
:=1 if
bσj<
1 for all
j<i
, and
θσi
:=0
otherwise. Moreover, the mechanism is upper-continuous, i.e.,
φi
:=
serve
for all
i
. Now
let
b
:= (0
,
1
,
1)and
b0
:= (
1
2,
1
,
1). Then
Q
(
b
) =
Q
(
b0
) =
{
2
,
3
}
but
x
(
b
) = (0
,
1
,
0)and
x(b0) = (0,0,1).ut
4.3 Egalitarian Mechanisms
Egalitarian mechanisms borrow an algorithmic idea proposed by Dutta and Ray
[22]
for computing egalitarian solutions. Given a set of players
Q⊆
[
n
], cost shares are
computed by doing the following iteratively: Find the most cost-efficient subset
S
of the
players that have not been assigned a cost share yet. That is, the quotient of the marginal
cost for including
S
divided by
|S|
is minimal. Then, assign each player in
S
this quotient
as his cost share. If players remain who have not been assigned a cost share yet, start a
new iteration.
Before discussing most cost-efficient subsets in Section 4.3.2, we generalize Dutta and
Ray’s idea by making use of a more general set selection function
σ
and price function
ρ
.
Specifically, let
Q⊆
[
n
]be the set of players to be served. For some fixed iteration, let
N(Q
be the subset of players already assigned a cost share. Then,
σ
(
Q,N
)selects the
players
S⊆Q\N
who are assigned the cost share
ρ
(
Q,N
). We require
σ
and
ρ
to be
valid (see discussion below):
Definition 4.3.1.
Set selection and price functions
σ
and
ρ
are valid if the following holds
for all N (Q0⊆Q⊆[n]:
W1) ;6=σ(Q,N)⊆Q\N,
W2) σ(Q,N)⊆Q0=⇒σ(Q,N) = σ(Q0,N)and ρ(Q,N) = ρ(Q0,N),
W3) ρ(Q,N)≤ρ(Q0,N),
W4) 0≤ρ(Q,N)≤ρ(Q,N∪σ(Q,N)).
Now egalitarian mechanisms are defined by Algorithm 4.1.
We shortly comment on validity: Property (W1) implies that any player is assigned a
cost share only once and that the algorithm terminates. Property (W2) is a consistency
property. It will ensure that the outcome does not change if a rejected player unilater-
ally modifies his bid such that he is rejected in a different iteration (see the proof of
Theorem 4.4.7). Finally, properties (W3) and (W4) imply that the assigned prices are
non-decreasing throughout the iterations of the algorithm.
Theorem 4.3.2. Egalitarian mechanism are SGSP.
We defer the proof of Theorem 4.3.2 to Section 4.4.1 where it will be an immediate
corollary of Theorems 4.4.6 and 4.4.7.
62
4.3 Egalitarian Mechanisms
Input: valid set selection and price functions σ,ρ; bid vector b∈Rn
Output: set of served players Q∈2[n]; vector of cost shares x∈Rn
≥0
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
Algorithm 4.1: Egalitarian mechanisms
4.3.1 Efficiency of Egalitarian Mechanisms
In order to show economic-efficiency bounds, a further property of price functions is
needed:
Definition 4.3.3.
Let
ρ
be a price function, and
β >
0. Then,
ρ
is called
β
-average for
costs C if for all N (Q⊆[n]and all ;6=A⊆Q\N, it holds that ρ(Q,N)≤β·C(A)
|A|.
Lemma 4.3.4.
Let
σ
and
ρ
be valid set selection and price functions such that
ρ
is
β
-
average for costs
C
. Moreover, let
;6
=
A⊆
[
n
]and
b∈Rn
be a bid vector with
bi≥β·C(A)
|A|
for all
i∈A
. Then, the egalitarian mechanism
M
= (
Q,x
)serves at least one player
i∈A
,
i.e., A∩Q(b)6=;.
Proof.
By way of contradiction, assume that
A∩Q
(
b
) =
;
. Consider the first iteration
k
in which Algorithm 4.1 rejects a player
i∈A
: This happens in line 4. We indicate all
variables in this iteration immediately before line 4 with a subscript k. Since player iis
dropped,
bi<ak=ρ(Qk,Nk)≤β·C(A)
|A|,
where the last inequality holds because
A⊆Qk\Nk
and
ρ
is
β
-average. A contradiction.
ut
Theorem 4.3.5.
Let
σ
and
ρ
be valid set selection and price functions such that
ρ
is
β
-average for non-decreasing costs
C
. Suppose the egalitarian mechanism
M
= (
Q,x
)
always recovers at least the actual cost C0. Then, M is (2β·Hn)-EFF.
Proof.
Let
v
contains the true valuations. Denote
Q
:=
Q
(
v
),
x
:=
x
(
v
), and let
vi
:=
max{vi,
0
}
. Moreover, let
P⊆
[
n
]be a set that minimizes
C
(
P
) +
Pi/∈Pvi
. We have
SC(Q) = C0(Q) + X
i∈[n]\Q
vi
≤X
i∈Q∩P
xi+X
i∈Q\P
xi+X
i∈[n]\Q
vidue to cost recovery
≤X
i∈Q∩P
xi+X
i∈P\Q
vi+X
i∈[n]\P
vidue to xi≤vifor i∈Q.
63
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Hence,
SC(Q)
C(P) + Pi/∈Pvi≤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 because the fraction on the left-hand side is at least 1 and
the same non-negative value is subtracted in both numerator and denominator. Now,
consider an arbitrary iteration
k
when Algorithm 4.1 decides to accept a player
i∈Q∩P
in line 5. Fix all variables just after line 3 in that iteration
k
and indicate them with a
subscript k. We have
xi=ak=ρ(Qk,Nk)≤β·C((Q∩P)\Nk)
|(Q∩P)\Nk|≤β·C(Q∩P)
|(Q∩P)\Nk|,
where the inequalities hold because (
Q∩P
)
\Nk⊆Qk\Nk
,
ρ
is
β
-average for costs
C
,
and
C
is non-decreasing. Now let
i1,...,i|Q∩P|
be the players in
Q∩P
ordered according
to the iteration in which they are accepted. Note that if a player
ij
is accepted in iteration
k
, then
|
(
Q∩P
)
\Nk|≥|Q∩P|− j
+1 because at most
j−
1 players from
Q∩P
can be
contained in Nk. Consequently, we get
xij≤β·C(Q∩P)
|Q∩P|− j+1
and thus Pi∈Q∩Pxi≤β·H|Q∩P|·C(Q∩P).
On the other hand, in
P\Q
, there is at least one player
i
with
vi< β ·C(P\Q)
|P\Q|
. Otherwise,
due to Lemma 4.3.4, 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\Q
with
vi< β ·C(P\Q)
|P\Q|−j. Therefore, Pi∈P\Qvi≤β·H|P\Q|·C(P\Q).
Combining the previous bounds and exploiting that Cis non-decreasing, we get
SC(Q)
C(P) + Pi/∈Pvi≤β·Hmax{|Q∩P|,|P\Q|} ·(C(Q∩P) + C(P\Q))
C(P)≤2β·Hn.
This completes the proof. ut
4.3.2 Most Cost-Efficient Set Selection
How can concrete set selection and price functions be defined so that they are valid and
the previous findings apply? This is what we answer next.
Definition 4.3.6.
A set selection function
σ
and its corresponding price function
ρ
are
called most cost-efficient with regard to optimal costs C if they satisfy (W2) and
σ(Q,N)∈arg min
;6=T⊆Q\NC(N∪T)−C(N)
|T|,
ρ(Q,N) = min
;6=T⊆Q\NC(N∪T)−C(N)
|T|.
64
4.3 Egalitarian Mechanisms
We remark that (W2) is not hard to achieve: A canonical way is, e.g., to always choose
the lexicographic maximum of all sets contained in argminT{(C(N∪T)−C(N))/|T|}.
Lemma 4.3.7.
Most cost-efficient set selection and price functions
σ
and
ρ
are valid. If
the costs C are subadditive then ρis also 1-average for C.
Proof.
It is a simple observation that
σ
and
ρ
fulfill properties (W1)–(W3) of Defini-
tion 4.3.1. To see property (W4), 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.
Now assume that Cis subadditive. Again, let N(Q⊆[n]and ; 6=A⊆Q\N. Then,
ρ(Q,N) = min
;6=T⊆Q\NC(Q∪T)−C(Q)
|T|≤C(Q∪A)−C(Q)
|A|≤C(A)
|A|.
Hence, ρis 1-average for Cif Cis subadditive. ut
As a corollary of Theorem 4.3.5 and Lemma 4.3.7 we get:
Theorem 4.3.8.
For arbitrary costs
C
, any egalitarian mechanism
M
induced by most
cost-efficient set selection and prices is 1-BB. If
C
is non-decreasing and subadditive, then
M
is also 2Hn-EFF .
Unfortunately, evaluating a most cost-efficient set selection function
σ
can take exponen-
tially many steps (in
n
). Furthermore, computing optimal costs
C
is often
NP
-hard. In
Section 4.5, we thus study how to pick “suitable” cost-efficient subsets in polynomial
time. We conclude this subsection by showing that our bound on the social cost is tight
up to a factor of 2.
Lemma 4.3.9.
For costs
C
defined by
C
(
S
):=1for all
; 6
=
S⊆
[
n
], any egalitarian
mechanism induced by most cost-efficient set selection and prices is no better than Hn-EFF.
Proof.
Let
v
:=
(1
i−ε)i=1...n
be the true valuation vector, where
ε∈
(0
,1
n
). Then,
Q
(
v
) =
;
because in Algorithm 4.1, line 4, one player after the other would be dropped.
Now, C([n]) = 1 while SC(;) = Hn−n·ε.ut
Lemma 4.3.10.
For any
α >
1, there is a non-decreasing cost function
C
: 2
[4]→R≥0
so that no egalitarian mechanism induced by most cost-efficient set selection and prices is
better than α-EFF.
65
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Proof.
Let
z
:=6
α
+1. Define
C
as follows:
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]) :=z.
Let
M
= (
Q,x
)be an egalitarian mechanism induced by most cost-efficient set selection
and prices, and suppose the true valuation vector is
v
= (1
,
1
,
2
,z−
5). Algorithm 4.1
first accepts
{
1
,
2
}
, each for a price of 1. Subsequently, it gives the service to 3 for a
price of 2 and in the next iteration, player 4 is rejected. Therefore,
Q
(
v
) =
{
1
,
2
,
3
}
and
SC(Q(v)) = 4+ (z−5) = 6α. However, C({2,3,4}) + v1=5+1=6. ut
4.3.3 Submodular and Supermodular Costs
We remark that if a cost function
C
is submodular, then the egalitarian mechanism
induced by most cost-efficient set selection and prices is unique. Moreover, it is also a
Moulin mechanism. This holds because by Definition 4.3.6, its cost-sharing method is
exactly the egalitarian solution by Dutta and Ray
[22]
—and for submodular costs, the
egalitarian solution is known to produce cross-monotonic cost shares [21].
On the other hand, when costs are supermodular, there is always a singleton set among
the most cost-efficient subsets: Suppose the set of remaining players is
Q
, and the set of
already accepted players is
N
. Consider now an arbitrary set
T
=
{t1,..., t|T|}⊆Q\N
.
Denote Ni:=N∪{ti}. Due to supermodularity, we get
C(N∪T) = C
|T|
[
i=1
Ni
≥C(N1) + C
|T|
[
i=2
Ni
−C(N)
≥···≥ |T|
X
i=1C(Ni)−C(N)+C(N).
By an averaging argument, there is at least one i∈[|T|]so that
C(N∪T)−C(N)
|T|≥P|T|
j=1C(Nj)−C(N)
|T|≥C(Ni)−C(N),
which proves the claim. Hence, when costs are supermodular, egalitarian mecha-
nisms based on most cost-efficient set selection are sequential mechanisms (cf. Def-
inition 4.2.10). They also coincide with Brenner and Schäfer’s singleton mechanisms [
11
].
In general, singleton mechanisms can be seen as egalitarian mechanism with set and
price selection functions that satisfy only conditions (W1) and (W2) of Definition 4.3.1,
but that additionally fulfill that σ(Q,N)is always a singleton set.
4.4 Acyclic Mechanisms and SGSP
Acyclic mechanisms have been introduced by Mehta et al.
[50]
as a generalization (from
an algorithmic point of view) of Moulin mechanism. Their outcome is likewise computed
66
4.4 Acyclic Mechanisms and SGSP
by simulating iterative ascending auctions. However, for any set of remaining players
S
,
there is a specific order in which prices are offered to the players. This order is specified
by an offer function
τ
: 2
[n]→R≥0
. Now, whenever a player cannot afford an offer, a new
iteration is started prematurely. Roughly speaking, acyclic mechanisms “conceal” the
lack of cross-monotonicity from the players and thus preserve truthfulness. Mehta et al.
[50]
proved that acyclic mechanisms are WGSP if they are driven by valid cost-sharing
methods and offer functions.
Definition 4.4.1 (Mehta et al. [50]).
Let
ξ
be a cost-sharing method and
τ
be an offer
function. For all i ∈S⊆[n]let
Ei(S):={j∈S|τj(S) = τi(S)},
Li(S):={j∈S|τj(S)< τi(S)},
Gi(S):={j∈S|τj(S)> τi(S)}
be the subsets of players in
S
with
e
qual,
l
esser, and
g
reater offer time compared to
i
. Then
τis called valid for ξif for all i ∈S⊆[n]:
i) ξi(S\T) = ξi(S)for all T ⊆Gi(S)and
ii) ξi(S\T)≥ξi(S)for all T ⊆Gi(S)∪(Ei(S)\{i}).
Now, acyclic mechanisms are defined by Algorithm 4.2.
Input: cost-sharing method ξ; valid offer function τ; bid vector b
Output: set of players Q, vector of cost shares x
1: Q:= [n]
2: while ∃i∈Q:bi< ξi(Q)do
3: Choose an arbitrary non-empty set T⊆argmini∈Q|bi<ξi(Q){τi(Q)}
4: Q:=Q\T
5: x:=ξ(Q)
Algorithm 4.2: Acyclic mechanisms
We remark that Algorithm 4.2 is more general than the original algorithm given by
Mehta et al.
[50]
. They proposed a special case of Algorithm 4.2 where
T
in line 3 is
always a singleton set, chosen deterministically according to some arbitrary tie breaking
scheme. For instance, such a deterministic tie breaking scheme could be to always pick
the singleton set
T
consisting only of the player with the smallest number; i.e., formally
T:=min(argmini∈Q|bi<ξi(Q){τi(Q)}).
4.4.1 Egalitarian Mechanisms Are Acyclic
In the following, we show that acyclic mechanisms are well-defined also by Algorithm 4.2.
As a welcome by-product, we immediately get that acyclic mechanisms are in fact SGSP
67
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
and thus notably stronger than only WGSP. Moreover, it is then a straightforward result
that egalitarian mechanisms are indeed acyclic.
We start with a simple proposition, which was already shown by Mehta et al.
[50]
.
Even though their proof was only for the special case of Algorithm 4.2 as described
above, it can be reused word by word. Like Mehta et al.
[50]
, we say that a player
j
is
offered the price
p
in iteration
i
if the following conditions hold immediately before line 3
of iteration
i
: First,
j∈Q
. Second, if
k
is a player who will be dropped in line 4, then
τj(Q)≤τk(Q). Third, p=ξj(Q).
Proposition 4.4.2 (Mehta et al. [50]).
Suppose Algorithm 4.2 offers the price
p
to player
j
in iteration
i
and the price
p0
in a subsequent iteration. Then
p≤p0
. Moreover, suppose
Q∗
is the set of players returned by Algorithm 4.2. Then, at the beginning of iteration
i
(i.e., immediately after line 2), it holds that
Lj
(
Q
)
⊆Q∗
and for all
k∈Lj
(
Q
)it holds that
ξk(Q∗) = ξk(Q).
The next two technical lemmata contain the main technique for showing that the
order in which players are dropped is irrelevant.
Lemma 4.4.3.
Let
S
be an output set of Algorithm 4.2 for the bid vector
b
. Suppose
A
is a
strict superset of S, i.e., S (A⊆[n]. Then the following holds:
i) There is a player k ∈A\S with bk< ξk(A).
ii)
Suppose there is a player
j∈S
with
ξj
(
S
)
< ξj
(
A
). Then there is a player
k∈Lj
(
A
)
\S
with bk< ξk(A).
Proof. i)
Consider the first iteration in which some
k∈A\S
is dropped. Immediately
before line 4 it holds that
bk< ξk
(
Q
)
≤ξk
(
A
). Here, the second inequality holds
because Proposition 4.4.2 implies A=Q\Bfor some B⊆Gk(Q)∪(Ek(Q)\{k}).
ii)
Due to Definition 4.4.1,
Lj
(
A
)
\S
is non-empty, and for all
`∈Lj
(
A
)
\S
it holds that
ξ`
(
A
) =
ξ`
(
S∪Lj
(
A
)) because
S∪Lj
(
A
) =
A\B
for some set
B⊆G`
(
S
). Define
now
A0
:=
S∪Lj
(
A
). Since
S(A0⊆A
, it follows by (i) that there is a player
k∈A0\S=Lj(A)\Swith bk< ξk(A0) = ξk(S∪Lj(A)) = ξk(A).ut
Lemma 4.4.4.
Let
K⊆
[
n
]be a set of players, and let
v,b
be
K
-variants. Consider two
distinct executions of Algorithm 4.2, the first for input
v
and the second for
b
. Let the
output sets be
R
and
S
, respectively. Suppose that
K∩S⊇K∩R
and
ξK
(
S
)
≤vK
(if
K
=
;
then this is a vacuous truth). Then, S ⊆R.
Proof.
Let
r
denote the number of iterations (i.e., repetitions of the body of the while-
loop) needed for the first execution (with input
v
). Define
Q0
:= [
n
], and for
i∈
[
r
],
define
Qi
and
Ti
as the values of
Q
and
T
at the end of iteration
i
(i.e., immediately after
line 4). Clearly, it holds for all i∈[r]that Qi= [n]\(T1∪···∪Ti). Moreover, Qr=R.
Now let
q∈N0
be maximal so that the first
q
iterations are identical for both exe-
cutions; i.e., in each iteration
i
=1
,...,q
of the second execution for input
b
the set
68
4.4 Acyclic Mechanisms and SGSP
Ti
is chosen, too. Consequently,
S⊆Qq
. In the following, we show by induction over
i∈{q+1... r}that S⊆Qi.
We first verify the base case
i
=
q
+1. Consider an arbitrary player
j∈Tq+1
. Also
during the second execution, he is offered price
ξj
(
Qq
)
>vj
in iteration
q
+1. Due to
Proposition 4.4.2, any price offered to him in a subsequent iteration cannot be smaller.
Consequently, j/∈S.
Finally, we verify the induction step
i→
(
i
+1). Due to the induction hypothesis,
S⊆Qi
.
Consider again an arbitrary player
j∈Ti+1
. By way of contradiction, assume
j∈S
.
Then,
ξj
(
Qi
)
>vj≥ξj
(
S
)where the first inequality is due to
j∈Ti+1
. Consequently,
due to Lemma 4.4.3 (ii), it holds that there is a player
k∈Lj
(
Qi
)
\S
with
bk< ξk
(
Qi
).
However, by Proposition 4.4.2, it holds for all players
k∈Lj
(
Qi
)that
k∈R
and
k/∈Ti+1
,
so vk≥ξk(Qi)>bk. Then, k∈Kbut k∈R\S. This is a contradiction. ut
Theorem 4.4.5.
The output of Algorithm 4.2 is independent of the way
T
in chosen in
line 3.
Proof.
This follows immediately from Lemma 4.4.4 for the special case
v
=
b
, i.e.,
K=;.ut
Theorem 4.4.6. Acyclic mechanisms are SGSP.
Proof.
Let
M
= (
Q,x
)be an acyclic mechanism, let
v
contain the true valuations, let
K
be a non-empty coalition, and let
b
be a
K
-variant. Define
R
:=
Q
(
v
)and
S
:=
Q
(
b
).
By way of contradiction, assume that
K
is SGSP-successful, i.e., (
uK(b)>uK(v)
and
K∩S⊇K∩R)or (uK(b)≥uK(v)and K∩S)K∩R).
By Lemma 4.4.4, it follows that
S⊆R
. Hence, there is a player
j∈K∩S
with
ξj
(
S
)
< ξj
(
R
), i.e., even
S(R
. Due to Lemma 4.4.3 (ii), there is then a player
k∈Lj(R)\Swith bk< ξk(R)≤vk, so k∈Kbut k∈R\S. This is a contradiction. ut
We now show that for valid set selection and price functions
σ
and
ρ
, Algorithm 4.1
gives the same result as running Algorithm 4.2 with cost-sharing method
ξ
and offer
function τas defined by Algorithm 4.3. Hence, every egalitarian mechanism is acyclic.
Input: valid set selection and price functions σ,ρ; set of players Q⊆[n]
Output: vector of cost shares ξ∈Rn
≥0; offer times τ∈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
Algorithm 4.3: Cost-sharing method and offer function of egalitarian mechanisms
Theorem 4.4.7. Egalitarian mechanisms are acyclic mechanisms.
69
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Proof.
Let
σ,ρ
be valid set and price selection functions, and let
ξ
and
τ
be the cost-
sharing method and offer functions defined by Algorithm 4.3. We first show that
τ
is
valid for
ξ
. Denote the values of all variables immediately after line 3 of iteration
k
with
a subscript
k
and with the input player set in parentheses. Let
Q⊆
[
n
]and
i∈Q
be
arbitrary. Fix now kas the (unique) iteration with i∈Sk(Q).
i)
Let
T⊆Gi
(
Q
)be arbitrary. We show by induction over
m∈
[
k
]that
Nm
(
Q
) =
Nm
(
Q\T
),
Sm
(
Q
) =
Sm
(
Q\T
), and
am
(
Q
) =
am
(
Q\T
). Then,
ξi
(
Q
) =
ak
(
Q
) =
ak
(
Q\T
) =
ξi
(
Q\T
). For the base case
m
=1, we have
N1
(
Q
) =
N1
(
Q\T
) =
;
. For
the induction step
m−
1
→m
, the induction hypothesis implies
Nm
(
Q
) =
Nm
(
Q\T
).
Now, both for the base case and for the induction step, we have
Sm
(
Q
)
⊆Q\T
because for all
j∈Sm
(
Q
):
τi
(
Q
) =
m
whereas for all
j∈T
:
τi
(
T
)
>k
. Hence,
(W2) implies Sm(Q) = Sm(Q\T)and am(Q) = am(Q\T).
ii)
Let
T⊆Gi
(
Q
)
∪
(
Ei
(
Q
)
\{i}
)be arbitrary. With exactly the same inductive argument
as for (i), we get for all
m∈
[
k−
1]that
Nm
(
Q
) =
Nm
(
Q\T
),
Sm
(
Q
) =
Sm
(
Q\T
), and
am
(
Q
) =
am
(
Q\T
). Moreover, also
Nk
(
Q
) =
Nk
(
Q\T
). Now, due to property (W3)
and
Q\T⊆Q
, we have
ak
(
Q
)
≤ak
(
Q\T
). Furthermore,
ak
(
Q\T
)
≤ξi
(
Q\T
)
since
a
is non-decreasing in Algorithm 4.3 due to property (W4). Thus,
ξi
(
Q
) =
ak(Q)≤ak(Q\T)≤ξi(Q\T).
Finally, we show that the egalitarian mechanism induced by
σ
,
ρ
yields the same
outcome as the acyclic mechanism induced by
ξ
,
τ
. Whenever Algorithm 4.1 accepts
a set
S
:=
σ
(
Q,N
)this means that the players in
S
have the minimum offering time
of those in
Q\N
and that
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 4.1 rejects players from
S
, the same players are also rejected by the acyclic
mechanism, due to Theorem 4.4.5. ut
4.4.2 Sequential Stand-Alone Mechanisms
We close this section by noting that SGSP and 1-BB alone is in fact not hard to achieve.
The following (sequential) mechanisms are called sequential stand-alone mechanisms by
Moulin
[51]
and work as follows: Start with the empty player set
Q
and do the following
iteratively, for every player
i
=1
,...,n
: If
i
can afford his marginal cost
C
(
Q∪{i}
)
−C
(
Q
),
then charge him this price and add him to
Q
. Otherwise, he will not be served. A formal
definition is given in Algorithm 4.4.
Lemma 4.4.8.
Sequential stand-alone mechanisms are 1-BB. Moreover, they are acyclic
mechanisms and thus SGSP.
Proof.
Define the cost-sharing method
ξ
and the offer function
τ
as follows. For
S⊆
[
n
]
and
i∈
[
n
], let
ξi
(
S
):=
C
(
S∩
[
i
])
−C
(
S∩
[
i−
1]). Furthermore,
τi
(
S
):=
i
. Note that
τ
is indeed valid for
ξ
. It is now a simple observation that Algorithm 4.2 with input
ξ
,
τ
,
and byields the same output as Algorithm 4.4 with input Cand b.ut
70
4.4 Acyclic Mechanisms and SGSP
Input: non-decreasing cost function C: 2[n]→R≥0; bid vector b∈Rn
Output: set of served players Q∈2[n]; vector of cost shares x∈Rn
≥0
1: Q:=;;x:=0
2: for i:=1,...,ndo
3: if bi≥C(Q∪{i})−C(Q)then
4: xi:=C(Q∪{i})−C(Q);Q:=Q∪{i}
Algorithm 4.4: Sequential stand-alone mechanisms
However, for many natural cost-sharing problems with subadditive costs, the economic
efficiency of sequential stand-alone mechanisms is poor. There is hence good reason to
use more advanced mechanisms like our egalitarian ones:
Lemma 4.4.9.
For the cost function
C
: 2
[n]→R≥0
with
C
(
S
) = 1for all
; 6
=
S⊆
[
n
],
sequential stand-alone mechanisms are no better than n-EFF.
Proof.
Let
M
= (
Q,x
)be the sequential stand-alone mechanism for
C
. Let
v
:=
(1−")i=1...n. Then, Q(v) = ;and thus SC(Q(v)) = (1−")·n. However, SC([n]) = 1. ut
Interestingly, sequential stand-alone mechanisms are useful for cost-sharing problems
with supermodular costs. Recall that supermodular costs imply that serving two disjoint
sets of players separately is never more costly than serving both groups at once. They can
best be seen as a result of congestion effects that occur in the underlying optimization
problem. This includes, for instance, traffic networks where the objective is the total (or
average) latency or min-sum scheduling problems (see, e.g., Schulz and Uhan [68]).
In order to show bounds on the economic efficiency, we slightly extend the notion of
subadditivity: We say a cost function
C
is
α
-subadditive (
α≥
1) if for all
A,B⊆
[
n
]it
holds that C(A∪B)≤α·(C(A) + C(B)).
Theorem 4.4.10.
For any costs
C
the sequential stand-alone mechanism
M
is 1-BB. If
C
is supermodular and
α
-subadditive, i.e., it always holds that
C
(
A
) +
C
(
B
)
−C
(
A∩B
)
≤
C(A∪B)≤α·(C(A) + C(B)), then M is also α-EFF.
In order to prove Theorem 4.4.10, we use a result by Brenner and Schäfer
[11]
. They
gave a bound on the economic efficiency of singleton mechanisms, which are clearly a
superclass of sequential stand-alone mechanisms. The proof bears some resemblance to
Theorem 4.3.5 and is not repeated here. Note that we state the result only in terms of
sequential stand-alone mechanisms.
Definition 4.4.11 (Brenner and Schäfer [11]).
A cost-sharing method
ξ
is said to be
weakly monotone with respect to costs
C
if for all sets of players
A⊆B⊆
[
n
]it holds that
Pi∈Aξi(B)≥C(A).
71
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Proposition 4.4.12 (Brenner and Schäfer [11]).
Let
M
be a sequential stand-alone
mechanism so that its induced cost-sharing method
ξ
is weakly monotone with respect to
costs
C
. Suppose that for all
A,B⊆
[
n
]it holds that
C
(
A∪B
)
≤α·
(
C
(
A
) +
C
(
B
)). Then,
M is α-EFF.
Proof (of Theorem 4.4.10).
Let
ξ
denote the cost-sharing method induced by
M
. It is
straightforward to see that
ξ
is “cross-monotonic with reversed signs” (formally,
−ξ
is cross-monotonic): Let
A⊆B⊆
[
n
]. Then it holds for all
i∈
[
n
]that
ξi
(
B
) =
C
(
B∩
[
i
])
−C
(
B∩
[
i−
1])
≥C
(
A∩
[
i
])
−C
(
A∩
[
i−
1]) =
ξi
(
A
). Consequently,
ξ
is
weakly monotone with respect to Cbecause
X
i∈A
ξi(B)≥X
i∈A
ξi(A) = C(A).
Now the proof follows by Proposition 4.4.12. ut
4.5 A Framework for Polynomial-Time Computation
In this section, we show how to solve cost-sharing problems in polynomial time by using
egalitarian mechanisms with a set selection function that picks the most cost-efficient set
with regard to costs of approximate solutions.
Formally, an optimization problem with the objective to minimize cost is a triple
Π=(
D,S
= (
SI
)
I∈D,f
= (
fI
)
I∈D
), where
D
is the set of problem instances (
d
omain)
such that for any
i
nstance
I∈D
,
SI
is the set of feasible
s
olutions, and
fI
:
SI→R≥0
is a
function mapping any solution to its cost.
We henceforth write a cost-sharing problem as a pair Φ = (Π
,INST
), where Πis the
underlying optimization problem and
INST
: 2
[n]→D
denotes the function mapping
any subset of the
n
players to the respective instance of Π. In particular, Φimplicitly
defines the optimal cost
C
: 2
[n]→R≥0
by
C
(
T
):=
minZ∈SINST(T){f
(
Z
)
}
. Moreover, for
any algorithm
ALG
that computes feasible solutions for Π, we define
CALG
: 2
[n]→R≥0
by
CALG(T):=f(ALG(INST(T))).
Resorting to approximate solutions does, of course, not yet remedy the need to iterate
through all available subsets in order to pick the most cost-efficient one. The basic idea
therefore consists of using an (approximation) algorithm
ALG
that is monotonic (see,
e.g., Murgolo
[53]
): Seemingly favorable changes to the input must not worsen the
algorithm’s performance. In the problems considered here, every player is endowed
with a size (e.g., processing requirement in the case of scheduling), and reducing a
player’s size must not increase the cost of the algorithm’s solution. Provided that this
property holds 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) mono-
tonic bound
Cmono
on
CALG
is needed whereas
ALG
itself does not need to be monotonic
any more.
72
4.5 A Framework for Polynomial-Time Computation
Definition 4.5.1.
Let Φ = (Π
,INST
)be a cost-sharing problem. Suppose
ALG
is an ap-
proximation algorithm for Π, and
Cmono
: 2
[n]→R≥0
is a cost function that satisfies the
following:
•For all T ⊆[n]: CALG(T)≤Cmono(T)≤β·C(T).
•For all U ⊆T⊆[n]:Cmono(MIN|U|T)≤Cmono(U).
Then, the pair (ALG,Cmono)is called a β-relaxation for Φ.
Since we do not require
Cmono
to be subadditive (as necessary to directly apply Theo-
rem 4.3.5), some additional care is needed as described in the following.
Given a
β
-relaxation
R
:= (
ALG,Cmono
), we define set selection and price functions
σR
and
ρR
recursively as follows. Suppose the set of remaining players is
Q
and the
set of already accepted players is
N
. Let
ξ
be the vector of cost shares computed by
Algorithm 4.3 for input σR,ρR, and N. Moreover, let
k:=maxarg min
i∈[|Q\N|]Cmono(N∪MINi(Q\N)) −Pi∈Nξi
i,
Cmono(MINi(Q\N))
i,
and S:=MINk(Q\N). Then, define
σR(Q,N):=Sand ρR(Q,N):=min¨Cmono(N∪S)−Pi∈Nξi
k,Cmono(S)
k«.
Note that this recursion is well-defined. Computing
σR
(
Q,N
)and
ρR
(
Q,N
)requires
ξ
for
which only
σR
(
N,·
)and
ρR
(
N,·
)are needed (unless
N
=
;
). Yet,
N(Q
by assumption.
Lemma 4.5.2.
Let
R
= (
ALG
,
Cmono
)be a
β
-relaxation for some cost-sharing problem Φ.
Then the following holds:
i) σRand ρRare valid.
ii) ρRis β-average for C.
Proof. i)
Let
σ
:=
σR
,
ρ
:=
ρR
. Let
ξ
be the cost-sharing method induced by
σ
and
ρ
. We show that Definition 4.3.1 holds. Clearly, properties (W1) and (W2) are
fulfilled. To see (W3), let
N(Q0⊆Q⊆
[
n
]. Define Σ(
N
):=
Pi∈Nξi
(
N
)and
S
:=
σ(Q,N),k:=|S|and S0:=σ(Q0,N),k0:=|S0|. Since 1 ≤k0≤|Q0\N|≤|Q\N|,
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.
73
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Since
ρ
(
Q0,N
)is equal to one of these upper bounds, we have
ρ
(
Q,N
)
≤ρ
(
Q0,N
).
Finally, to see property (W4), 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, we have
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 (W2), similarly as in first part of the
proof of Theorem 4.4.7. 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
)is the minimum of the upper bounds, and therefore
ρ
(
Q,N
)
≤
ρ(Q,N∪σ(Q,N)).
ii) Let N(Q⊆[n]and A⊆Q\N. Then,
ρR(Q,N)≤Cmono(MIN|A|(Q\N))
|A|≤Cmono(A)
|A|≤β·C(A)
|A|.ut
To also compute a feasible solution for the instance of the optimization problem that
corresponds to the players served by an egalitarian mechanism, we need:
Definition 4.5.3.
Let Φ = (Π
,INST
)be a cost-sharing problem where Π = (
D,S,f
). Then,
Φis called mergable if for all disjoint
T,U⊆
[
n
]and for all
X∈SINST(T)
and
Y∈SINST(U)
,
there is a
Z∈SINST(T∪U)
with
f
(
Z
)
≤f
(
X
)+
f
(
Y
). We denote this operation by
Z
=
X⊕Y
.
Based on
σR
and
ρR
, Algorithm 4.5 completely solves the cost-sharing problem,
including computing a feasible solution for the underlying optimization problem. In the
following, we verify correctness.
Lemma 4.5.4.
Let
R
= (
ALG
,
Cmono
)be a
β
-relaxation for a mergable cost-sharing prob-
lem Φ. The following holds:
i)
At the end of each iteration of Algorithm 4.5, it holds that
x
=
ξ
(
N
)where
ξ
is the
cost-sharing method defined by Algorithm 4.3 for input σRand ρR.
ii) In every iteration, line 3 of Algorithm 4.5 needs at most 2n evaluations of Cmono.
iii) The mechanism defined by Algorithm 4.5 is β-BB.
74
4.5 A Framework for Polynomial-Time Computation
Input: β-relaxation R= (ALG,Cmono); bid vector b∈Rn
Output: set of served players Q∈2[n], vector of cost shares x∈Rn
≥0,
solution Z∈SINST(Q)
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⊆Nthen
6: if Cmono(N∪S)−Pi∈Nxi≤Cmono(S)then
7: Z:=ALG(INST(N∪S))
8: else
9: Z:=Z⊕ALG(INST(S))
10: N:=N∪S;xi:=afor all i∈S
Algorithm 4.5: Egalitarian mechanisms with β-relaxations
Proof.
Consider the execution of Algorithm 4.5 for input
R
and
b
. Let
m∈N
be the
number of iterations needed. For all
k∈
[
m
]
0
, indicate the value of all variables at the
end of the
k
-th iteration (i.e., immediately after line 10 if
k>
0, and immediately before
line 2 if
k
=0) with a superscript
k
. Moreover, let
p
(
k
)be the number of times line 5
has evaluated to true before the end of iteration k.
i)
We show that
ξ
(
Nk
)is computed in exactly the same way as
xk
was. Consider
therefore the execution of Algorithm 4.3 for input
σR
,
ρR
, and
Nk
. Indicate the
value of all variables at the end of the
j
-th iteration with a tilde and a superscript
j
.
Again, superscript 0 refers to the variable values immediately before the while-loop.
We prove by induction over
k∈
[
m
]
0
that
e
Np(k)
=
Nk
and
e
ξp(k)
=
xk
. Clearly, the
base case
k
=0 is fulfilled because
e
N0
=
N0
=
;
and
e
ξ0
=
x0
=0. In the following,
we verify the induction step (
k−
1)
→k
. If line 5 of Algorithm 4.5 evaluated to
false then
p
(
k
) =
p
(
k−
1),
Nk
=
Nk−1
, and
xk
=
xk−1
, which proves the induction
step for this case.
Consider therefore the case that line 5 evaluated to true. Then,
p
(
k
)
−
1=
p
(
k−
1).
We have that
e
Sp(k)
=
σ
(
e
Qp(k),e
Np(k)−1
) =
σ
(
Nk,Nk−1
)where the last inequality
is due to the induction hypothesis. Moreover, we have
Sk
=
σ
(
Qk−1,Nk−1
) =
σ
(
Nk,Nk−1
)where the last equality is due to (W2) and
Sk⊆Nk
. Hence,
e
Sp(k)
=
Sk
and likewise eap(k)=ak. The induction step follows.
ii) This follows directly from the definition of σRand ρR.
iii)
Define Σ(
Nk
):=
Pi∈Nkxk
i
. We show by induction over
k∈
[
m
]
0
that
f
(
Zk
)
≤
Σ(Nk)≤Cmono(Nk).
75
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
The base case
k
=0 holds trivially. For the induction step (
k−
1)
→k
, we only
need to consider the case that line 5 evaluated to true in iteration
k
. Otherwise,
Zk
=
Zk−1
and
Nk
=
Nk−1
, so we would be done. Now, if
Cmono
(
Nk−1∪Sk
)
−
Σ(Nk−1)≤Cmono(Sk)then f(Zk) = CALG(Nk)≤Cmono(Nk) = Σ(Nk). Otherwise,
f(Zk) = f(Zk−1⊕ALG(INST(Sk))
≤f(Zk−1) + CALG(Sk)≤Σ(Nk−1) + Cmono(Sk),
where the last inequality is due to the induction hypothesis and because
CALG
is a
lower bound for
Cmono
. Now, Σ(
Nk
) = Σ(
Nk−1
)+
Cmono
(
Sk
)
.
Since line 6 evaluated
to false, we moreover have Σ(
Nk−1
)+
Cmono
(
Sk
)
<Cmono
(
Nk−1∪Sk
) =
Cmono
(
Nk
)
.
Hence, the induction step follows.
Clearly, the output of Algorithm 4.5 is
Qm
=
Nm
,
xm
, and
Zm
. We have shown that
C0(Qm) = f(Zm)≤Σ(Qm)≤Cmono(Qm)≤β·C(Qm). This completes the proof.ut
As a corollary of Lemmata 4.5.2 and 4.5.4, we obtain:
Theorem 4.5.5.
Let Φbe a mergable cost-sharing problem, and let (
ALG,Cmono
)be a
β
-relaxation for Φ. Then the mechanism defined by Algorithm 4.5 is SGSP,
β
-BB, and
(2
β·Hn
)-EFF. Moreover, Algorithm 4.5 evaluates
Cmono
for no more than 2
n2
subsets of
[
n
], makes no more than
n
(direct) calls to
ALG
, and the number of merge operations is no
more than n.
4.6 Applications
We use three approaches for obtaining
β
-relaxations that are polynomial-time com-
putable in the size of the succinct representation of the cost-sharing problem plus the
bid vector: Monotonic approximation algorithms (Section 4.6.2), a non-monotonic
approximation algorithm with a polynomial-time computable monotonic bound
Cmono
(Section 4.6.3), and optimal costs that are monotonic and polynomial-time computable
(Section 4.6.4). Subsequently, we also give some remarks about applying sequential
stand-alone mechanisms to cost-sharing problems with supermodular optimal costs
(Section 4.6.5).
4.6.1 Makespan Minimization and Bin Packing Cost-Sharing Problems
A makespan cost-sharing problem
Q||Cmax
is succinctly represented by a pair (
p,s
)where
p∈Nn
contains the processing requirements of the
n
jobs, and
s∈Nm
contains the
speeds of the
m
machines. Recall that each player owns exactly one job and that for any
set of players
S⊆
[
n
],
C
(
S
)is the value of a minimum-makespan schedule for the jobs
from S.
A bin packing cost-sharing problem is succinctly represented by a vector of item sizes
ς∈
(0
,
1]
n
. Each player owns exactly one item, and for any set of players
S⊆
[
n
],
C
(
S
)
is the minimum number of bins with capacity 1 that are needed for S.
76
4.6 Applications
In order to keep our notation clean when designing
β
-relaxations that fulfill Defini-
tion 4.5.1, we assume in this section that players’ indices are always sorted in ascending
order of their processing requirements (in the case of scheduling) or item sizes (in the
case of bin packing). This is without loss of generality: Otherwise, players could be
sorted (in a deterministic way) before Algorithm 4.5 is called, which adds only
O
(
logn
)
to the running time and is thus always negligible.
Lemma 4.6.1.
Any bin packing or makespan cost-sharing problem Φ = (Π
,INST
)is mer-
gable in time O(n). Moreover, INST is computable in linear time (in the size of the succinct
representation of Φ).
Proof.
For bin packing with disjoint item/player sets
T
and
U
, we obtain a bin packing
for
T∪U
by taking both the bins with items from
T
and the bins with items from
U
. The
costs (number of bins) simply add up. For scheduling disjoint job/player sets
T
and
U
,
we obtain a schedule for
T∪U
by assigning each job to the machine assigned before.
The resulting makespan doesn’t exceed the sum of the two makespans. ut
4.6.2 Monotonic Approximation Algorithms
Makespan Costs on Identical Machines
We start by considering identical-machine
makespan cost-sharing problems (
P||Cmax
). Their succinct representation is (
p,m
)where
p∈Nn
and
m∈N
. The
LPT
(longest processing time first) algorithm [
27
]is known to
be a
4m−1
3m
-approximation algorithm for this problem. Recall that it processes the jobs
in decreasing order and assigns each job to the machine on which its completion time
will be smallest. Its running time is
O
(
n·logn
)for the sorting phase and
O
(
n·logm
)for
the job assignment phase. For identical machines, we show that
LPT
is monotonic with
regard to processing requirements. In order to do so, we first need a technical lemma.
Let SORT denote a function that sorts the components of a vector in ascending order.
Lemma 4.6.2.
Let
a,b∈Rn
be vectors whose components are sorted in ascending order.
Moreover, let
c,d∈R
and define
a0
:=
SORT
(
a−1,a1
+
c
)and
b0
:=
SORT
(
b−1,b1
+
d
).
Suppose that a≤band c ≤d; then it holds that a0≤b0.
Proof.
Let
j,k∈
[
n
]be arbitrary with
a0j
=
a1
+
c
and
b0
k
=
b1
+
d
. Note that a value
may occur several times in the vector. By definition,
a0= (a2,a3,..., aj,a1+c,aj+1, ..., an)and
b0= (b2,b3,..., bk,b1+d,bk+1,..., bn).
Now let i∈[n]be arbitrary. We verify that a0
i≤b0
i. Note that
a0
i=
ai+1if i<j
a1+c∈[ai,ai+1]if i=j
aiif i>j
and b0
i=
bi+1if i<k
b1+d∈[bi,bi+1]if i=k
biif i>k.
Consequently, b0
i≥biand, if i<n, then ai+1≥a0
i. By case analysis, we get:
77
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
•Case i≥kand i≤j: Then b0
i≥b0
k=b1+d≥a1+c=a0j≥a0
i.
•Case i≥kand i>j: Then b0
i≥bi≥ai=a0
i.
•Case i<k: Then b0
i=bi+1≥ai+1≥a0
i.ut
Lemma 4.6.3.
Let
n,m∈N
,
i∈
[
n
], and
p,p0∈Nn
be
i
-variants with
pi<p0
i
. Then
f(LPT(p,m)) ≤f(LPT(p0,m)).
Proof.
We will compare the executions of LPT for input
p
and
p0
. For input
p
, denote by
l
(
i
)
∈Rm
the vector that, at the end of iteration
i∈
[
n
]
0
, contains the completion times
of all
m
machines, sorted in ascending order. For input
p0
, define
l0
(
i
)correspondingly.
By iteration 0 we denote the precondition l(0) = l0(0) = (0,...,0)of LPT’s inner loop.
Define
q
:=
SORT
(
p
)and
q0
:=
SORT
(
p0
). By Lemma 4.6.2, it holds that
q≤q0
. We
prove by induction that for all iterations
i∈
[
n
]
0
it holds that
l
(
i
)
≤l0
(
i
). Obviously,
the base case
i
=0 is fulfilled by definition. For the induction step
i→
(
i
+1), note that
by definition of LPT, it holds that
l
(
i
+1) =
SORT
(
l−1
(
i
)
,l1
(
i
) +
qn−i
)and
l0
(
i
+1) =
SORT(l0
−1(i),l1(i) + q0
n−i). Hence, l(i+1)≤l0(i+1)due to Lemma 4.6.2.
Now, the makespan of the solution for input
p
is
f
(
LPT
(
p,m
)) =
lm
(
n
)
≤l0
m
(
n
) =
f(LPT(p0,m)), i.e., no greater than the makespan of the solution for input p0.ut
Remark.
For non-identical machines,
LPT
is not monotonic with regard to processing re-
quirements. Let
s
:= (200
,
99),
p
:= (205
,
200
,
150
,
150), and
p0
:= (200
,
200
,
150
,
150).
Then, the corresponding costs are f(LPT(p,s)) = 101
40 and f(LPT(p0,s)) = 110
40 .
200
150
205
150
200
99
200
150
200
150
200
99
Figure 4.3:
For non-identical machines,
LPT
is not monotonic with regard to processing
requirements
As a corollary of Lemma 4.6.3 we get:
Theorem 4.6.4.
For any identical-machine makespan cost-sharing problem (
P||Cmax
) with
succinct representation (
p,m
), where
p1≤ ··· ≤ pn
, it holds that (
LPT,CLPT
)is a
4m−1
3m
-
relaxation and Algorithm 4.5 runs in time O(n3·logm).
Bin Packing and Makespan Costs on Related Machines
We also obtain 2-relaxations
for bin packing and for makespan cost-sharing problems on related machines (
Q||Cmax
).
The key ingredient here is the following monotonic approximation algorithm
RFFD
(“rounded first fit decreasing”) for bin backing:
78
4.6 Applications
Given the vector of item sizes
ς∈
(0
,
1]
n
, round each size up to the next power of 2,
i.e., let item
i
’s rounded size be
ς0
i
:=2
dlog2ςie
for all
i∈
[
n
]. Then, run the
FFD
(first fit
decreasing) algorithm, which is known to produce an optimal packing for this modified
instance
ς0
(see Coffman, Jr. et al.
[15]
). Clearly,
RFFD
is a 2-approximation algorithm
running in time
O
(
n·logn
). Since
RFFD
is optimal for the rounded sizes, it is monotonic.
Lemma 4.6.5.
For any bin packing cost-sharing problem with succinct representation
ς
, where
ς1≤ ··· ≤ ςn
, there is a 2-relaxation for
C
and Algorithm 4.5 runs in time
O(n3·logn).
Proof.
Since
RFFD
is a monotonic 2-approximation algorithm, it holds that (
RFFD,CRFFD
)
is a 2-relaxation. The overall running-time when given as input to Algorithm 4.5 is
O(n3·logn).ut
Remark.
The tight bound of
FFD
is
11
9·opt
+
6
9
,[
20
]. However,
FFD
is not monotonic:
Let
ς
:= (
9
17,9
17,5
17,5
17,5
17
,
4
17,4
17,4
17,3
17,3
17
)and
ς0
:= (
ς−2,8
17
). Then, the corresponding
costs are f(FFD(ς)) = 3 and f(FFD(ς0)) = 4.
33 4
4
4
5
5 5
9 9
3
4
4
4
5
5
5
9
8
3
Figure 4.4: FFD is not monotonic
Remark.
It is known that the
NFD
(next fit decreasing) algorithm is monotonic [
53
]and
a 2-approximation algorithm for the bin packing problem. Hence, also (
NFD,CNFD
)is a
2-relaxation.
Theorem 4.6.6.
For any related-machine makespan cost-sharing problem (
Q||Cmax
) with
succinct representation (
p,s
), where
p1≤···≤ pn
, there is a 2-relaxation. Algorithm 4.5
runs in time O(n3·logm·logPi∈[n]pi).
Proof.
Consider the decision variant of the following modified bin packing problem:
Given
n∈N
items with sizes
ς∈Qn
>0
and
m∈N
bins with capacities
c∈Qm
>0
, decide
whether all
n
items fit into the
m
bins. Let
FFD∗
be the following algorithm: Run
FFD
.
That is, in descending order of item sizes, put every item into the first bin where it fits.
Note here that no changes are necessary to account for the variable bin capacities. If, at
some point an item does not fit any more, return “false”. Otherwise, return “true”.
We show that
FFD∗
is optimal when item sizes are divisible, meaning that every item
size is exactly divided by any smaller item size (cf. also Coffman, Jr. et al.
[15]
). Let
n,m∈N
,
ς∈Qn
>0
be a vector of divisible item sizes, and
c∈Qm
>0
be the capacity vector.
79
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
W.l.o.g., assume
ς1≥···≥ςn
here. Suppose item
j
is the first item that does not fit any
more into one of the
m
bins, i.e.,
FFD∗
(
ς,c
) =
false
. This means that the remaining space
in any bin is less than ςj.
Now note that, for every packing of the first (
j−
1)items, the filling level in each bin is
always a multiple of
ςj
. Consequently, if in some other bin packing there was still space
for item
j
, there would also be a bin for which the filling level exceeds the bin capacity.
This is a contradiction and proves optimality of FFD∗for divisible item sizes.
Now let
RFFD∗
denote the algorithm that first rounds each item size up to the next
power of 2 and then calls
FFD∗
. Due to the observation that
FFD∗
is optimal for divisible
item sizes, we know that
RFFD∗
is monotonic in the item sizes. In order to obtain a
2-relaxation for makespan minimization, we can employ
RFFD∗
together with binary
search (compare also Algorithm 4.7): Within trivial upper and lower bounds, search for
the minimum makespan dso that RFFD∗(p
d,s) = true. ut
4.6.3 Non-Monotonic Approximation Algorithms with a Polynomial-Time
Computable Monotonic Bound
Besides the previous result, we also show how to adapt the PTAS for identical machines
(
P||Cmax
) by Hochbaum and Shmoys
[34]
. Although the running time of the PTAS is
prohibitive for any small
ε
, the result is theoretically interesting: First, any fixed budget
balance greater than 1 can be achieved in polynomial time. Second, the approach here is
different to before: Not the PTAS itself is monotonic but only a bound computed inside
the algorithm.
The basic idea of the PTAS is a reduction to bin packing (see Algorithm 4.7): Given
processing requirements
p∈Nn
, binary search between trivial upper and lower bounds
is employed in order to find a makespan
d
such that the bin packing instance
p
d
does
not need more than
m
bins of capacity (1+
"
), whereas the bin packing instance
p
d−1
does need more than
m
bins. Specifically, the PTAS makes use of
BPDUAL"
, which is an
ε
-dual approximation algorithm for the bin packing problem [
34
, pp.149–151]. For
completeness, it is shown in Algorithm 4.6.
BPDUAL"
outputs solutions that are dual
feasible; this means that
BPDUAL"
uses bins of capacity (1+
ε
)but never needs more bins
than the feasible optimal solution (with capacity 1).
Now, for any bin packing instance
ς∈
(0
,
1]
n
, let
S∗
ς⊇Sς
be the set of all dual-
feasible solutions and
f∗
ς
:
S∗
ς→N
be a function mapping each dual-feasible solution
to its cost, i.e., to the number of used bins. We define
g∗
ς
:
S∗
ς→N
by
g∗
ς
(
Z
):=
max{f∗
ς
(
Z
)
,dPi∈[n]ςie}
. Hence, the crucial property of
g∗
ς
is to guarantee that
g∗
ς
is
never less than the total size of all items. We show that g∗is monotonic.
Lemma 4.6.7.
Let
ς,ς0∈Qn
≥0
be two vectors of item sizes,
i∈
[
n
],
ςi> ς0
i
, and
ς−i
=
ς0
−i
.
Then b :=g∗
ς(BPDUALε(ς)) ≥g∗
ς0(BPDUALε(ς0)) =:b0.
Proof.
By way of contradiction, assume
b<b0
. Now consider an execution of
BPDUALε
for input ς0(see Algorithm 4.6). All items of size < ε are called “small”. Other items of
size
≥ε
are called “large”. In the first phase, round each of the sizes of the large items
80
4.6 Applications
Input: approximation ε∈(0,1); item size vector ς∈(0,1]n
Output: allocation a∈Nn
1:
Partition the interval (
ε,
1]of large sizes into
s
:=
d1
ε2e
equal-length subintervals
(li,li+1]. Use lias rounded size for all original sizes in this interval.
2:
Determine all feasible configurations (
x1,..., xs
)
∈Ns
0
defined by
Ps
i=1xi·li≤
1
(where xiis number of items with size in the interval (li,li+1])
3:
Use dynamic programming to find an allocation of the large items (using rounded
sizes; excluding original sizes ≤ε), based on following the recurrence:
Bins(y1,..., ys):=1+min
(x1,...,xs)
is feasible
configuration
{Bins(y1−x1,..., ys−xs)}
Bins
(
y1,..., ys
)is the minimum number of bins needed when there are
yi
pieces of
size li.
4: Enlarge bins to 1 +εand go back to original sizes
5:
Pack small items with original size
≤ε
into an arbitrary bin containing
≤
1. If no
such bin exists, open a new bin. Let adenote the final allocation of items to bins.
Algorithm 4.6: ε-dual approximation algorithm for bin packing
to one of constantly many sizes and solve this rounded instance optimally without the
small items. Afterwards in the second phase, go back to original sizes and pack small
items one after the other into an arbitrary bin containing
≤
1. If no such bin exists, open
a new bin.
Since the first phase computes optimal solutions of rounded instances, it is monotonic.
Hence,
b<b0
implies that only in the last phase where the small items are packed, the
(
b
+1)-th bin is opened. More precisely, according to line 5 of Algorithm 4.6, there must
be a point where
b
bins are used but all bins contain more than 1. However, this means
that
Pi∈[n]ς0
i>b
, i.e., the total size of all items of instance
ς0
is more than
b
. Specifically,
b=g∗
ς(BPDUALε(ς)) ≥dPi∈[n]ςie≥Pi∈[n]ςi>Pi∈[n]ς0
i>b. A contradiction. ut
Algorithm 4.7 contains the PTAS, together with a crucial extension in line 6. Note that
this line is not necessary for the approximation guarantee, but only needed for mono-
tonicity. For
n,m∈N
and
p∈Nn
, define
SIZE
(
p,m
):=
max{1
m·Pi∈[n]pi,p1,p2,..., pn}
.
Note that
lower
in Algorithm 4.7 is always a lower bound on the optimal makespan:
Since
BPDUALε
is an
ε
-dual approximation algorithm, this holds at the beginning and
also whenever
lower
is updated in line 8. On the other hand,
upper ·
(1+
ε
)is always an
upper bound both on the optimal makespan as well as on the makespan of the schedule
a
. Our crucial extension of the PTAS is as follows: Letting
ς
:=
p
d
, we use the check
g∗
ς
(
BPDUAL"
(
ς
))
≤m
in the binary search (instead of testing
f∗
ς
as in the original PTAS).
Let
lower"
(
p,m
)denote the final value of
lower
returned by Algorithm 4.7 for input
"
,
p
, and
m
. That is,
lower"
(
p,m
)is the minimum
d
for which the check
g∗
ς
(
BPDUAL"
(
ς
))
≤
81
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Input: approximation ε∈(0,1);
vector p∈Nnof processing requirements; number of machines m∈N
Output: allocation a∈[m]n, lower bound on optimum makespan lower
1: upper :=2·SIZE(p,m)
2: lower :=SIZE(p,m)
3: while upper 6=lower do
4: d:=b(upper +lower)/2c
5: a:=BPDUALε(p
d); set bto number of bins used in a
6: b:=max{b,dPi∈[n]
pi
de} .Crucial extension for monotonicity
7: if b>mthen
8: lower :=d+1.Afterwards, still lower ≤d≤upper
9: else
10: upper :=d.Afterwards, still lower ≤d≤upper
11: a:=BPDUALε(p
lower).Not necessary if b≤m
Algorithm 4.7: Modified PTAS for the minimum makespan problem
m
evaluates to true. Moreover, let
HS"
denote the adapted PTAS. Now,
lower"
(
p,m
)is a
lower bound on the optimal makespan and (1+
"
)
·lower"
(
p,m
)is an upper bound on
the makespan of the schedule found by
HS"
. Moreover,
lower"
(
p
)is computed within
HS"
in polynomial time because monotonicity of
g∗
ensures that indeed the minimum
d
is found by the binary search. As a corollary of Lemma 4.6.7, we get:
Theorem 4.6.8.
Let Φbe an identical-machine makespan cost-sharing problem (
P||Cmax
)
with succinct representation (
p,m
), where
p1≤ ··· ≤ pn
. Define the monotonic cost
function
Cmono
(
A
):= (1+
"
)
·lower"
(
INST
(
A
)). Then, (
HSε,Cmono
)is a (1+
"
)-relaxation
for Φ, and Algorithm 4.5 runs in time O(n2+1
"2·logPi∈[n]pi).
4.6.4 Makespan Problems with Monotonic Optimal Costs
There are several mergable makespan problems for which optimal costs are monotonic
and computable in polynomial time. For instance, for the problem of scheduling identical
jobs on identical parallel machines (
P|pi
=
p|Cmax
), it holds that (
LPT,CLPT
)is a 1-
relaxation and Algorithm 4.5 runs in time
O
(
n3·logm
). In the following, we give a
selection of further such problems (see, e.g., Brucker [12]):
•Symmetric costs:
–Q|pi=p|Cmax
•Variable release dates:
–Q|pi=p,ri|Cmax
–Q|pmtn, pi=p,ri|Cmax
82
4.6 Applications
•Variable processing requirements:
–Q|pmtn|Cmax
It is straightforward to see that all of the induced (optimal) cost functions are subadditive
and the problems are mergable. This holds as well for the preemptive case. Moreover, the
optimal costs are always monotonic in the variable property (release dates or processing
requirements) so that determining the most cost-efficient set can always be done in
polynomial time by only checking a single set for each cardinality (see Section 4.5): If
jobs are ordered by increasing value of the variable property, the first
k
jobs minimize
the cost over all sets of cardinality
k
. Consequently, we get that 1-relaxations exist for all
of the above problems.
Theorem 4.6.9.
For sharing the (optimal) cost induced by any of the above makespan
problems, there is a 1-BB and 2
Hn
-EFF egalitarian mechanism. Its outcome can be computed
in polynomial time.
4.6.5 Scheduling Problems with Supermodular Costs
We find it interesting to note that Brenner and Schäfer’s singleton mechanisms [
11
]for
P||PCi
and 1
||PwiCi
are in fact egalitarian mechanisms based on most cost-efficient
set selection: For these problems, the induced (optimal) cost functions are supermodular
(see, e.g., Schulz and Uhan
[68]
). Moreover, for any set of remaining players
Q
and any
set of already accepted players
N
, a player
i∈Q\N
with minimal
wi/pi
constitutes a
most cost-efficient set—and indeed, singleton mechanisms always choose one of these
singleton sets. In particular, the assigned cost shares of each job are equal to the
completion time under Smith’s rule [
73
], an algorithm which assigns the jobs in the
order of increasing ratios
wi/pi
and which is known to deliver optimal schedules for the
above problems in polynomial time.
For
P||PwiCi
, computing optimal costs is
NP
-hard, but Smith’s rule guarantees an
approximation ratio of (1+
p2
)
/
2
≈
1
.
21, [
40
]. It is easy to verify that the costs
induced by Smith’s rule are supermodular. Somewhat unsurprisingly now, the egalitarian
mechanisms induced by always choosing the most cost-efficient set with respect to this
approximation cost are again equivalent to the singleton mechanisms by Brenner and
Schäfer [11]for this problem.
Since the order in which players are offered prices is constant, the above mechanisms
are in fact even sequential stand-alone mechanisms. Thus, all of the previously mentioned
subclasses of the acyclic-mechanism framework coincide here in a natural way.
For the objective to minimize the completion time on identical parallel machines,
Brenner and Schäfer
[11]
showed that optimal costs are 2-subadditive. Yet, it is a simple
observation that the very same proof holds also for related parallel machines:
Proposition 4.6.10 (Brenner and Schäfer [11]).
Let
C
be the optimal cost function in-
duced by Q||PwiCi. Then, C is 2-subadditive.
83
Chapter 4 Cost Sharing Without Indifferences: To Be or Not to Be (Served)
Now recall Theorem 4.4.10 stating that when
β
-approximate costs are supermodular
and
α
-subadditive, simple sequential stand-alone mechanisms guarantee
β
-BB and
(
α·β
)-EFF (regardless of the order of the players). This implies, of course, that the
above mechanisms for
P||PCi
and 1
||PwiCi
are 1-BB and 2-EFF and the mechanisms
for
P||PwiCi
are 1.21-BB and 2.42-EFF, as previously shown by Brenner and Schäfer
[11]
. Another polynomial-time solvable problem with supermodular optimal costs is
Q|pi=1|PwiCi,[68]. We therefore obtain the following new result:
Theorem 4.6.11.
For sharing the (optimal) cost induced by
Q|pi
=1
|PwiCi
, every se-
quential stand-alone cost-sharing mechanism is 1-BB, 2-EFF, and computable in polynomial
time.
As a last remark, the recovered costs of a cross-monotonic cost-sharing methods are
always subadditive. Consequently, it is not surprising that Moulin mechanisms suffer bad
budget balance if the underlying optimization problem severely violates subadditivity.
E.g., for the problem 1
||PiCi
, no cross-monotonic cost-sharing method can be better
than
n+1
2
-BB [
10
]. Obviously, the SGSP mechanisms discussed above tremendously
improve on this.
4.7 Conclusion
The pivotal point of this chapter was to study cost-sharing scenarios where the case that
a player feels indifferent about being served is negligible. We believe that SGSP (or
one of the other collusion-resistance properties without indifferences) is often a viable
replacement for the often too limiting GSP requirement. We consider the main asset of
our work to be fourfold: (a) Characterizing the relationship between the new collusion-
resistance properties. (b) Egalitarian mechanisms; showing existence of SGSP, 1-BB, and
2
Hn
-EFF mechanisms for any non-decreasing subadditive costs. (c) Our framework for
polynomial-time computability that reduces constructing SGSP,
O
(1)-BB, and
O
(
logn
)-
EFF mechanisms to finding monotonic approximation algorithms. (d) Showing that
acyclic mechanisms are robust against the scheme underbidders are removed; as a
consequence, they comprise egalitarian mechanisms and are SGSP—i.e., in a precise
sense, they are remarkably stronger than was known before.
Of course, several immediate issues are left often by our work:
•
Which other combinatorial optimizations problems can our polynomial-time frame-
work be applied to?
•
It is easy to see that rooted Steiner tree cost-sharing problems are mergable and
their costs non-decreasing and subadditive; but do they allow for a β-relaxation?
•
What are lower bounds on the performance guarantees by polynomial-time acyclic
mechanism?
84
Chapter 5
Does Coalition Size Matter?
5.1 Overview of Contribution
In this chapter, we concentrate on the question whether reducing the maximum coalition
size that a mechanism should withstand allows for a richer set of possible mechanisms.
We say a mechanism is
k
-GSP (or
k
-WGSP, respectively) if it ensures collusion resistance
up to coalition size k. In detail, our results are:
•
While we give (arguably artificial) cost-sharing mechanisms that are
k
-GSP but
not
(k+1)
-GSP, we obtain as our main result that already 2-GSP is equivalent
to GSP once we require mechanisms to be separable, i.e., cost shares must only
depend on the set of served players (and not directly on the bids). We remark that
no general technique for the design of truthful cost-sharing mechanisms is known
that violates separability. Our result can be seen as a generalization of the main
theorem in an article by Mutuswami [55].
•
In contrast to the previous result, WGSP is not equivalent to 2-WGSP plus separa-
bility.
•Even without separability, 2-GSP implies WGSP.
We regard the chief asset of our work to be threefold: First, our results indicate that the
substantial “jump” in collusion resistance seems to occur from 1-GSP =SP to 2-GSP and
not from Θ(1)-GSP to
ω
(1)-GSP. Second, GSP is often felt to be too strong an axiom with
unrealistic implications on players’ capabilities and behavior; now, the fact that GSP is
equivalent to merely 2-GSP plus separability gives some a posteriori justification for GSP.
Third and last, we firmly believe that our characterizations will facilitate devising and
understanding new GSP cost-sharing mechanisms.
5.2 Notions of Non-Manipulability by Small Coalitions
Demand for a certain collusion resistance implies assumptions on players’ behavior and
their coalition-forming capabilities: For instance, if (a) side-payments are unlikely but
(b) players yet have virtually unlimited means to communicate and (c) one expects them
to help others even for no personal reward (e.g., by voluntary non-participation in case
of indifference), then GSP is an appropriate axiom. Similarly, when players have no
85
Chapter 5 Does Coalition Size Matter?
MoneyNone
None
Each
with all
Communication
Transfers
Service
GSP
k-GSP
SP
WGSP UGSP
bribe-proof
Figure 5.1: Two dimensions of coalition-forming capabilities
means to communicate at all, then simple SP is probably sufficient. One can also think
of collusion resistance at the other end of the spectrum: We use the term “ultimate
group-strategyproofness” (UGSP) here if a mechanism even prevents that coalitions can
improve their total utility by manipulation. Essentially, WGSP, GSP, and UGSP imply
different levels of transfers that coalitions might accomplish in order to be successful.
Figure 5.1 provides a schematic illustration.
Since it seems unlikely that all players can communicate with each other and make
binding agreements on collective manipulation, this gives rise to the following natural
question: Can we increase the degree of freedom for designing cost-sharing mechanisms
by relaxing the GSP requirement with respect to coalition sizes? Surprisingly, we prove
in the rest of this chapter that the answer is essentially “no”.
Definition 5.2.1.
A cost-sharing mechanism
M
is
k
-GSP (or
k
-WGSP) if for all true valua-
tions
v∈Rn
and all non-empty coalitions
K⊆
[
n
]with
|K|≤ k
there there is no
K
-variant
bof vwith uK(b)>uK(v)(or uK(b)uK(v), respectively).
We remark that 2-GSP is equal to what Serizawa
[70]
called “pairwise SP”. Moreover,
note that 2-GSP immediately implies 2-WGSP, SP, and WUNB.
5.2.1 Resistance Against Coalitions with Side-Payments
We show in the following that resistance against coalitions is essentially infeasible if
we assume that players are capable of organizing side-payments. This is an immediate
corollary of a result by Schummer [69].
Definition 5.2.2 (Schummer [69]).
A mechanisms
M
is bribe-proof if for all all players
i,j∈
[
n
], all side-payments
p∈R
, and all true valuations
v∈Rn
there is no
i
-variant
b
of
vso that ui(b) + p>ui(v)and uj(b)−p>uj(v).
Note that neither
i
=
j
nor
p
=0 are excluded in the definition, hence bribe-proof
implies SP. Intuitively, in a bribe-proof mechanism, player
j
cannot bribe
i
with
p
units
of money into misreporting his valuation.
Schummer
[69]
calls bribe-proof the “weakest intuitive condition” that rules out
misreports of coalitions of size two, when assuming that players are capable of side-
payments. Indeed, the following is an easy observation:
86
5.2 Notions of Non-Manipulability by Small Coalitions
Lemma 5.2.3.
Let
M
be a cost-sharing mechanism. Then
M
is bribe-proof if and only
if for all true valuations
v
and all players
i,j∈
[
n
]there is no
i
-variant
b
of
v
so that
ui(b) + uj(b)>ui(v) + uj(v).
Proof.
Verifying necessity (“
⇐
”) is trivial, so we only consider sufficiency (“
⇒
”). Suppose
ui(b)
+
uj(b)
=
ui(v)
+
uj(v)
+
"
for some
" >
0. Define
p
:=
uj(b)−uj(v)−"
2
. Then
ui(b)
+
p
=
ui(b)
+
uj(b)−uj(v)−"
2
=
ui(v)
+
uj(v)−uj(v)
+
"
2>ui(v).
Moreover,
uj(b)−p=uj(b)−uj(b) + uj(v) + "
2>uj(v).ut
One of Schummer’s results is that if the domain of players’ types is path-connected
1
and the set of alternatives is finite, then each player’s utility does not depend on the
other players’ bids. For our purposes, since the domain of valuations (i.e., types) is the
Euclidean space and thus path-connected, and since there are only finitely many subsets
of players, it is sufficient to note:
Proposition 5.2.4 (Schummer [69]).
Let
M
be a bribe-proof cost-sharing mechanism
and
i∈
[
n
]be an arbitrary player with true valuation
vi
. Suppose
b,b0
are ([
n
]
\i
)-variants.
Then ui(b) = ui(b0).
As an immediate consequence we get that bribe-proofness rules out all but trivial cost-
sharing mechanisms:
Corollary 5.2.5.
Suppose
M
is a bribe-proof mechanism. Then, for all players
i∈
[
n
], the
threshold value θi(b−i)is a constant.
5.2.2 Some Preliminary Implications by SP and WUNB
We start with some immediate consequences of SP and WUNB that will be needed
throughout this chapter. Since service-allocation vectors will be more convenient than
sets of served players, a mechanism will henceforth be denoted
M
= (
q,x
)where
q:Rn→{0,1}n.
Lemma 5.2.6.
Let
M
= (
q,x
)be a SP cost-sharing mechanism,
v∈Rn
contain the true
valuations, i ∈[n]be an arbitrary player, and bbe an i-variant of v. Then:
i) ui(b)<ui(v)and qi
(
v
) = 1=
⇒qi
(
b
) = 0
,ui(b)
=0
<ui(v)
, and
bi≤θi
(
v−i
) =
xi(v)<vi
ii) ui(b)<ui(v)and qi
(
v
) = 0=
⇒qi
(
b
) = 1
,ui(b)<
0=
ui(v)
, and
bi≥θi
(
v−i
) =
xi(b)>vi
Proof. i)
If
bi>xi
(
v
), then
i
could manipulate and improve at
b
by bidding
vi
; hence
bi≤xi
(
v
)due to SP. If
qi
(
b
) = 1, then
bi≥xi
(
b
)due to VP and
xi
(
b
)
>xi
(
v
)
because
ui(b)<ui(v)
by assumption; a contradiction. Hence,
qi
(
b
) = 0. Then,
ui(b) = 0 and vi>xi(v)because ui(v)>0.
1
A topological space
X
is path-connected if every two points
x,y∈X
can be be connected by a path, i.e.,
there is a continuous function f:[0,1]→Xwith f(0) = xand f(1) = y.
87
Chapter 5 Does Coalition Size Matter?
ii) This follows immediately because ui(b)<ui(v) = 0 due to VP. ut
Lemma 5.2.7.
Let
M
be a WUNB cost-sharing mechanism,
v∈Rn
contain the true
valuations,
i∈
[
n
]be an arbitrary player, and
b
be an
i
-variant of
v
. Then:
Mi
(
b
) =
Mi(v) =⇒u(b) = u(v).
Proof.
If there was as player
j
so that, w.l.o.g.,
uj(b)>uj(v)
, then player
i
could help
j
at vby bidding bi. A contradiction to WUNB. ut
Lemma 5.2.8.
Let
M
= (
q,x
)be a SP and WUNB cost-sharing mechanism,
v∈Rn
contain
the true valuations,
i∈
[
n
]be an arbitrary player, and
b
be an
i
-variant of
v
. Moreover, let
j∈[n]\i. Then:
i) uj(b)>uj(v) =⇒ui(b)<ui(v)
ii) uj(b)<uj(v)and qi(v) = 1=⇒qi(b) = 0and bi< θi(v−i) = xi(v)≤vi
iii) uj(b)<uj(v)and qi(v) = 0=⇒qi(b) = 1and bi> θi(v−i) = xi(b)≥vi
iv) vi=θi(v−i) =⇒uj(b)≤uj(v)
v) u−i(b)≤u−i(v)or u−i(b)≥u−i(v)
vi) uj(b)>uj(v) =⇒θj(b−j)< θj(v−j)
Proof. i) This is a trivial consequence of WUNB and SP.
ii)
By WUNB and the threshold property, it holds that
bi<xi
(
v
)because otherwise
player icould help jat bby bidding vi. Hence, qi(b) = 0.
iii)
By WUNB, it holds that
qi
(
b
) = 1 and
bi>xi
(
b
)because otherwise
ui(b|bi)
=0
and player icould help jat bby bidding vi.
iv)
By the threshold property, it holds that
ui(v)
=
ui(b)
=0. Hence, the proof follows
by (i).
v)
By way of contradiction, assume there are
j,k∈
[
n
]
\i
with
uj(b)<uj(v)
and
uk(b)>uk(v)
. Due to (ii) and (iii), player
i
gets the service for either
v
or
b
, but
not for both. We may assume w.l.o.g. that
qi
(
v
) = 1 and
qi
(
b
) = 0. Let now
b0
be
another
i
-variant of
v
and
b
with
b0
i
:=
xi
(
v
). Then the threshold property and
(iv) ensure
uj(v)≤uj(b0)
and
uk(b)≤uk(b0)
. It follows that if
qi
(
b0
) = 1, then
i
can help
k
at
v
by bidding
b0
i
. Correspondingly, if
qi
(
b0
) = 0, then
i
could help
j
at
bby bidding b0
i. A contradiction to WUNB.
vi)
Note that
ui(b)<ui(v)
by (i). Since
uj(b)>uj(v)≥
0, it holds that
qj
(
b
) = 1
and
xj
(
b
)
<vj
. Now, if
qj
(
v
) = 1, then
θj
(
b−j
) =
xj
(
b
)
<xj
(
v
) =
θj
(
v−j
). On
the other hand, if
qj
(
v
) = 0, then
θj
(
b−j
) =
xj
(
b
)
<bj
=
vj≤θj
(
v−j
), where the
last inequality is due to the threshold property. ut
88
5.3 Group-Strategyproofness Against Only Two Players
5.2.3 k-GSP Is Strictly Weaker Than GSP
Before establishing the link between 2-GSP and GSP in the next sections, we give an
example showing that
k
-GSP is not equivalent to GSP, for arbitrary
k<n
. Consider the
mechanism defined by Algorithm 5.1.
Input: bid vector b∈R3
Output: service allocation q∈{0,1}3; cost shares x∈R3
≥0
1: if b= (1,1,1)then q:= (1,1,1);x:= (1,1,1)
2: else
3: q:= (0,0,0);x:= (0,0,0);θ:= (1,1,1)
4: if b1>1 and b2>1then θ3:=2
5: for all i∈{1,2,3}with bi> θido qi:=1; xi:=θi
Algorithm 5.1: 3-Player mechanism that is 2-GSP but not GSP
It is easy to see that this mechanism is 2-GSP because for any true valuations
v
the
only player that could ever improve is player 3. In this case, however,
v1>
1 and
v2>
1,
so in order to help player 3 both players 1 and 2 have to deviate. We remark that
Algorithm 5.1 can be generalized for nplayers so that it is (n−1)-GSP but not n-GSP.
5.3 Group-Strategyproofness Against Only Two Players
5.3.1 Upper Continuity and 2-GSP Together Imply GSP
As a simple starting point, we first consider upper-continuous mechanisms. Recall that
upper continuity is a straightforward way for dealing with indifferent players, in that
they are always included in the set of served players. Upper-continuity is, e.g., fulfilled
by Moulin mechanism and by acyclic mechanisms, but not by our symmetric mechanisms
from Chapter 3. We first need the following simple observation that is essentially
2
a
corollary of Lemma 4.2.5.
Lemma 5.3.1.
Let
M
= (
q,x
)be an upper-continuous 2-GSP cost-sharing mechanism.
Then M is also ONB.
Thus, combined with a result by Mutuswami
[55]
, upper continuity and 2-GSP together
imply GSP:
Proposition 5.3.2 (Mutuswami [55]).
Let
M
be a SP, ONB, and WUNB cost-sharing
mechanism. Then, M is also GSP.
2
In detail, Lemma 4.2.5 states that WSGSP and upper continuity together imply ONB. It is a simple
observation that WSGSP could be replaced by “2-WSGSP” without invalidating the proof. Clearly, 2-GSP
implies “2-WSGSP”.
89
Chapter 5 Does Coalition Size Matter?
Corollary 5.3.3.
Let
M
be an upper-continuous 2-GSP cost-sharing mechanism. Then,
M
is also GSP.
We remark here that Mutuswami
[55]
assumes non-negative bids in his work, i.e., that
the domain of valuations is restricted to
Rn
≥0
. Yet, his proof can be used without changes
also for the more general setting where negative bids are allowed. In other words, the
previous results of this subsection only require CS but not strong CS.
5.3.2 Separability and 2-GSP Together Imply GSP
We now generalize Corollary 5.3.3 to arbitrary separable mechanisms. Specifically, we
will obtain as our main result that a 2-GSP cost-sharing mechanism is GSP if and only if
it is separable. We start with an auxiliary lemma, stating that every 2-GSP cost-sharing
mechanism is at least resistant against coalitions where deviators either do not participate
(submit a negative bid) or bid very much.
Lemma 5.3.4.
Let
M
= (
q,x
)be a 2-GSP cost-sharing mechanism,
v∈Rn
contain the
true valuations,
K⊆
[
n
]be a non-empty coalition, and
b
be a
K
-variant of
v
so that for
all
i∈K
:
bi∈{−
1
,b∞}
. Then, either
ui(b)
=
ui(v)
for all
i∈K
or
ui(b)<ui(v)
for at
least one i ∈K.
In particular, Lemma 5.3.4 implies that a 2-GSP mechanism always computes a Pareto-
optimal outcome when given truthful bids, meaning that all other outcomes either
provide all players with the same utility or make at least one player worse off.
Proof.
By way of contradiction, assume that
uK(b)>uK(v)
. Roughly speaking, we will
look at what happens when players adopt the bids bin a sequential fashion.
W.l.o.g., we may assume that players are numbered so that
K
= [
k
],
uk(b)>uk(v)
,
and there is an
m∈ {
0
...k−
1
}
so that
bi
=
−
1 for
i∈
[
m
]and
bi
=
b∞
for
i∈
{m
+1
...k}
. Note that
m
=0 is possible, but
m
=
k
is not. For
i∈ {
0
...k}
, define
bi:= (v−[i],b[i]). Clearly, b0=vand bk=b.
Our assumptions, together with VP, imply for all
i∈
[
m
]that 0 =
ui(bi)
=
ui(b)
=
ui(v). Now an inductive argument yields for all i∈[m]that
∀j∈{i+1...n}:uj(bi)≤uj(v). (5.3.5)
Clearly, this holds for the base case
i
=1 due to WUNB. Now for the induction step
i→
(
i
+1), suppose the induction hypothesis
(5.3.5)
holds for
i
. Then 0
VP
≤ui+1(bi)IH
≤
ui+1(v)
=0. Since also
ui+1(bi+1)
=0, we have for all
j∈{i
+2
...n}
that
uj(bi+1)≤
uj(bi)IH
≤uj(v)
, where the first inequality is again due to WUNB. Consequently,
(5.3.5)
holds also for i+1.
Let now
p:=max{i∈{m+1...k}|ui(bi)<ui(bi−1)}(5.3.6)
90
5.3 Group-Strategyproofness Against Only Two Players
be the “last” player who loses utility when adopting bid
bp
. Since
uk(b)>uk(v)(5.3.5)
≥
uk(bm)
and due to Lemma 5.2.8 (i), we have that
(5.3.6)
is well-defined. Lemma 5.2.6
together with qp(bp) = 1 implies
qp(bp−1) = 0 and up(bp)<up(bp−1) = 0VP
≤up(v). (5.3.7)
By definition of
p
and again by Lemma 5.2.8 (i), it must hold that
up(b|bp)≤up(bp|bp)
because otherwise
p
would not have been maximal. Now recall that
qp
(
b
) = 1 due to
bp
=
b∞
. Hence,
xp
(
b
)
≥xp
(
bp
)
(5.3.7)
>vp
and so
up(b)<
0. This is a contradiction to
p∈K.ut
Theorem 5.3.8.
Let
M
= (
q,x
)be a separable 2-GSP cost-sharing mechanism. Then
M
is
also GSP.
Proof.
We show for all
k∈ {
3
...n}
that if
M
is (
k−
1)-GSP, then
M
is also
k
-GSP. The
proof of this statement is by contradiction. Let
v
be the true valuation vector, suppose
k∈{
3
...n}
, and assume there are a coalition
K⊆
[
n
]with
|K|
=
k
and a
K
-variant
b
of
vso that uK(b)>uK(v).
Outline of contradiction
Roughly speaking, we proceed as follows: Starting from the
true valuations
v
, we let the players in
K
adopt the bid vector
b
in a sequential fashion.
This process is divided into two phases: First, all those players deviate who gain the
service for
b
but not increased utility (compared to
v
). In the second phase, all other
players switch to the bids as in
b
. It will turn out that the utilities in the second phase
are essentially stagnant, so the crucial changes in utility have to occur during the first
phase. This yields a contradiction, both when the first phase is short (at most one player)
and when it is long.
Remaining Details
We first note that our assumption implies
k<n
and
∃i∈
[
n
]
\K
:
ui
(
b
)
<ui
(
v
). Otherwise, due to having a cost-sharing method, the grand coalition [
n
]
would also be successful by bidding
b0∈Rn
defined by
b0
i
=
b∞
if
qi
(
b
) = 1 and
b0
i
=
−
1
otherwise. This is a contradiction to Lemma 5.3.4. Moreover, we have for all i∈Kthat
∃j∈K\i:uj(v−i,bi)>uj(v), (5.3.9)
so
∀i∈K
:
ui(v−i,bi)<ui(v)
due to Lemma 5.2.8 (i). Otherwise, if for some
i∈K
there was no
j∈K\i
with
uj(v−i,bi)>uj(v)
, then the coalition
K\i
could improve at
(v−i,bi)by bidding as in b, which contradicts (k−1)-GSP. Now Lemma 5.2.6 implies:
Claim 1. For each player i ∈K, exactly one of the following two conditions holds:
i) bi≥θi(v−i)>vi, qi(v) = 0, ui(v) = 0
ii) bi≤θi(v−i)<vi, qi(v) = 1, ui(v)>0, and qi(b) = 1
91
Chapter 5 Does Coalition Size Matter?
For notational convenience and w.l.o.g., we assume that players are numbered such
that K= [k],un(b)<un(v), and there is a λ∈[k]with
•for all i∈{1...λ−1}:qi(v) = 0, qi(b) = 1, and xi(b) = vi,
•for all i∈{λ...k}:ui(b)>ui(v)or Mi(b) = Mi(v).
This is not a restrictive assumption, because the case
qi
(
v
) = 1 but
qi
(
b
) = 0 cannot
occur by Claim 1.
We now look at what happens if players adopt the bid vector
b
in a sequential fashion.
As an abbreviating notation, we define for all
S⊆
[
n
]the vector
bS
:= (
v−S,bS
). Roughly
speaking, the following technical claim says that utilities stay fixed in the second phase,
i.e., once the players in
{
1
...λ−
1
}
(those who gain the service for
b
but not increased
utility) have deviated to the bids as in b.
Claim 2. Suppose S is a set of players with {1...λ−1}⊆S⊆[k]. If |S|≥2, then
∀i∈[n]:ui(b|bS
i) = ui(bS|bS
i). (5.3.10)
Moreover, if |S|=1then
∀i∈[n]:ui(b|bS
i)≥ui(bS|bS
i). (5.3.11)
Proof (of Claim 2).
We prove by induction on the size
s∈{max{
1
,λ−
1
}...k}
of
S
that
(5.3.10)
and
(5.3.11)
hold (provided that the respective constraints on
S
are fulfilled).
The base case
s
=
k
holds trivially because it implies
S
= [
k
]and hence
bS
=
b
. We
therefore only need to consider the induction step.
Induction Step (s→s−
1
)
Assume that
(5.3.10)
is fulfilled for all sets
S
with
|S|≥s
.
Fix now some set
S
with
|S|
=
s−
1. We show, by a sequence of substeps, that
(5.3.10)
and
(5.3.11)
hold. Let
`∈
[
k
]
\S
be a player and define
T
:=
S∪`
. Note that
`∈{λ...k}
.
i) If |S|≥2, then it holds that
∀i∈[n]\`:ui(bT|bS
i)≤ui(bS|bS
i).
By way of contradiction, assume that
ui(bT|bS
i)>ui(bS|bS
i)
for some
i∈
[
n
]
\
`
. Then
u`(bT)<u`(bS)
by Lemma 5.2.8 (i). If
b`>v`
then
q`
(
bS
) = 0 by
Lemma 5.2.6. On the other hand, if
b`<v`
, then
q`
(
bS
)
L5.2.6
=
1
C1
=q
(
b
)and
x`(b)VP
≤b`
P2.1.8
≤x`(bS). In both cases, u`(b)≥u`(bS).
Now, due to the induction hypothesis, we have for all
j∈
[
n
]
\`
that
uj(b|bS
j)
=
uj(b|bT
j)IH
=uj(bT|bT
j)
=
uj(bT|bS
j)L5.2.8(v)
≥uj(bS|bS
j)
. For player
i
, the last
inequality is strict. Hence, the coalition
K0
:= ([
k
]
\S
)
∪i
can manipulate and
help
i
at
bS
, by bidding as in
b
. This is a contradiction to (
k−
1)-GSP because
|K0|≤ k−|S|+1≤k−1.
92
5.3 Group-Strategyproofness Against Only Two Players
ii) If |S|≥1, then it also holds that
∀i∈[n]\`:ui(bT|bS
i)≥ui(bS|bS
i).
Again, assume by way of contradiction that
ui(bT|bS
i)<ui(bS|bS
i)
for some
i∈
[
n
]
\`
. By Lemma 5.2.8 (ii) and (iii), we have
q`
(
bT
)
6
=
q`
(
bS
). Consider the
two cases:
•Case b`<v`:
Then,
q`
(
bS
)
P2.1.8
=
1
C1
=q`
(
b
),
b`
L5.2.8(ii)
<x`
(
bS
), and
q`
(
bT
) = 0. Since we
also have
u`(b|b`)IH
=u`(bT|b`)
=0 by the induction hypothesis, it follows
that x`(b) = b`. Altogether, u`(b)>u`(bS).
Now, there must be a player j∈[k]\Twith
0≤uj(b)<uj(bS), (5.3.12)
so
qj
(
bS
) = 1. Otherwise, the coalition
K0
:= [
k
]
\S
could manipulate and
help
`
at
bS
, by bidding as in
b
. This is a contradiction to (
k−
1)-GSP because
|K0|=k−|S|≤ k−1.
Define
W
:=
S∪j
. Since
b`<v`
, we have that
u`(bW)IH
=u`(b)C1
>
0, so
M`(bW) = M`(b).
Since
u`(bW)
=
u`(b)>u`(bS)
, it follows by Lemma 5.2.8 (i) that
uj(bW)<
uj(bS)
. Together with
qj
(
bS
)
(5.3.12)
=
1, Lemma 5.2.6 (i) implies now
qj
(
bW
) =
0 and
bj<vj
, so
qj
(
b
)
C1
=
1. Then, since
uj(b|bj)IH
=uj(bW|bj)
=0, it must
hold that
bj
=
xj
(
b
)
(5.3.12)
>xj
(
bS
). This is a contradiction to SP because
player jcould improve at bWby bidding vj.
•Case b`>v`:
Then,
q`
(
bT
) = 1 and
v`≤x`
(
bT
)
<b`
due to Lemma 5.2.8 (iii). Since
u`(b|b`)IH
=u`(bT|b`)>
0, we have
q`
(
v
)
C1
=
0,
q`
(
b
) = 1 and
x`
(
b
) =
x`
(
bT
)
≥v`
. However, this contradicts
`∈{λ...k}
because neither
u`(b)>
u`(v)nor M`(v) = M`(b).
iii)
We can now complete the induction step and show that
(5.3.10)
and
(5.3.11)
hold.
Consider first a player
i∈
[
n
]
\`
. Then
ui(b|bS
i)
=
ui(b|bT
i)IH
=ui(bT|bT
i)
=
ui(bT|bS
i). Hence, if |S|≥2 then the previous substeps (i) and (ii) imply
ui(b|bS
i) = ui(bS|bS
i).
If |S|=1, then substep (ii) implies
ui(b|bS
i)≥ui(bS|bS
i).
93
Chapter 5 Does Coalition Size Matter?
Now consider player
`
. It holds that
u`(b)≤u`(bS)
because otherwise
K0
:= [
k
]
\S
could manipulate and help
`
at
bS
by bidding as in
b
. Since
|K0|
=
k−|S|≤ k−
1,
this would be a contradiction to (
k−
1)-GSP. Now, if the inequality was strict, i.e.,
if 0 ≤u`(b)<u`(bS), then q`(bS) = 1 and x`(bS)<v`. Consider the two cases:
•Case q`(b) = 0:
Then
b`
C1
>v`
, so
M`
(
bT
)
P2.1.8
=M`
(
bS
)and, in particular,
x`
(
bT
) =
x`
(
bS
)
<
v`<b`.
Consequently,
u`(b|b`)IH
=u`(bT|b`)>
0. A contradiction to
q`(b) = 0.
•Case q`(b) = 1:
Then
x`
(
bS
)
<x`
(
b
)
VP
≤b`
, so
M`
(
bT
)
P2.1.8
=M`
(
bS
). However, due to
u`(b|b`)IH
=u`(bT|b`)>
0, we have then
x`
(
b
) =
x`
(
bT
) =
x`
(
bS
). A
contradiction.
Hence, it must hold that u`(b) = u`(bS). This completes the proof of the claim.
We now consider the first phase and show that, as long as only players in
{
1
...λ−
1
}
have deviated to the bids as in
b
, there is always a player who strictly benefits when also
the remaining players switch to b.
Claim 3. Suppose `∈{1...λ−1}. Then
∃i∈[`]:ui(b|bi)>ui(b[`]|bi).
Proof (of Claim 3).
For the base case
`
=1, note that Claim 1 and the fact that 1
∈
{
1
...λ−
1
}
imply
q1
(
v
) = 0,
q1
(
b1
) =
q1
(
b
) = 1, and
b1≥x1
(
b1
)
>v1
=
x1
(
b
); so
u1(b|b1)>u1(b1|b1).
For the induction step (
`→`
+1), assume the induction hypothesis holds for
`
, i.e.,
there is some i∈[`]with ui(b|bi)>ui(b[`]|bi). Now either
•ui(b|bi)>ui(b[`]|bi)≥ui(b[`+1]|bi)or
•ui(b[`+1]|bi)>ui(b[`]|bi)
, in which case we have
u`+1(b[`+1])<u`+1(b[`])
due
to Lemma 5.2.8 (i) and therefore
q`+1
(
b[`+1]
)
6
=
q`+1
(
b[`]
)due to Lemma 5.2.6.
Together with
b`+1
C1
>v`+1
because of the fact that
`
+1
∈{
1
...λ−
1
}
, this implies
q`+1
(
b[`+1]
)
P2.1.8
=
1,
q`+1
(
b
)
C1
=
1,
q`+1
(
b[`]
) = 0, and
b`+1
VP
≥x`+1
(
b[`+1]
)
L5.2.6
>
v`+1=x`+1(b). Consequently, u`+1(b|b`+1)>u`+1(b[`+1]|b`+1).
We now have everything ready to disprove that [
k
]is a successful coalition. If
λ≥
3,
then Claim 2 implies that
∀i∈
[
λ−
1]:
ui(b|bi)
=
ui(b[λ−1]|bi)
whereas Claim 3 says
∃i∈[λ−1]:ui(b|bi)>ui(b[λ−1]|bi). This is a contradiction.
On the other hand, if
λ≤
2, then Claim 2 implies that
un(b)≥un(b1)
. Since also
u−1(b1)≥u−1(v)
by
(5.3.9)
and Lemma 5.2.8 (v), it holds that
un(b)≥un(v)
. Again a
contradiction. This proves the theorem. ut
94
5.4 Weak Group-Strategyproofness and Non-Bossiness
Now recall that every GSP cost-sharing mechanism is separable. This has first been
observed by Moulin
[51]
and is also the result of our more general Theorem 4.2.9. We
thus obtain the following characterization:
Corollary 5.3.13.
Let
M
be a cost-sharing mechanism. Then,
M
is GSP if and only if it is
2-GSP and separable.
5.4 Weak Group-Strategyproofness and Non-Bossiness
5.4.1 Separability and 2-WGSP Do Not Imply WGSP
A natural question is whether a statement similar to Theorem 5.3.8 holds also for WGSP.
We give an example showing that this not the case. Consider the mechanism
M
= (
q,x
)
defined by Algorithm 5.2.
Input: bid vector b∈R6
Output: service allocation q∈{0,1}6; cost shares x∈R6
≥0
1: q:= (0,...,0),x:= (0,...,0)
2: for all i∈{4,5,6}with bi>1 or (bi=1 and b1+(i−3 mod 3)≥2)do
3: qi:=1, xi:=1
4: for all i∈{1,2,3}with bi≥1+qi+3do
5: qi:=1, xi:=1+qi+3
Algorithm 5.2: Separable mechanism that is 2-WGSP but not WGSP
The unique cost-sharing method ξ:{0,1}n→Rn
≥0of mechanism Mis given by
ξi(s):=
0 if si=0
1 otherwise, if i∈{4,5,6}or (i∈{1,2,3}and si+3=0)
2 otherwise, if i∈{1,2,3}and si+3=1
Note that the threshold property is fulfilled: For players
i
=1
,
2
,
3 the threshold value is
θi
(
b−i
) = 2 if
bi+3>
1 or (
bi+3
=1 and
b1+(imod 3)≥
2). It is
θi
(
b−i
) = 1 otherwise.
For players i=4,5,6, the threshold value is constant, θi(b−i) = 1.
The only players who could ever improve are 1, 2, and 3. However, no subset
S⊂ {
1
,
2
,
3
}
of size
|S|
=2 can jointly improve because there is always a player
i∈S
whose threshold value does not depend on
bS
. Hence,
M
is 2-WGSP. However,
it is not 3-WGSP: Let
v
= (2
,
2
,
2
,
1
,
1
,
1)be the true valuations vector and consider
b
= (1
,
1
,
1
,
1
,
1
,
1). Then,
Q
(
v
) =
{
1
...
6
}
and
Q
(
b
) =
{
1
,
2
,
3
}
, so
{
1
,
2
,
3
}
is a successful
coalition.
95
Chapter 5 Does Coalition Size Matter?
5.4.2 2-GSP Implies WGSP
We now completely drop separability and show that already 2-GSP alone implies WGSP.
Hence, this is a case where a stronger notion of collusion resistance, yet only for players
with limited communication abilities, implies a weaker collusion resistance against
coalitions of arbitrary size.
Theorem 5.4.1.
Let
M
= (
q,x
)be a 2-GSP cost-sharing mechanism. Then,
M
is also
WGSP.
Proof.
The proof is by induction over the size
m∈
[
n
]of successful coalitions. That is,
we show for all
m∈
[
n
]that
M
is
m
-WGSP. Clearly, the base cases
m
=1 and
m
=2 are
fulfilled by definition. In the remainder of the proof we therefore show the induction
step m−1→m.
Induction step
Assume
M
is (
m−
1)-WGSP. W.l.o.g., let players be numbered such that
a successful
m
-coalition consists of the first
m
players, i.e., [
m
]. Due to the induction
hypothesis, we have that
bi6
=
vi
for all
i∈
[
m
]as otherwise there would be a successful
coalition of size (
m−
1). By way of contradiction, assume now that
M
is not
m
-WGSP,
i.e., there are true valuations
v∈Rn
and an [
m
]-variant
b
of
v
such that for all players
i∈[m]it holds that ui(b)>ui(v).
Outline of contradiction
For
i∈
[
m
], denote
B
(
i
):=
{j∈
[
m
]
|uj
(
v−i,bi
)
≥uj
(
b
)
}
,
i.e., all players in
B
(
i
)benefit when player
i
deviates to bid
bi
. Define the binary relation
/
:=
{
(
i,j
)
∈
[
m
]
2|j∈B
(
i
)
}
. We will show that
/
is irreflexive, transitive, and serial
(i.e., without maximum elements). That is,
∀i∈[m]:i6/i, (IRR)
∀i,j,k∈[m]:i/jand j/k=⇒i/k, (TRA)
∀i∈[m]:∃j∈[m]:i/j. (SER)
This is a contradiction. Intuitively, consider the directed graph with node set [
m
]and
edge set
/
. There is an edge (
i,j
)whenever player
i
’s deviation to
bi
would make
player
j
at least as happy as at
b
. Now, irreflexivity
(IRR)
requires ([
m
]
,/
)to be free of
self-loops. Yet, transitivity (TRA) and seriality (SER) imply that self-loops do exist.
Irreflexivity and seriality
Obviously,
(IRR)
holds due to SP. Moreover,
(SER)
holds by
the induction hypothesis: Otherwise, if for some
i∈
[
m
]there was no
j∈
[
m
]
\i
with
uj
(
v−i,bi
)
≥uj
(
b
), then the coalition [
m
]
\i
could improve at (
v−i,bi
)by bidding as
in
b
, which contradicts (
m−
1)-WGSP. The remaining main part of the proof is thus to
show (TRA).
96
5.4 Weak Group-Strategyproofness and Non-Bossiness
Transitivity
Assume there are
i,j,k∈
[
m
]so that
i/j
and
j/k
. Let
vi
be the
i
-variant
of
v
with
vi
i
:=
θi
(
v−i
). Define
vj
correspondingly. Moreover, for
i,j∈
[
m
], let
vi,j
be
the {i,j}-variant of vwith vi,j
i:=θi(v−i)and vi,j
j:=θj(v−j).
•By definition of vi, it holds that
uj(vi)L5.2.8(iv)
≥uj(v−i,bi)≥uj(b)>uj(v). (5.4.2)
Consequently, it follows by Lemma 5.2.8 (i) that
ui(v−i,bi)<ui(v)
and
ui(vi)<
ui(v)
. Then, Lemma 5.2.6 implies
qi
(
v
)
6
=
qi
(
vi
) =
qi
(
v−i,bi
). By the threshold
property, we then have
Mi(v−i,bi) = Mi(vi)and ui(v−i,bi) = ui(vi)<ui(v). (5.4.3)
Now Lemma 5.2.7 implies
uk(v−i,bi) = uk(vi). (5.4.4)
•
By
(5.4.2)
, we have
qj
(
vi
) = 1 and
θj
(
vi
−j
) =
xj
(
vi
)
<vj
. By Lemma 5.2.8 (vi),
we have θj(vi
−j)< θj(v−j) = vi,j
j. So by the threshold property,
Mj(vi,j) = Mj(vi). (5.4.5)
Then
ui(vi,j|vi
i)L5.2.7
=ui(vi|vi
i)and uk(vi,j)L5.2.7
=uk(vi). (5.4.6)
•
Due to
uj
(
vi,j
)
(5.4.5)
=uj
(
vi
)
(5.4.2)
>uj
(
v
)it must hold by 2-GSP that
ui
(
vi,j
)
<ui
(
v
).
Consider now the two cases:
–Case qi(v) = 1:
Then
vi,j
i
=
θi
(
v−i
) =
xi
(
v
)
<vi
due to Lemma 5.2.6 (i) with
(5.4.3)
. Hence,
2-GSP implies
qi
(
vi,j
) = 0, because otherwise
ui(vi,j)≥ui(v)
due to VP.
We have
ui
(
vj
)
L5.2.8(v)
≥ui
(
v
)
>
0, so
qi
(
vj
) = 1 and
θi
(
vj
−i
) =
xi
(
vj
)
≤
xi
(
v
) =
θi
(
v−i
). Now if the inequality was strict, then
i
could improve at
vi,j
by bidding
vi
, because
qi
(
vi,j
) = 0 and
vi,j
i
=
θi
(
v−i
). Hence,
θi
(
vi,j
−i
) =
θi(vj
−i) = θi(v−i).
–Case qi(v) = 0:
Then
ui(vi,j)<ui(v)
=0 and thus
qi
(
vi,j
) =
qi
(
vi
)
(5.4.3)
=
1. Consequently,
θi(vi,j
−i) = xi(vi,j)(5.4.6)
=xi(vi) = θi(v−i).
We have shown that θi(vi,j
−i) = θi(v−i) = vi,j
i, so
uk(vj)L5.2.8(iv)
≤uk(vi,j). (5.4.7)
97
Chapter 5 Does Coalition Size Matter?
Putting everything together, we get
uk(v−i,bi)(5.4.4)
=uk(vi)(5.4.6)
=uk(vi,j)(5.4.7)
≥uk(vj)L5.2.8(iv)
≥uk(v−j,bj)≥uk(b).
Recall that the last inequality stems from our assumption that
j/k
. Hence,
k∈B
(
i
), i.e.,
i/k. This completes the proof. ut
5.4.3 Relationship Between Collusion-Resistance and Non-Bossiness
Properties
Lemma 5.4.8.
Let
M
= (
q,x
)be a SP and ONB cost-sharing mechanism. Then it is
separable.
Proof.
Let
i∈
[
n
]. Suppose
b,b0
are
i
-variants so that
b0
i
=
b∞
if
qi
(
b
) = 1 and
b0
i
=
−
1
otherwise. The threshold property implies
Mi
(
b
) =
Mi
(
b0
), so
M
(
b
) =
M
(
b0
)by ONB.
This argument can be used repeatedly: Let b∗∈Rnbe defined by b∗
j:=b∞if qj(b) = 1
and b∗
j=−1 otherwise. Then also M(b) = M(b∗).ut
Consequently, Theorem 5.3.8 can be seen as a generalization of Proposition 5.3.2 because
the requirements of the latter (SP, ONB, and WUNB) imply 2-GSP and separability in a
relatively straightforward manner.
3
The following example shows that Theorem 5.3.8
is strictly more general because ONB is not a necessary condition for GSP: Define
mechanism M= (q,x)by
q(b):=
(1,1)if (b1≥1 and b2>1)or b= (1,1)
(1,0)if (b1≥1 and b2≤1)and b6= (1,1)
(0,1)if b1<1 and b2>1
(0,0)if b1<1 and b2≤1
and x(b):=q(b).
Obviously, the threshold value for both players is constantly 1, so neither of the two
players could ever improve and
M
is GSP. However, the mechanism is not ONB because
M1(1,1) = M1(2,1)but M2(1,1)6=M2(2,1).
We conclude by stating another result by Mutuswami
[55]
, which completes our
overview of the various notions of non-manipulability and many of their implications
(see Figure 5.2).
Proposition 5.4.9 (Mutuswami [55]).
Let
M
be a SP, ONB cost-sharing mechanism.
Then it is also WGSP.
3
The fact that SP, ONB, and WUNB together imply also 2-GSP follows, of course, from Proposition 5.3.2.
However, it can be easily seen directly: By way of contradiction, assume that
{i,j}
is a GSP-successful
coalition at
v
for some
{i,j}
-variant
b
. W.l.o.g., let
uj(b)>uj(v)
. Now, SP implies that
uj(v−i,bi)≥
uj(b)>uj(v)≥
0. Hence
qj
(
b
) =
qj
(
v−i,bi
) = 1, and
xj
(
b
) =
xj
(
v−i,bi
)due to the threshold property.
ONB implies
M
(
b
) =
M
(
v−i,bi
). However, since SP and WUNB are fulfilled, Lemma 5.2.8 (i) implies
ui(v−i,bi)<ui(v). This contradicts that player iis part of a successful coalition.
98
5.5 Conclusion
‘Regularity’
conditions
Non-bossiness
Incentive-compatibility
GSP
2-GSP
WGSP
+
SP
ONB
WUNBseparable
+
upper
continuous +
2-WGSP +
+
T5.3.8
T5.4.1
P5.4.9
T4.2.9
L5.4.8
L5.3.1
Figure 5.2: Overview of the various non-manipulability properties
5.5 Conclusion
GSP is a very strong axiom. It implies that players have full information of all other
players’ valuations and essentially unbounded ability to communicate and make bindings
agreements. In particular, players would abandon a dominant strategy—telling the
truth—even if they did not benefit from the deviation themselves. There are scenarios,
in particular when the number of players is large, where these assumptions seem not
appropriate.
In this chapter, we proposed relaxing GSP to
k
-GSP, which implies that the players’
ability to coordinate deviations is limited to small coalitions (of size at most
k
). Some-
what surprisingly, however, we showed that already 2-GSP is equivalent to GSP once
we require cost-sharing mechanisms to be separable—which is a very natural property
and fulfilled by all known general techniques for the design of truthful cost-sharing
mechanisms. Hence, our result gives some justification that GSP may, after all, still be
desirable in several scenarios. Moreover, we proved that even without separability, 2-GSP
implies WGSP. While, to the best of our knowledge, restrictions on coalition sizes have
not been considered before in the cost-sharing literature, 2-GSP bears some resemblance
to notions of non-bossiness. We also shed light on the relationship to these notions.
Finally, various open problems remain.
•
We believe that our results facilitate developing and understanding new GSP
mechanisms. So, how can the fact that we only have to counter coalitions of size
two be exploited for the open questions from Section 3.7?
•
Characterizations of WGSP mechanisms are still needed. Are there “reasonable”
o
(
log
(
n
))-WGSP mechanisms that are computable in polynomial time and perform
better than acyclic mechanisms?
99
Chapter 6
Generalizing the Model
6.1 Overview of Contribution
In this last chapter, we generalize cost-sharing problems to a general-demand setting
where each player may have demand for multiple levels of service. Correspondingly, a
mechanism now has to output a multiset of served players. This is particularly useful in
scenarios where multiple levels of service correspond to increased fault tolerance and a
higher quality of service.
We show that the idea of Moulin mechanisms, i.e., serving the largest feasible set, can
be generalized to serving the largest feasible multiset. However, cross-monotonic cost
shares are not alone sufficient any more to imply GSP: Instead, constraints have to be
imposed also on the marginal cost shares of the players. In fact, getting these constraints
“right” is the major difficulty here. We therefore define valid marginal cost-sharing
methods and thus obtain the first general technique for the design of general-demand
cost-sharing mechanisms that are GSP.
6.2 General-Demand Cost Sharing
Formally, a general-demand cost-sharing problem is specified by the maximum service level
L∈N
available to each player and a cost function
C
:[
L
]
n
0→R≥0
. Each player
i∈
[
n
]
is characterized by a valuation vector
vi∈RL
where
vi,l
indicates the marginal valuation
of receiving level
l
additionally to levels 1
,...,l−
1. For technical reasons that we will
discuss later, we always require non-increasing marginal valuations
vi,1 ≥ ··· ≥ vi,L
.
An outcome now consists of a service-allocation vector
q∗∈
[
L
]
n
0
, which represents the
multiset of served players, and a vector of cost shares
x∗∈Rn
. The utility of a player
i∈[n]for outcome (q∗,x∗)is
q∗
i
X
l=1
vi,l−x∗
i.
Note that a cost-sharing problem is binary-demand if L=1.
Ageneral-demand mechanism
M
= (
q,x
)is a pair of functions
q
:
Rn×L→
[
L
]
n
0
and
x
:
Rn×L→Rn
. Truthfulness, budget-balance, and economic efficiency are all generalized
in the obvious way. A marginal cost-sharing method
χ
:[
L
]
n
0→Rn×L
≥0
associates service
101
Chapter 6 Generalizing the Model
allocations to their marginal cost shares, where for all service allocations
s∈
[
L
]
n
0
, all
players
i∈
[
n
]and all service levels
l>si
, we require that
ξi,l
(
s
) = 0. Now given a
marginal cost-sharing method
χ
and a bid vector
b∈Rn×L
, we say a service allocation
s
is
b
-feasible if for all
i∈
[
n
]and all
l∈
[
si
]it holds that
χi,l
(
s
)
≤bi,l
. Trivially, the
empty service allocation (0,...,0)is always b-feasible.
6.2.1 Generalized Moulin Mechanisms
The following notation will be convenient. For any two service-allocation vectors
s,t∈
[
L
]
n
0
, we define
s∨t
:= (
max{si,ti}
)
i∈[n]
.
1
Moreover, for
l∈
[
L
]
0
, we define
s≤l:= (min{si,l})i∈[n].
Now, we define a property that will take the role of cross-monotonicity in the binary-
demand setting.
Definition 6.2.1. A marginal cost-sharing method χis
•
cross-monotonic if for all service-allocation vectors
s≤t
, all players
i
, and all service
levels l ≤siit holds that χi,l(s)≥χi,l(t);
•
non-decreasing if for all service-allocation vectors
t
and all players
i
it holds that
χi,1(t)≤χi,2(t)≤···≤χi,ti(t).
•
level-restricted if for all service-allocation vectors
t
, all players
i
, and all service
levels l it holds that χi,l(t) = χi,l(t≤l);
We say χis valid if it satisfies all of the above three properties.
We briefly comment on the new properties “non-decreasing” and “level-restricted”: Non-
decreasing marginal cost shares imply that the more service levels a player gets, the more
expensive each additional level becomes. This is in contrast to the marginal valuations,
which are non-increasing. It will turn out that this reversed growth is crucial for ensuring
that generalized Moulin mechanisms are GSP.
Level-restrictedness has an interesting implication: It ensures that one could iteratively
invoke binary-demand cost-sharing mechanisms: In each iteration
i
=1
,..., L
, determine
an allocation of service level
i
among those players that have also received service level
i−
1. By level-restrictedness, a later iteration cannot change the marginal cost shares
assigned in previous iterations.
Definition 6.2.2.
A mechanism
M
is a generalized Moulin mechanism if it has a valid
marginal cost-sharing method and it associates all bid vectors
b∈Rn×L
to the maximal
b-feasible service allocation.
We remark that generalized Moulin mechanisms are well-defined because there is always
aunique maximal
b
-feasible allocation. This can be verified as follows: Assume that both
s
and
t
are maximal
b
-feasible allocations. Define
r
:=
s∨t
. Then,
r
is also
b
-feasible
1This is the usual notation when considering [L]ntogether with the partial order “≤” as a lattice.
102
6.2 General-Demand Cost Sharing
because
M
has cross-monotonic cost shares. Moreover, it holds that
r≥s
and
r≥t
.
Now, since both sand tare maximal by assumption, it hence follows that r=s=t.
Equivalently to Definition 6.2.2,
M
= (
q,x
)is a generalized Moulin mechanism if and
only if there is a marginal cost-sharing method χso that for all b∈Rn×Lit holds that
q(b) = max¦s∈[L]n
0|∀i∈[n]:∀l∈[si]:bi,l≥χi,l(s)©and
x(b) = (PL
l=1χi,l(q(b)))i∈[n].
Theorem 6.2.3. Generalized Moulin mechanisms are GSP.
Proof.
Let
M
= (
q,x
)be a generalized Moulin mechanism. Let
v
contain the true
valuations,
K⊆
[
n
]be a non-empty coalition, and
b
be a
K
-variant of
v
with
uK(b)≥
uK(v). We will show that then uK(b) = uK(v).
Let
s
:=
q
(
v
)and
t
:=
q
(
b
). If
t≤s
, then cross-monotonicity implies
u(b)≤u(v)
and we are done. Hence, by way of contradiction, assume that
∃i∈
[
n
]:
ti>si
. Define
level l:=min{si|i∈[n]and ti>si}and A:={i∈[n]|ti>land l=si}. Define rby
ri:=(l+1 if i∈A
siotherwise.
Note that for all players
i∈A
and for all levels
k∈
[
l
], we have
s≤k
=
r≤k
= (
s∨t
)
≤k
and
thus
χi,k
(
s
) =
χi,k
(
r
) =
χi,k
(
s∨t
)because
χ
is level-restricted. Similarly,
χi,l+1
(
r
) =
χi,l+1(s∨t).
We now show for all
i∈A
that
vi,l+1≥χi,l+1
(
t
). By way of contradiction, assume
this is not the case. Then, due to non-increasing marginal utilities, non-decreasing cost
shares, and cross-monotonicity, there is an
i∈A
so that for all
k∈{l
+1
... ti}
:
vi,k<
χi,k(t)≤bi,k. Consequently, i∈Kand
ui(b) =
ti
X
k=1vi,k−χi,k(t)<
l
X
k=1vi,k−χi,k(t)as explained
≤
l
X
k=1vi,k−χi,k(s∨t)due to cross-monotonicity
=
l
X
k=1vi,k−χi,k(s)=ui(v)as explained.
This is a contradiction to i∈K.
Now, we have shown for all
i∈A
that
vi,l+1≥χi,l+1
(
t
)
≥χi,l+1
(
s∨t
) =
χi,l+1
(
r
).
Therefore, by cross-monotonicity,
r
is
v
-feasible. This is again a contradiction because
r>s
, but
s
is the maximal
v
-feasible allocation due to the definition of generalized
Moulin mechanisms. ut
Similar to the binary-demand case, the outcome of generalized Moulin mechanisms
can be computed efficiently. Specifically, using the same arguments as for Lemma 2.2.6,
103
Chapter 6 Generalizing the Model
Input: marginal cost-sharing method χ:[L]n
0→Rn×L
≥0, bid vector b∈Rn×L
Output: service allocation q∈[L]n
0, cost distribution x∈Rn
≥0
1: q:= (L,..., L)
2: while there is a player iwith qi>0 and bi,qi< χi,qi(q)do
3: qi:=qi−1
4: x:= (PL
l=1χi,l(q))i∈[n]
Algorithm 6.1: Generalized Moulin mechanisms
it is easy to see that for every marginal cost-sharing method
χ
and for all bid vectors
b
,
Algorithm 6.1 computes the outcome of the respective generalized Moulin mechanism.
In our SAGT’08 paper [
7
], we design valid marginal cost-sharing methods for the
fault-tolerant facility location problem. An instance of this problem is specified as for the
usual facility location problem (see Section 2.3.3); yet in addiction, we are also given
the number of facilities each customer wants to be connected to.
Theorem 6.2.4 (Bleischwitz and Schoppmann [7]).
For sharing the cost induced by
fault-tolerant facility location, there is a generalized Moulin mechanism that guarantees
3L-BB and (3L·(1+Hn))-EFF. Its outcome can be computed in polynomial time.
We remark that the only other truthful cost-sharing mechanisms for this problem are due
to Mehta et al.
[50]
. However, their mechanisms are only WGSP, and depending on the
size of
n
and
L
, both the budget balance and economic efficiency of our mechanisms are
better.
6.3 Conclusion
Obviously, generalizing the cost-sharing model to general demand opens the door to
numerous new problems—essentially all research questions addressed in this thesis are
also interesting in the generalized setting.
The requirements of generalized Moulin mechanisms are somewhat limiting: While
the motivation for cross-monotonicity is similar to the binary-demand case [
74
,
51
,
52
],
also all the arguments against it (see Section 1.3) still hold. Similarly, the properties
“non-decreasing” and “level-restricted” may seem implausible in various practical set-
tings. Finally, the requirement that marginal valuations are non-increasing is a strong
restriction: E.g., players cannot express that they need to be connected to at least two
facilities—possibly due to security concerns—and that only one connection would be
almost useless to them. We therefore state as open problems:
•
Are there other techniques for designing GSP general-demand cost-sharing mech-
anisms? In particular, are there techniques when marginal valuations may be
increasing?
104
6.3 Conclusion
•
What are lower bounds on the performance of generalized Moulin mechanisms,
similar to the results by Immorlica et al.
[36]
? Are there general lower bounds for
GSP mechanisms, independent of the design technique?
Fault-tolerant facility location is the only problem for which both acyclic mechanisms
and generalized Moulin mechanisms are known. Even though acyclic mechanisms
are only WGSP, there is no clear indication that they allow for significantly improved
performance.
•
Can the approximation guarantees for general-demand cost-sharing be improved,
possibly by sacrificing some collusion resistance?
•Would relaxing GSP to 2-GSP help for general-demand cost sharing?
105
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Index
Note: Bold page numbers refer to (formal) definitions.
2P-CSF, see two-price cost-sharing form
acyclic mechanism, see mechanism, acyclic
algorithmic mechanism design, 1–3
β-average, 63, 73
BB, see budget-balanced
bin packing, see optimization problem, bin
packing
binary-demand, 9, 17, 101
bribe-proof, 13, 86
budget-balanced, 4, 20–21
completion time, 25
consumer sovereignty, 18
strong, 7, 18, 60, 90
cost function, 17, 20–22
cost-sharing mechanism, 4, 18
cost-sharing method, 6, 19
cost-sharing problem, 4, 17
cross-monotonic, 6, 23, 43
CS, see consumer sovereignty
disadvantaged player, 29
economically efficient, 4, 21
EFF, see economically efficient
egalitarian mechanism, see mechanism, egal-
itarian
egalitarian solution, 6
excludable public good, 11, 23, 65, 71
facility location, see optimization problem,
facility location
b-feasible, 8, 19,102
FFD,see first fit decreasing
first fit decreasing, 26, 79
general-demand, 101
generalized Moulin mechanism, see mecha-
nism, generalized Moulin
group-strategyproof, 5, 18
against coalitions of size
k
, 9, 12,
85–
86, 89–98
against service-aware players, 53,
55–
59
on a restricted domain of valuations,
53, 55–59
GSP, see group-strategyproof
k
-GSP, see group-strategyproof against coali-
tions of size k
incentive-compatible, 4
indifferent, 7, 20
longest processing time first, 25, 41, 77–78
lower-continuous, 20
LPT,see longest processing time first
makespan, 3, 8, 25
marginal cost, 6, 12, 48, 70
marginal cost-sharing method, 101
marginal valuation, 101
mechanism
acyclic, 7, 66–69
egalitarian, 62
generalized Moulin, 102–103
Moulin, 6, 23–24
precedence, 28
sequential, 61
sequential stand-alone, 48, 70, 83
singleton, 12, 66, 83
symmetric, 30–32
Vickrey-Clark-Groves, 5, 22–23
mergable, 74
monotonic algorithm, 72
most cost-efficient, 6, 62, 64–65, 72, 83
Moulin mechanism, see mechanism, Moulin
next fit decreasing, 26, 79
NFD,see next fit decreasing
115
Index
no positive transfers, 18
NPT, see no positive transfers
offer function, 67
ONB, see outcome non-bossy
optimization problem, 72
bin packing, 26, 78–82
facility location, 11, 26, 104
machine scheduling, 8, 11, 13,
24
, 41–
44, 77–84
scheduling, 3
Steiner tree, 4, 11, 26, 54
outcome, 18
outcome non-bossy, 12, 19, 89, 98
Pareto-optimal, 90
polynomial-time approximation scheme, 25,
80
precedence mechanism, see mechanism, prece-
dence
price of anarchy, 1, 15
strong, 13
PTAS, see polynomial-time approximation
scheme
quasi-linear, 18
rank, 17
β-relaxation, 73
scheduling, see optimization problem, ma-
chine scheduling
segment, 29
separable, 19, 60, 61, 98
sequential mechanism, see mechanism, se-
quential
sequential stand-alone mechanism, see mech-
anism, sequential stand-alone
service-aware, 55
set selection function, 62
SGSP, see group-strategyproof against service-
aware players
Shapley value, 6
singleton mechanism, see mechanism, sin-
gleton
social cost, 4, 5, 21
social welfare, 5
SP, see strategyproof
Steiner tree, see optimization problem, Steiner
tree
strategyproof, 4, 18, 87–89, 98
effective pairwise, 12
subadditive, 8, 11, 21, 35–41, 65, 71
submodular, 6, 21, 66
successful coalition, 19
α-summable, 43
superadditive, 12, 21
supermodular, 21, 66, 71, 83–84
symmetric
cost-sharing method, 29–30
costs, 8, 21, 35–41, 44–48
symmetric mechanism, see mechanism, sym-
metric
threshold property, 20
threshold value, 7, 20
truthful, 4
two-price cost-sharing form, 35–38, 41–42
β-uniform, 40, 43
upper-continuous, 12, 20, 56, 89–90
utility, 4, 18
valid
marginal cost-sharing method, 102
offer function, 67
set selection and price function,
62
, 73
symmetric cost-sharing method,
29
, 48
valuation, 2, 18
value, 6
K-variants, 17
VCG, see mechanism, Vickrey-Clark-Groves
vector, 17
voluntary participation, 18
VP, see voluntary participation
weakly group-strategyproof, 5, 19, 61–62
against coalitions of size k,86, 95
against service-aware players, 53,
55–
61
weakly monotone, 71
weakly utility non-bossy, 12, 19, 87–89
WGSP, see weakly group-strategyproof
k
-WGSP, see weakly group-strategyproof against
coalitions of size k
WSGSP, see weakly group-strategyproof against
service-aware players
WUNB, see weakly utility non-bossy
116
