Selsh Routing in Networks
Dissertation
von
Martin Gairing
Schriftliche Arbeit zur Erlangung des Grades
Doktor der Naturwissenschaften
an der Fakultät für Elektrotechnik, Informatik und Mathematik
der Universität Paderborn
Paderborn, 10. Oktober 2006
To My Family
Acknowledgements
I have been very lucky to work together and learn from a number of people
during the last years. First and foremost, I'm deeply indebted to my adviser,
Burkhard Monien, for always oering invaluable help and encouragement. His
advice, knowledge and support have been the key for achieving my goals.
Moreover, I would like to express my thanks to Thomas Lücking, Marios
Mavronicolas and Karsten Tiemann for the great collaboration. Thank you to all
my other co-authors on the topic of selsh routing; those are: Sebastian Aland,
Dominic Dumrauf, Robert Elsässer, Rainer Feldmann, Manuel Rode, Florian
Schoppmann, and Paul Spirakis. I am indebted to Ulf-Peter Schroeder for the
careful reading of a preliminary version of this dissertation and for numerous
suggestions for improvement. I thank all my colleagues for many discussions
on scientic and non-scientic topics, for the great atmosphere, and for tech-
nical and administrative support. To name those who I did not mention yet:
Ulrich Ahlers, Bernard Bauer, Yvonne Bleischwitz, Torsten Fahle, Sven Grothk-
lags, Sigrid Gundelach, Georg Kliewer, Michael Laska, Ulf Lorenz, Henning Mey-
erhenke, Tomá² Plachetka, Marion Rohlo, Thomas Sauerwald, Stefan Scham-
berger, Norbert Sensen, Meinolf Sellmann, Thomas Thissen, Karsten Tiemann,
Tobias Tscheuschner, Andreas Woclaw, and Oleksiy Znamenshchykov.
Before coming to Paderborn, I beneted from a number of mentors at Clem-
son University. I am especially grateful to Steven Hedetniemi for his continuous
support and for introducing me to the joy of doing research. Together with David
Jacobs, Renu Laskar, and Pradip Srimani, he provided support for my masters
degree and encouraged me to start a PhD project.
Finally I want to thank my family and my friends for their love, support and
encouragement. In particular, I am indebted to my mother and grandmother for
encouraging a love of learning at an early age. Last, but not least, my love goes
to my wife Ulrike and my sons Noah and Simon.
Paderborn, October 2006
Martin Gairing
Contents
1 Introduction
................................................. 1
1.1 Motivation Framework ...................................... 1
1.2 Contribution .............................................. 2
1.2.1 Routing Games on Parallel Links ....................... 2
1.2.2 Weighted Congestion Games ........................... 2
1.2.3 Selsh Routing with Incomplete Information ............. 3
1.3 Related Models for Selsh Routing ........................... 3
1.4 Publications ............................................... 4
1.5 Organization .............................................. 4
2 Preliminaries
................................................. 5
2.1 Notation .................................................. 5
2.2 Gamma Function .......................................... 5
2.3 Falling Factorials, Stirling Numbers and Bell Numbers .......... 6
2.4 Binomial Cost Function ..................................... 6
3 Models
....................................................... 9
3.1 Routing Games on Parallel Links ............................. 9
3.1.1 Instance ............................................. 9
3.1.2 Strategies and Strategy Proles......................... 10
3.1.3 Load and Latency .................................... 10
3.1.4 Private Cost ......................................... 10
3.1.5 Social Cost Measures.................................. 11
3.1.6 Nash Equilibria ...................................... 13
3.1.7 Price of Anarchy ..................................... 13
3.1.8 Selsh Steps and Nashication ......................... 14
3.2 Weighted Congestion Games ................................. 14
3.2.1 Instance ............................................. 14
3.2.2 Strategies and Strategy Proles......................... 14
3.2.3 Private Cost ......................................... 15
3.2.4 Nash Equilibria ...................................... 15
3.2.5 Social Cost .......................................... 15
VIII Contents
3.2.6 Price of Anarchy ..................................... 16
3.2.7 Network Congestion Games ............................ 16
3.3 Bayesian Routing Games .................................... 16
3.3.1 Instance ............................................. 16
3.3.2 Strategies and Strategy Proles......................... 17
3.3.3 Private Cost ......................................... 18
3.3.4 Bayesian Nash Equilibria .............................. 19
3.3.5 Social Cost and Price of Anarchy ....................... 20
3.3.6 Weighted Bayesian Congestion Games ................... 21
4 Selsh Routing on Parallel Links
............................. 23
4.1 Introduction ............................................... 23
4.1.1 Summary of Results .................................. 23
4.1.2 Related Work ........................................ 25
4.1.3 Organization ......................................... 27
4.2 Identical Links ............................................. 27
4.2.1 Sequences of Selsh Steps.............................. 27
4.2.2 Nashication ......................................... 30
4.3 Related Links.............................................. 32
4.3.1 Nashication ......................................... 32
4.3.2 Price of Anarchy ..................................... 34
4.4 Restricted Strategy Sets..................................... 47
4.4.1 Computation of Pure Nash Equilibria ................... 47
4.4.2 Price of Anarchy ..................................... 56
4.5 Polynomial Social Cost ..................................... 66
4.5.1 Identical Players...................................... 68
4.6 Conclusion and Discussion................................... 81
5 Weighted Congestion Games
................................. 83
5.1 Introduction ............................................... 83
5.1.1 Summary of Results .................................. 83
5.1.2 Related Work ........................................ 85
5.1.3 Organization ......................................... 86
5.2 Price of Anarchy for Unweighted Congestion Games ............ 86
5.2.1 Upper Bound ........................................ 86
5.2.2 Lower Bound ........................................ 93
5.3 Price of Anarchy for Weighted Congestion Games .............. 95
5.3.1 Upper Bound ........................................ 95
5.3.2 Lower Bound ........................................ 97
5.4 Conclusion and Discussion................................... 101
6 Bayesian Routing Games
..................................... 103
6.1 Introduction ............................................... 103
6.1.1 Summary of Results .................................. 104
6.1.2 Related Work ........................................ 106
Contents IX
6.1.3 Organization ......................................... 107
6.2 Pure Bayesian Nash Equilibria ............................... 107
6.2.1 Existence ............................................ 107
6.2.2 Computation ........................................ 112
6.3 Properties of Fully Mixed Bayesian Nash Equilibria............. 114
6.4 Social Cost and Price of Anarchy............................. 120
6.4.1 Makespan Social Cost ................................. 121
6.4.2 Social Cost as Sum of Private Costs..................... 129
6.4.3 Social Cost as Maximum of Private Costs ................ 131
6.5 Conclusion and Discussion................................... 134
References
....................................................... 135
1
Introduction
1.1 Motivation Framework
Large-scale trac and communication networks, like e.g. the Internet, telephone
networks, or road trac systems often lack a central regulation for several rea-
sons: The size of the network may be too large, the networks may be dynam-
ically evolving over time, or the users of the network may be free to act ac-
cording to their private interests, without regard to the overall performance of
the system. Besides the lack of central regulation even cooperation among the
users may be impossible due to the fact that the users may not even know each
other. Networks with non-cooperative users have already been studied in the early
1950's in the context of road trac systems [7,97]. Nowadays, modern computer
artifacts, like e.g. the Internet, are modeled as communication networks with
non-cooperative users. For such communication networks, combining ideas from
game theory and computer science has become increasingly important [29,63,
82,83,86].
An environment, which lacks a central control unit due to its size or oper-
ational mode, can be modeled as a non-cooperative game [85]. Here, the users
are assumed to be selsh players that selshly choose their private strategies,
which in our environment correspond to paths (or probability distributions over
the paths) from their sources to their destinations. When routing their trac ac-
cording to the strategies chosen, the players will experience an expected latency
caused by the trac of all players sharing edges. Each player tries to minimize
its private cost, expressed in terms of its expected latency. This often contradicts
the goal of optimizing the social cost which measures the global performance of
the whole network. The degradation of the global performance due to the selsh
behavior of its players is often termed as price of anarchy [86] or coordination
ratio [67]. The theory of Nash equilibria [80,81] provides us with an important
concept for environments of this kind: A Nash equilibrium is a state of the sys-
tem in which no player can decrease its private cost by unilaterally changing its
strategy. It has been shown by Nash that a Nash equilibrium exists under fairly
broad circumstances.
2 1 Introduction
The concept of Nash equilibria has become an important mathematical tool
in analyzing the behavior of selsh players in non-cooperative systems [86]. Many
algorithms have been developed to compute a Nash equilibrium in a general game
(see [76] for an overview). Although the theorem of Nash [80,81] guarantees the
existence of a Nash equilibrium, the complexity of computing a Nash equilibrium
was open for a long time, even for
2
-player games. Only recently, Chen and Deng
[15] settled the complexity of computing a Nash equilibrium for
2
-player games.
1.2 Contribution
In this work, we study dierent models for selsh routing in non-cooperative
networks. Our models dier in the structure of the underlying network and the
information accessible to the players. We now give a high-level description for the
models considered in this thesis and for our contributions.
1.2.1 Routing Games on Parallel Links
In a
routing game on parallel links
a set of
n
players wishes to assign their
trac
w1, . . . , wn
to one of
m
parallel links connecting a single source node to a single
destination node. Each link has a certain
capacity
, that represents the rate at
which the link processes trac. So the
latency
for a link is the total trac through
this link divided by its capacity. A
pure strategy
for a player is some specic link,
while a
mixed strategy
is a probability distribution over its pure strategies. A
pure (resp. mixed) strategy prole
species a pure (resp. mixed) strategy for each
player. Each player chooses a strategy in order to minimize its
private cost
, which
is dened as its
expected latency
. A strategy prole is a
Nash equilibrium
if no
player can decrease its private cost by unilaterally changing its strategy.
Associated with a strategy prole, there is also a global objective function,
called
social cost
. For routing games on parallel links we consider two dierent
social cost measures:
makespan social cost
and
polynomial social cost
. On the one
hand, makespan social cost is dened as the expected maximum latency on a
link [67]. On the other hand, polynomial social cost is dened as the sum of a
certain polynomial, evaluated at the incurred link loads [42]. The maximum ratio
between the maximum social cost of a Nash equilibrium and the minimum social
cost of a pure strategy prole is called
price of anarchy
.
In this dissertation, we present results concerning the computational complex-
ity of pure Nash equilibria. Furthermore, we prove a multitude of results that are
related to the price of anarchy in various sub-models.
1.2.2 Weighted Congestion Games
The class of
congestion games
has been introduced by Rosenthal [88]. In a con-
gestion game there is a set of resources and the strategy set of each player is a
subset of the power set of these resources. The
latency
on a resource is determined
1.3 Related Models for Selsh Routing 3
by a
latency function
in the number of players sharing this resource. Each player
aims to minimize its
private cost
, which is dened as the sum of the latencies
of its chosen resources. Milchtaich [77] considered
weighted congestion games
as
an extension to congestion games in which the players have weights and thus
dierent inuence on the latency of the resources. Weighted congestion games
provide us with a general framework for modeling any kind of non-cooperative
resource sharing problem. A typical resource sharing problem is that of routing.
In a routing game the strategy sets of the players correspond to paths in a net-
work. Routing games where the demand of the players cannot be split among
multiple paths are also called
(weighted) network congestion games
.
For weighted congestion games we use the
total latency
[92] as our social cost
measure. For the case of network congestion games, the total latency is a measure
for the weighted total travel time of the players. Given this social cost measure,
the price of anarchy is dened as before.
In this dissertation, we show exact values for the price of anarchy of weighted
and unweighted congestion games with polynomial latency functions. The given
values also hold for weighted and unweighted network congestion games.
1.2.3 Selsh Routing with Incomplete Information
In his seminal work, Harsanyi [58] introduced an elegant approach to study non-
cooperative games with
incomplete information
. In our work, we use this ap-
proach to dene a new selsh routing game with incomplete information that we
call
Bayesian routing game
. Here, each of
n
selsh
players
wishes to assign its
trac
to one of
m
parallel
links
. Again, the rate at which links process trac
is given by their
capacities
. However, this time players do not know each other's
trac. Following Harsanyi's approach, we introduce, for each player, a set of pos-
sible
types
. In our model, each type of a player corresponds to some trac and
the players' uncertainty about each other's trac is described by a probability
distribution over all possible
type proles
.
In this dissertation, we prove results on the existence and computational com-
plexity of pure Bayesian Nash equilibria, we study structural properties of a cer-
tain class of mixed Bayesian Nash equilibria, and we prove bounds on the price
of anarchy for various social cost measures.
1.3 Related Models for Selsh Routing
The
Wardrop model
has already been studied in the 1950's in the context of road
trac systems by Wardrop [97] and by Beckmann, McGuire and Winsten [7].
Moreover, it was already discussed earlier by Pigou [87] in the 1920's. For a
survey of the early work on this model see [8]. In the Wardrop model, traf-
c has to be sent through a shared network and trac is allowed to be split
into arbitrary pieces. In this environment, unregulated trac is modeled as net-
work ow. Wardrop [97] introduced the concept of
Wardrop equilibria
to describe
4 1 Introduction
user behavior in this kind of trac networks. Given an arbitrary network with
edge latency functions, Wardrop equilibria have been classied as ows with all
ow paths used between a given source-destination pair having equal latency. A
Wardrop equilibrium can be interpreted as a Nash equilibrium in a game with
innitely many players, each carrying an innitesimal amount of trac from a
source to a destination.
A lot of work (see [92, Sec. 1.2] for a brief survey) on this model has been
motivated by Braess's Paradox [13]. Inspired by the arisen interest in the price
of anarchy, Roughgarden and Tardos re-investigated the Wardrop model [89,92].
Other recent work on the price of anarchy in the Wardrop model and its variations
include [18,20,24,50,75]. Recently, the convergence towards a Wardrop equi-
librium was studied by Fischer et al. [33,34,35]. Many results on the Wardrop
model have been collected in the book of Roughgarden [90].
Another model for selsh routing was rst discussed by Orda et al. [84] and
further studied by Roughgarden [91] and Comminetti et al. [19]. In this model,
the number of players is nite and each player controls a non-negligible amount
of ow that can be split over dierent paths. In contrast to the Wardrop model,
each player centrally controls its ow share, seeking to minimize the total latency
of its ow share.
1.4 Publications
The results presented in this thesis are published in parts as joint work in the
Proceedings of the
International Colloquium on Automata, Languages, and Pro-
gramming (ICALP)
[30,45,48], the Proceedings of the
Italian Conference on
Theoretical Computer Science (ICTCS)
[46], the Proceedings of the
International
Symposium on Mathematical Foundations of Computer Science (MFCS)
[31,42],
the Proceedings of the
International Symposium on Theoretical Aspects of Com-
puter Science (STACS)
[4], the Proceedings of the
Annual ACM Symposium on
Theory of Computing (STOC)
[41], the Proceedings of the
Annual ACM Sympo-
sium on Parallel Algorithms and Architectures (SPAA)
[49],
Parallel Processing
Letters
[44], and
Theoretical Computer Science
[47].
1.5 Organization
After a brief description of some basic notations, denitions and technical results
in Chapter 2, we formally introduce the considered models for selsh routing in
Chapter 3. In Chapter 4, we study routing games on parallel links. Chapter 5
holds our results for weighted congestion games and Chapter 6comprises our
ndings for Bayesian routing games.
2
Preliminaries
This section presents some basic notations, denitions and preliminary technical
results which are needed throughout this thesis.
2.1 Notation
For any integer
k≥1
, denote
[k] = {1, . . . , k}
and
[k]0={0, . . . , k}
. Furthermore,
for any two integers
`, k
with
0≤`≤k
, denote
[`, k] = {`, . . . , k}
.
For a vector
v= (v1, . . . , vn)
, let
v−i= (v1, . . . , vi−1, vi+1, . . . , vn)
and let
(v−i, v0
i) = (v1, . . . , vi−1, v0
i, vi+1, . . . , vn)
.
For an event
E
in the sample space, denote by
Pr(E)
the probability of event
E
happening.
For a random variable
X
with associated distribution
P
, denote by
EP(X)
the the expectation of
X
.
2.2 Gamma Function
Denote
Γ
the
Gamma function
, that is, for any natural number
N
,
Γ(N+1) = N!
,
while for any arbitrary real number
x > 0
,
Γ(x) = Z∞
t=0
txe−tdt.
The Gamma function is invertible, both
Γ
and its inverse
Γ−1
are increasing. It
is well known (see e.g. Gonnet [55]) that
Γ−1(N) = log(N)
log log(N)·(1 + o(1)).
We will use the facts that
Γ(x+ 1) = x·Γ(x)
for all
x > 0
and that
Γ(x)≤x
for
all
1≤x≤3
. For an introduction to the Gamma function we refer to [60].
6 2 Preliminaries
2.3 Falling Factorials, Stirling Numbers and Bell Numbers
For any pair of integers
k≥1
and
t≥1
, the
t
'th falling factorial of
k
,
denoted
as
kt
, is given by
kt=k·(k−1) ·. . . ·(k−(t−1)),
when
k≥t
. Otherwise (
t≥k+ 1
),
kt= 0
.
For any pair of integers
d≥1
and
t∈[d]0
, the
Stirling number of the second
kind
[94], denoted as
S(d, t)
, counts the number of partitions of a set with
d
elements into exactly
t
blocks
(non-empty subsets). In particular,
S(d, 1) = 1
.
Also, for all integers
d≥2
,
S(d, 2) = 2d−1−1
. Stirling numbers of the second
kind satisfy the recurrence relation
S(d, t) = X
q∈[t,d]d−1
q−1·S(q−1, t −1)
for all integers
d≥2
and
t∈[d]
(see, e.g., [57, Table 265, Identity (6.15)]). It is
also known that for all integers
d≥2
and
k≥1
,
kd=Pt∈[d]S(d, t)·kt
.
For any integer
d≥1
, the
Bell number
of order
d
[10], denoted as
Bd
, counts
the number of partitions of a set with
d
elements into blocks. So, clearly,
B0= 1
and
Bd=Pt∈[d]S(d, t)
.
2.4 Binomial Cost Function
Denition 2.1.
For any integer
r≥1
, consider a probability vector
p=
(p1, . . . , pr)
. Fix a function
g(λ) : R→R
. Then, the
binomial function
BF(p, g)
is given by
BF(p, g) = X
A⊆[r] Y
k∈A
pk·Y
k/∈A
(1 −pk)·g(|A|)!.
Strictly speaking, Denition 2.1 denes a
functional
. If all probabilities have
the same value
p
, then we (abuse notation to) write
BF(p, r, g)
. Clearly, in this
case,
BF(p, r, g) = X
k∈[r]0r
kpk(1 −p)r−kg(k).
We show that when
g
is monomial, the binomial function takes a special form.
Proposition 2.2.
For each integer
d≥1
,
BF(p, r, λd) = X
t∈[d]
pt·S(d, t)·rt.
2.4 Binomial Cost Function 7
Proof.
By induction on
r
. For the basis case, let
r= 1
. Then,
BF p, 1, λd=
1
1p11d=p
and
Pt∈[d]ptS(d, t)1t=p1S(d, 1)11=p
, so that the claim follows.
Assume inductively that for some integer
r≥2
, for each integer
d≥1
,
BF(p, r −1, λd) = X
t∈[d]
pt·S(d, t)·(r−1)t.
For the induction step, we derive that
BF(p, r, λd)
=X
k∈[r]0r
kpk(1 −p)r−kkd
=X
k∈[r]r
kpk(1 −p)r−kkd
=X
k∈[r]
r
kr−1
k−1pk(1 −p)r−kkd
=p·r·X
k∈[r]r−1
k−1pk−1(1 −p)r−kkd−1
=p·r·X
k∈[r−1]0r−1
kpk(1 −p)r−1−k(k+ 1)d−1
=p·r·X
k∈[r−1]0r−1
kpk(1 −p)r−1−k
X
q∈[d−1]0d−1
qkq
=p·r·X
q∈[d−1]0d−1
q
X
k∈[r−1]0r−1
kpk(1 −p)r−1−kkq
=p·r·X
q∈[d−1]0d−1
qBF(p, r −1, λq)
=p·r·d−1
0BF(p, r −1,1) + p·r·X
q∈[d−1] d−1
qBF(p, r −1, λq)
=p·r+p·r·X
q∈[d−1] d−1
qBF(p, r −1, λq)
=p·r+p·r·X
q∈[d−1] d−1
q
X
t∈[q]
pt·S(q, t)·(r−1)t
=p·r+X
q∈[d−1] d−1
q
X
t∈[q]
pt+1 ·S(q, t)·rt+1
8 2 Preliminaries
=p·r+X
t∈[d−1]
pt+1 ·rt+1 ·
X
q∈[t,d−1] d−1
q·S(q, t)
=p·r+X
t∈[2,d]
pt·rt·
X
q∈[t,d]d−1
q−1·S(q−1, t −1)
=p·r+X
t∈[2,d]
pt·rt·S(d, t)
=X
t∈[d]
pt·rt·S(d, t),
as needed.
Proposition 2.2 implies that for a
constant
probability vector and a monomial
function, the binomial function is a combinatorial sum of Stirling numbers of the
second kind.
It is known [45, Lemma 3] that in case
g
is
convex,
the binomial function does
not decrease when replacing all probabilities in the probability vector
p
by the
average probability
ep=Pi∈[r]pi
r
.
Lemma 2.3 (Gairing et al. [45]).
For a convex function
g
,
BF(p, g)≤
BF(ep, r, g)
.
3
Models
We now introduce the considered models for selsh routing. First, we dene
rout-
ing games on parallel links
in Section 3.1. Afterwards, we introduce the class of
weighted congestion games
in Section 3.2. Weighted congestion games provide us
with a general framework for modeling any kind of non-cooperative resource shar-
ing problem (including that of routing). In Section 3.3, we introduce
Bayesian
routing games on parallel links
. In this model, players have only incomplete in-
formation about each others trac.
Each model is introduced in a self-contained fashion. A reader, who is only
interested in one of the models, might skip the other two sections here.
3.1 Routing Games on Parallel Links
3.1.1 Instance
In a
routing game on parallel links
, we have a simple network consisting of a set of
m
parallel links connecting a
source
node to a
destination node
. Each of
n
players
wishes to route a particular amount of trac along a (non-xed) link from source
to destination. Assume throughout that
n≥2
and
m≥2
. Associated with each
player
i
is a
strategy set
Si⊆[m]
, as the set of allowed links for player
i
. If
Si= [m]
for all players
i∈[n]
, then we have
unrestricted strategy sets
, otherwise
restricted strategy sets
. Denote
wi
the
trac
of player
i∈[n]
. Dene the
n×1
trac vector
w
in the natural way. In the model of
identical players
, all players'
trac is equal to
1
. The players' trac may vary arbitrarily in the model of
arbitrary players
. Without loss of generality assume that
w1≥. . . ≥wn
. Let
W=Pi∈[n]wi
.
Denote
cj>0
the
capacity
of link
j∈[m]
, representing the rate at which
the link processes trac. So, the
latency
for trac
w
through link
j
equals
w/cj
.
Dene the
m×1
capacity vector
c
in the natural way. In the model of
identical
links,
all link capacities are equal to
1
. Link capacities may vary arbitrarily in
the model of
related links
. Let
C=Pj∈[m]cj
.
An
instance
is described by a tuple
hw,ci
. In case of identical players, we
replace
w
by
n
and in case of identical links, we replace
c
by
m
.
10 3 Models
3.1.2 Strategies and Strategy Proles
A
pure strategy
for player
i
is some specic link
`i
from its strategy set
Si
. A
mixed strategy
for player
i∈[n]
is a probability distribution over pure strategies.
Thus, a mixed strategy is a probability distribution over the set of links.
A
pure strategy prole
is represented by an
n
-tuple
L= (`1, . . . , `n)∈[m]n
while a
mixed strategy prole
is represented by an
n×m
probability matrix
P
of
nm
probabilities
pij
,
i∈[n]
and
j∈[m]
, where
pij
is the probability that player
i
chooses link
j
. The
support
of player
i∈[n]
in the mixed strategy prole
P
,
denoted by
supporti(P)
, is the set of links to which player
i
assigns its trac with
positive probability. Thus,
supporti(P) = {j∈[m]|pij >0}.
If
supporti(F) = [m]
, for a mixed strategy prole
F
and for all players
i∈[n]
,
then we say that
F
is a
fully mixed strategy prole
. In other words,
F
is a fully
mixed strategy prole, if
fij >0
for all players
i∈[n]
and links
j∈[m]
.
3.1.3 Load and Latency
Fix a mixed strategy prole
P
. Denote by
δj(P)
the
expected load
on link
j∈[m]
.
Thus,
δj(P) = X
i∈[n]
pijwi.
In the same way, denote by
δ−k
j(P)
the expected load of all players
i∈[n], i 6=k
on link
j∈[m]
. Thus,
δ−k
j(P) = X
i∈[n],i6=k
pijwi.
Denote by
Λj(P)
the
expected latency
on link
j∈[m]
. Clearly,
Λj(P) = δj(P)
cj
.
The
maximum expected latency
Λ(P)
is the maximum, over all links, of the ex-
pected latency
Λj(P)
on a link
j∈[m]
, that is,
Λ(P) = max
j∈[m]Λj(P).
3.1.4 Private Cost
For a pure strategy prole
L= (`1, . . . , `n)
, the
latency cost for player
i
,
denoted
by
λi(L)
, is
λi(L) = Pk∈[n]:`k=`iwk
c`i
,
3.1 Routing Games on Parallel Links 11
that is, the latency cost for player
i
is the latency of the link it chooses.
Fix now a mixed strategy prole
P
. The
expected latency cost
for player
i∈[n]
on link
j∈Si
, denoted by
λij(P)
, is the expectation, over all random choices of
the remaining players, of the latency cost on link
j
given that player
i
assigns its
trac to link
j∈Si
. Thus,
λij(P) = wi+Pk∈[n],k6=ipkjwk
cj
=(1 −pij)wi+δj(P)
cj
.
For each player
i∈[n]
, the
minimum expected latency cost,
denoted by
λi(P)
, is
the minimum, over all links
j∈Si
, of the expected latency cost for player
i
on
link
j
. Thus,
λi(P) = min
j∈Si
λij(P).
The
private cost
of player
i∈[n]
, denoted by
PCi(P)
, is the expected latency
of player
i
. Thus,
PCi(P) = X
j∈[m]
pij ·λij(P).
Denote by
IC(w,c,P)
the
maximum individual cost
which is the maximum, over
all players, of the private costs. Thus,
IC(w,c,P) = max
i∈[n]PCi(P).
3.1.5 Social Cost Measures
Associated with an instance
hw,ci
and mixed strategy prole
P
is the
social
cost
[67, Section 2]. For routing games on parallel links we consider two dierent
social cost measures.
3.1.5.1 Makespan Social Cost
In their seminal work, Koutsoupias and Papadimitriou [67] introduced the follow-
ing social cost measure. Associated with an instance
hw,ci
and a mixed strategy
prole
P
is the
makespan social cost
, denoted by
SCMSP(w,c,P)
, which is the
expectation, over all random choices of the players, of the maximum (over all
links) latency of trac through a link. Thus,
SCMSP(w,c,P) = EP max
j∈[m]Pi∈[n]:`i=jwi
cj!
=X
(`1,...,`n)∈[m]nY
k∈[n]
pk`k· max
j∈[m]Pi∈[n]:`i=jwi
cj!.
The displayed formulas for makespan social cost refer to a pure strategy prole
L= (`1, . . . , `n)
drawn according to the probability distribution induced by the
12 3 Models
mixed strategy prole
P
. Note that
SCMSP(w,c,P)
reduces to the maximum
latency through a link in the case of pure strategies.
Associated with an instance
hw,ci
is the
makespan optimum
[67, Section 2],
denoted by
OPTMSP(w,c)
, which is the least possible maximum (over all links)
latency of trac through a link. Thus,
OPTMSP(w,c) = min
L∈[m]nSCMSP(w,c,L).
Note, that the optimum refers to a pure strategy prole. Call a pure strategy
prole
L
with
SCMSP(w,c,L) = OPTMSP(w,c)
optimal
.
3.1.5.2 Polynomial Social Cost
For the model of
identical links
, another social cost function was introduced by
Gairing et al. [42]. Let
πd(λ) = X
t∈[d]0
at·λt
be a polynomial of degree
d > 0
with non-negative coecients. So
at≥0
for all
t∈[d]0
and
ad>0
. Consider the model of
identical links
. Associated with an
instance
hw, mi
, a
polynomial cost function
πd(λ)
and a mixed strategy prole
P
is
the
polynomial social cost
, denoted by
SCπd(λ)(w, m, P)
, which is the expectation
of the sum, over all links, of the polynomial cost function
πd(λ)
evaluated at the
incurred link loads. Thus, by linearity of expectation,
SCπd(λ)(w, m, P) = EP
X
j∈[m]
πd
X
k∈[n] : `k=j
wk
=X
j∈[m]
EP
πd
X
k∈[n] : `k=j
wk
=X
j∈[m]X
A⊆[n] Y
i∈A
pij!
Y
i6∈A
(1 −pij)
πd
X
k∈[n] : `k=j
wk
.
The displayed formulas for polynomial social cost refer to a pure strategy prole
L= (`1, . . . , `n)
drawn according to the probability distribution induced by the
mixed strategy prole
P
. If we restrict to the polynomial cost function
πd(λ) =
λd
, then we write
SCλd(w, m, P)
. Note that
SCπd(λ)(w, m, P) = X
0≤t≤d
at·SCλt(w, m, P).
So, polynomial social cost is a linear combination (with non-negative coecients)
of
monomial social costs
.
3.1 Routing Games on Parallel Links 13
Associated with an instance
hw, mi
and a polynomial cost function
πd(λ)
is
the
polynomial optimum,
denoted by
OPTπd(λ)(w, m)
, which is the least possible,
over all pure strategy proles, polynomial social cost. Thus,
OPTπd(λ)(w, m) = min
L∈[m]nSCπd(λ)(w, m, L).
A (pure) strategy prole
L
such that
SCπd(λ)(w, m, L) = OPTπd(λ)(w, m)
will be
called
optimal
(for the instance
hw, mi
and the polynomial cost function
πd(λ)
).
The
monomial optimum
is dened as the natural special case of polynomial op-
timum.
3.1.6 Nash Equilibria
We are interested in a special class of mixed strategy proles called Nash equi-
libria [80,81] that we describe below. Say that a player
i∈[n]
is
satised
in the
mixed strategy prole
P
, if
λij(P) = λi(P)
for all links
j∈supporti(P)
, and
λij(P)≥λi(P)
for all links
j∈Si\supporti(P)
. Thus, a satised player has
no incentive to unilaterally deviate from its mixed strategy. A player
i∈[n]
is
unsatised
in the mixed strategy prole
P
if
i
is not satised for in the mixed
strategy prole
P
.
The mixed strategy prole
P
is a
Nash equilibrium
[67, Section 2], if each
player
i∈[n]
is satised. In other words
P
is a Nash equilibrium, if and only
if
PCi(P)≤PCi(P−i, li)
for all players
i∈[n]
and all links
li∈Si
. Thus, each
player assigns its trac with positive probability only to links (possibly more
than one of them) for which its expected latency cost is minimized. The
fully
mixed Nash equilibrium
[74], denoted by
F
, is a Nash equilibrium that is a fully
mixed strategy prole. We will often consider fully mixed Nash equilibrium for
routing games with unrestricted strategy sets on identical links. Here, the fully
mixed Nash equilibrium
F
uniquely exists and has probabilities
fij =1
m
for all
players
i∈[n]
and links
j∈[m]
[74].
3.1.7 Price of Anarchy
Let
?∈ {MSP, πd(λ)}
. The
price of anarchy
(also known as
coordination ratio
[67,
Section 2]), denoted by
PoA∗
, is the supremum, over all instances
hw,ci
and Nash
equilibria
P
, of the ratio
SC?(w,c,P)
OPT?(w,c)
. Thus,
PoA?= sup
hw,ci,P
SC?(w,c,P)
OPT?(w,c).
Similarly, for the
pure price of anarchy
, denoted by
pPoA∗
, we take the supremum
over all instances
hw,ci
and
pure
Nash equilibria
L
. Thus,
pPoA?= sup
hw,ci,L
SC?(w,c,L)
OPT?(w,c).
14 3 Models
In the same way, the
individual price of anarchy
is the supremum, over all in-
stances
hw,ci
and Nash equilibria
P
, of the ratio
IC(w,c,P)
OPTMSP(w,c).
3.1.8 Selsh Steps and Nashication
Given a pure strategy prole
L= (`1, . . . , `n)
, a
selsh step
of player
i∈[n]
is a
deviation to a strategy prole
(L−i, `0
i)
where
PCi(L−i, `0
i)<PCi(L)
and
`0
i∈Si
.
Such a selsh step is a
greedy selsh step
if there is no strategy
`00
i∈Si
for player
i
such that
PCi(L−i, `00
i)<PCi(L−i, `0
i)
.
For makespan social cost Fotakis et al. [37] showed that selsh steps can be
used for computing a pure Nash equilibrium with non-increased social cost. We
will use the term
nashication
to denote the process of converting a pure strategy
prole into a pure Nash equilibrium with non-increased social cost.
3.2 Weighted Congestion Games
3.2.1 Instance
A
weighted congestion game
Γ
is a tuple
Γ=n, E, (wi)i∈[n],(Si)i∈[n],(fe)e∈E.
Here,
n
is the number of
players
and
E
is the nite set of
resources
. For every
player
i∈[n]
,
wi∈R+
is the
weight
and
Si⊆2E
is the
strategy set
of player
i
. Denote
S=S1×. . . ×Sn
and
S−i=S1×. . . ×Si−1×Si+1 . . . ×Sn
. For
every resource
e∈E
, the
latency function
fe:R+→R+
describes the
latency
on resource
e
. We consider
polynomial latency functions
with maximum degree
d
and non-negative coecients, that is, for all resources
e∈E
, the latency function
is of the form
fe(x) = Pd
j=0 ae,j ·xj
with
ae,j ≥0
for all
j∈[d]0
.
In a (unweighted)
congestion game
, the weights of all players are equal to
1
.
Thus, the latency on a resource only depends on the
number
of players choosing
this resource.
3.2.2 Strategies and Strategy Proles
A
pure strategy
for player
i∈[n]
is some specic
si∈Si
whereas a
mixed strategy
Pi= (p(i, si))si∈Si
is a probability distribution over
Si
, where
p(i, si)
denotes the
probability that player
i
chooses the pure strategy
si
.
A
pure strategy prole
is an
n
-tuple
s= (s1, . . . , sn)∈S
whereas a
mixed
strategy prole
P= (P1, . . . , Pn)
is represented by an
n
-tuple of mixed strategies.
For a mixed strategy prole
P
denote by
p(s) = Y
i∈[n]
p(i, si)
the probability that the players choose the pure strategy prole
s= (s1, . . . , sn)
.
3.2 Weighted Congestion Games 15
3.2.3 Private Cost
Fix any pure strategy prole
s
, and denote by
δe(s) = Pi∈[n]:e∈siwi
the
load
on
resource
e∈E
. The
private cost
of player
i∈[n]
in a pure strategy prole
s
is
dened by
PCi(s) = X
e∈si
fe(δe(s)) .
For a mixed strategy prole
P
, the
private cost
of player
i∈[n]
is
PCi(P) = X
s∈S
p(s)·PCi(s).
3.2.4 Nash Equilibria
We are interested in a special class of (mixed) strategy proles called Nash equi-
libria [80,81] that we describe here. Given a weighted congestion game and an
associated mixed strategy prole
P
, player
i∈[n]
is
satised
if the player can not
improve its private cost by unilaterally changing its strategy. Otherwise, player
i
is
unsatised
. The mixed strategy prole
P
is a
Nash equilibrium
if and only if
all players
i∈[n]
are satised, that is,
PCi(P)≤PCi(P−i, si)
for all
i∈[n]
and
si∈Si
.
Note, that if this inequality holds for all pure strategies
si∈Si
of player
i
, then it also holds for all mixed strategies over
Si
. Depending on the type of
strategy prole, we dier between
pure
and
mixed
Nash equilibria.
3.2.5 Social Cost
Associated with a weighted congestion game
Γ
and a mixed strategy prole
P
is the
social cost
SCTL(Γ, P)
as a measure of social welfare. In particular we use
the expected total latency [92], that is,
SCTL(Γ, P) = X
s∈S
p(s)X
e∈E
δe(s)·fe(δe(s))
=X
s∈S
p(s)X
i∈[n]X
e∈si
wi·fe(δe(s))
=X
i∈[n]
wi·PCi(P).
The
optimum
associated with a weighted congestion game
Γ
is the least pos-
sible social cost, over all pure strategy proles
s∈S
. Thus,
OPTTL(Γ) = min
s∈SSCTL(Γ, s).
16 3 Models
3.2.6 Price of Anarchy
The
price of anarchy
, also called
coordination ratio
and denoted by
PoATL
, is
the supremum, over all instances
Γ
and Nash equilibria
P
, of the ratio
SCTL(Γ,P)
OPTTL(Γ)
.
Thus,
PoATL = sup
Γ,P
SCTL(Γ, P)
OPTTL(Γ)
3.2.7 Network Congestion Games
In a
(weighted) network congestion game
the set of resources
E
corresponds to
edges in a graph
G= (V, E)
. For each player
i∈[n]
we have given an origin
destination pair
(oi, di)
, where
oi, di∈V
. The strategy set
Si
of player
i∈[n]
is
then the set of simple paths connecting its origin
oi
to its destination
di
.
3.3 Bayesian Routing Games
3.3.1 Instance
A
Bayesian routing game
is a tuple
Γ= (n, m, c, T, Ψ)
. Here, each of
n
players
wishes to assign a particular amount of trac to one of
m
parallel
links
connecting
a source node to a destination node. Assume throughout that
n≥2
and
m≥2
.
Denote
c= (c1, . . . , cm)
, where
cj>0
is the
capacity
of link
j∈[m]
. In the case
of
identical links
, all capacities equal
1
. In this case, we write
Γ= (n, m, 1, T, Ψ)
.
Link capacities vary arbitrarily in the case of
related links
. For each player
i∈[n]
,
there is a nite set of possible types
Ti
. For each type
t∈Ti
, denote by
w(t)
the
trac
of type
t
,
w(t)≥0
. Denote
T=T1×. . . ×Tn
, the set of all possible
type
proles
. For each player
i∈[n]
, dene
τi=|Ti|
as the number of types of player
i
. Dene
τ=Pi∈[n]τi
as the total number of types of the players. For simplicity,
we assume that the trac of all types of the players
(w(ti))ti∈Ti,i∈[n]
is encoded
in
T
, so we do not include them in the game tuple. We use the term
type agent
(i, t)
to refer to the type
t∈Ti
of player
i∈[n]
.
There is a joint probability distribution
Ψ= (Ψ(t1, . . . , tn))(t1,...,tn)∈T
, called
type distribution
, over the set of type proles
T
. Thus,
Ψ
is a function
Ψ:T→
[0,1]
and
P(t1,...,tn)∈TΨ(t1, . . . , tn)=1
. Denote by
Ψ(i, t)
the probability that
player
i
is of type
t
. So,
Ψ(i, t) = X
(t1,...,tn)∈T:ti=t
Ψ(t1, . . . , tn).
We say that
Ψ
is
independent
if
Ψ(t1, . . . , tn) = Y
i∈[n]
Ψ(i, ti)
for all
(t1, . . . , tn)∈T,
3.3 Bayesian Routing Games 17
otherwise,
Ψ
is
correlated
. By the denition of conditional probability,
Ψ(t1, . . . , tk−1, tk+1, . . . , tn|tk=t) = Ψ(t1, . . . , tk−1, t, tk+1, . . . , tn)
Ψ(k, t),
that is, the probability of a type prole
(t1, . . . , tn)
given that
tk=t
is the
probability of type prole
(t1, . . . , tn)
divided by the probability that player
k
is
of type
t
. Throughout we only consider instances where
Ψ(k, t)>0
for all players
k∈[n]
and all types
t∈Tk
. Denote by
W(i)
the
expected trac
of player
i∈[n]
.
Clearly,
W(i) = X
(t1,...,tn)∈T
Ψ(t1, . . . , tn)·w(ti)
=X
t∈Ti
Ψ(i, t)·w(t).
Furthermore, dene the
expected total trac
as
W=X
i∈[n]
W(i).
For any pair of players
i, s ∈[n]
and for any type
t∈Ti
, dene
W(s|ti=t)
as
the
conditional expected trac
of player
s
, given that player
i
has type
t
. So,
W(s|ti=t) = X
(t1,...,tn)∈T:
ti=t
Ψ(t1, . . . , ti−1, ti+1, . . . , tn|ti=t)w(ts).
For the case of independent type distribution, we have
W(s|ti=t) = W(s)
for
all types
t∈Ti
of player
i
.
A special instance of our Bayesian routing game in which each player has
only a single type is a
complete information routing game
. For such a game, we
write
ΓCI = (n, m, c, T, 1)
. Here, the set
T
contains only one type vector
t
that
is used with probability 1. Complete information routing games are exactly the
games introduced in Section 3.1. However, in order to emphasize the connection
to Bayesian routing games, we call them dierently here.
3.3.2 Strategies and Strategy Proles
A
pure strategy
σi
for player
i∈[n]
is a mapping of the set of possible types
Ti
to the set of links
[m]
. So,
σi
is a function
σi:Ti→[m]
. Denote as
Σi
the set of
all possible pure strategies for player
i∈[n]
. Denote
Σ=Σ1×. . .×Σn
. A
mixed
strategy
Pi= (p(i, σi))σi∈Σi
for player
i∈[n]
is a probability distribution over
Σi
. Here,
p(i, σi)
denotes the probability that player
i
chooses the pure strategy
σi
.The
support
of player
i∈[n]
in the mixed strategy prole
P
, denoted by
supporti(P)
, is the set of links to which player
i
assigns at least one type
t∈Ti
with positive probability, that is,
18 3 Models
supporti(P) = {j∈[m]|∃σi∈Σi,∃t∈Ti
with
p(i, σi)>0
and
σi(t) = j}.
Similarly, the support of any type
t∈Ti
of player
i∈[n]
is dened by
supportt(P) = {j∈[m]|∃σi∈Σi
with
p(i, σi)>0
and
σi(t) = j}.
Note that
supporti(P) = [
t∈Ti
supportt(P).
A
pure strategy prole
σ
is an
n
-tuple
(σ1, . . . , σn)∈Σ
. Call
σ
normal
if
σi(t) =
σi(t0)
for all types
t, t0∈Ti
and for all players
i∈[n]
. So, each player
i∈[n]
does
not distinguish among its types in a normal pure strategy prole.
A
mixed strategy prole
P= (P1, . . . , Pn)
is an
n
-tuple of mixed strategies.
Call a mixed strategy prole
F= (F1, . . . , Fn)
fully mixed
if each player assigns
strictly positive probability to each of its pure strategies, that is,
f(i, σi)>0
for
all players
i∈[n]
and all strategies
σi∈Σi
. Notice that
supporti(F) = [m]
for
all players
i∈[n]
and
supportt(F) = [m]
for all players
i∈[n]
and types
t∈Ti
.
3.3.3 Private Cost
3.3.3.1 Pure Strategy Proles
Fix any type distribution
Ψ
and a pure strategy prole
σ= (σ1, . . . , σn)
. The
expected load
on link
j∈[m]
, denoted by
δj(σ,Ψ)
, is dened by
δj(σ,Ψ) = X
(t1,...,tn)∈T
Ψ(t1, . . . , tn)X
i∈[n]:
σi(ti)=j
w(ti).
In the same way, denote by
δ−k
j(σ,(Ψ|tk=t))
the
conditional expected load
of
all players
i∈[n]
other than
k
on link
j∈[m]
given that
tk=t
. So,
δ−k
j(σ,(Ψ|tk=t)) = X
(t1,...,tn)∈T:
tk=t
Ψ(t1, . . . , tk−1, tk+1, . . . , tn|tk=t)X
i∈[n]\{k}:
σi(ti)=j
w(ti).
Denote by
λj
(i,t)(σ,Ψ)
the private cost of type agent
(i, t)
when its trac is
assigned to link
j∈[m]
. So,
λj
(i,t)(σ,Ψ) = δ−i
j(σ,(Ψ|ti=t)) + w(t)
cj
.
Denote by
v(i,t)(σ,Ψ)
the
conditional private cost
of player
i∈[n]
, given that
player
i
is of type
t
; this is also the private cost of type agent
(i, t)
; so,
v(i,t)(σ,Ψ) = λσi(t)
(i,t)(σ,Ψ).
Note that
v(i,t)(σ,Ψ)
does not depend on the other types
t0∈Ti\{t}
of player
i
.
Finally, denote by
PCi(σ, Ψ)
the
private cost
of player
i
. Clearly,
PCi(σ, Ψ) = X
t∈Ti
Ψ(i, t)·v(i,t)(σ, Ψ).
3.3 Bayesian Routing Games 19
3.3.3.2 Mixed Strategy Proles
Fix any type distribution
Ψ
and a mixed strategy prole
P
. The
expected load
on link
j∈[m]
, denoted by
δj(P,Ψ)
, is dened by
δj(P,Ψ) = X
σ∈ΣY
i∈[n]
p(i, σi)·δj(σ,Ψ)
In the same way, denote by
δ−k
j(P,(Ψ|tk=t))
the
conditional expected load
of
all players
i∈[n]
other than
k
on link
j∈[m]
given that
tk=t
. So,
δ−k
j(P,(Ψ|tk=t)) = X
σ∈ΣY
i∈[n]
p(i, σi)·δ−k
j(σ,(Ψ|tk=t)).
For the case of an independent type distribution
Ψ
, we get that for all types
t, t0∈Tk
,
δ−k
j(P,(Ψ|tk=t)) = δ−k
j(P,(Ψ|tk=t0))
. Therefore, to simplify
notation, we write in this case
δ−k
j(P,Ψ)
.
Denote by
λj
(i,t)(P,Ψ)
the private cost of type agent
(i, t)
when its trac is
assigned to link
j∈[m]
. So,
λj
(i,t)(P,Ψ) = δ−i
j(P,(Ψ|ti=t))+w(t)
cj
.
Denote by
v(i,t)(P,Ψ)
the
conditional private cost
of player
i∈[n]
, given that
player
i
is of type
t
; this is also the private cost of type agent
(i, t)
; so,
v(i,t)(P,Ψ) = X
σi∈Σi
p(i, σi)·λσi(t)
(i,t)(P,Ψ).
Note that
v(i,t)(P,Ψ)
does not depend on the other types
t0∈Ti\{t}
of player
i∈[n]
. Finally, denote by
PCi(P,Ψ)
the
private cost
of player
i∈[n]
. Clearly,
PCi(P,Ψ) = X
t∈Ti
Ψ(i, t)·v(i,t)(P,Ψ).
3.3.4 Bayesian Nash Equilibria
A strategy prole
P
is a Bayesian Nash equilibrium, if no player has an incentive
to deviate from its (mixed) strategy, that is, no player can possibly decrease
its private cost when other players are sticking to their strategies. Formally, the
mixed strategy prole
P= (P1, . . . , Pn)
is a
Bayesian Nash equilibrium
if
PCi(P,Ψ)≤PCi(P0,Ψ)
for all mixed strategy proles
P0= (P1, . . . , P0
i, . . . , Pn)
and for all players
i∈[n]
.
Moreover, since
v(i,t)(P,Ψ)
does not depend on the other types
t0∈Ti\{t}
of
player
i
, the above condition is equivalent to
20 3 Models
v(i,t)(P,Ψ)≤v(i,t)(P0,Ψ)
for all mixed strategy proles
P0= (P1, . . . , P0
i, . . . , Pn)
and for all players
i∈[n]
and types
t∈Ti
. Note that
P
is a Bayesian Nash equilibrium if and only if for
all players
i∈[n]
and types
t∈Ti
,
v(i,t)(P,Ψ) = λj
(i,t)(P,Ψ),
for
j∈supportt(P)
, and
v(i,t)(P,Ψ)≤λj
(i,t)(P,Ψ),
for
j6∈ supportt(P).
We refer to these conditions as the
Bayesian Nash equilibrium conditions
.
3.3.5 Social Cost and Price of Anarchy
Associated with a Bayesian routing game
Γ= (n, m, c, T, Ψ)
and a mixed strat-
egy prole
P
is the
social cost
as a measure of social welfare. We consider three
dierent measures for social cost:
•
the
makespan social cost
, which is the expectation over all player choices and
type proles, of the maximum latency on a link. So,
SCMSP(Γ, P)
=X
(σ1,...,σn)∈ΣY
i∈[n]
p(i, σi)X
(t1,...,tn)∈T
Ψ(t1, . . . , tn)·max
j∈[m]
1
cjX
i∈[n],
σi(ti)=j
w(ti)
=X
(t1,...,tn)∈T
Ψ(t1, . . . , tn)X
(σ1,...,σn)∈ΣY
i∈[n]
p(i, σi)·max
j∈[m]
1
cjX
i∈[n],
σi(ti)=j
w(ti)
;
•
the
sum of private costs
,
SCSUM(Γ, P) = X
i∈[n]
PCi(P,Ψ);
•
the
maximum of private costs
,
SCMAX(Γ, P) = max
i∈[n]PCi(P,Ψ).
Let
∗ ∈ {MSP,SUM,MAX}
. Denote the corresponding
optimum social cost
by
OPT∗(Γ) = minPSC∗(Γ, P)
. The
price of anarchy
PoA∗
is the supremum, over
all instances
Γ
and Bayesian Nash equilibria
P
, of the ratio
SC∗(Γ,P)
OPT∗(Γ)
, that is,
PoA∗= sup
Γ,P
SC∗(Γ, P)
OPT∗(Γ).
3.3 Bayesian Routing Games 21
3.3.6 Weighted Bayesian Congestion Games
A generalization of Bayesian routing games are
weighted Bayesian congestion
games with linear latency functions
. In a congestion game [88], each player
i∈[n]
can assign its trac to a subset
si
of the resources out of a given set
Si⊆2[m]
of subsets of resources. The latency function of resource
e∈[m]
is given by an
arbitrary, non-decreasing linear cost function
ge(x) = aex+be
. For a Bayesian
congestion game, a pure strategy prole
σ
is dened by
σ= (σ1, . . . , σn)
with
σi:Ti→Si
for all
i∈[n]
. Thus, a pure strategy of player
i∈[n]
maps each type
t∈Ti
to a
set
of resources, while for a Bayesian routing game a pure strategy of
player
i∈[n]
maps each type
t∈Ti
to a
single
link.
For a pure strategy prole
σ
, the
conditional expected load
of all players
i∈[n]
other than
k
, on resource
e∈[m]
given that
tk=t
is then
δ−k
e(σ,(Ψ|tk=t)) = X
(t1,...,tn)∈T:
tk=t
Ψ(t1, . . . , tk−1, tk+1, . . . , tn|tk=t)X
i∈[n]\{k}:
e∈σi(ti)
w(ti),
whereas the
conditional private cost
of player
i
, given that player
i
is of type
t∈Ti
is then dened by
v(i,t)(σ,Ψ) = X
e∈σi(t)
ge(δ−i
e(σ,(Ψ|ti=t))+w(t)).
4
Selsh Routing on Parallel Links
4.1 Introduction
In this chapter, we consider routing games on parallel links. Such games have
been formally introduced in Section 3.1. Here,
n
non-cooperative players wish to
route their trac
w1, . . . , wn
through a simple network of
m
parallel links with
capacities
c1, . . . , cm
. In the model of
identical players
, all players have equal
trac. The players' trac may be dierent in the model of
arbitrary players
. In
the model of
identical links
, all links have equal capacity. Link capacities may
vary arbitrarily in the model of
related links
.
Each player is allowed to route its trac along links from its strategy set. If the
strategy set of each player consists of all links, then we have
unrestricted strategy
sets
, otherwise
restricted strategy sets
. We assume unrestricted strategy sets, if we
do not explicitly state the contrary. A
pure strategy
for a player is some specic
link from its strategy set, while a
mixed strategy
is a probability distribution over
pure strategies. Each player utilizes a (mixed) strategy, trying to minimize its
private cost
, which is dened as its
expected latency
. A
strategy prole
species a
strategy for each player. Such a strategy prole is a
Nash equilibrium
, if no player
can improve its expected latency by unilaterally changing its strategy. Depending
on the employed player strategies, we distinguish between
pure
and
mixed
Nash
equilibria. We also consider
fully mixed Nash equilibria
, where each player uses
each link with strictly positive probability.
Associated with a strategy prole is also a global objective function, called
social cost. In this chapter we consider two dierent denitions of social cost. The
rst one, called
makespan social cost
, is dened as the
expected maximum latency
on a link
. The second one, called
polynomial social cost
, is the sum (over all links)
of a certain polynomial cost function evaluated at the incurred link loads. The
maximum ratio between the maximum social cost of a Nash equilibrium and the
minimum social cost of a pure strategy prole is called
price of anarchy
.
4.1.1 Summary of Results
We present a multitude of results for routing games on parallel links. Our results
are partitioned into two groups. The rst one consists of results that are con-
24 4 Selsh Routing on Parallel Links
cerned with the computational complexity of pure Nash equilibria. The second
one comprises our ndings that are related to the price of anarchy.
4.1.1.1 Computation of Pure Nash Equilibria
It is easy to see (cf. Fotakis et al. [37]) that any selsh step decreases the
lexico-
graphical ordering
of the link latencies. This implies that any sequence of selsh
steps eventually reaches a pure Nash equilibrium. However, this does not say
anything about the length of such a sequence. For the model of identical links,
we obtain to following results:
•
The length of a sequence of selsh steps is at most
2n−1
, if the players always
deviate to their best link (Theorem 4.2).
•
It is
NP
-complete to decide, whether a given pure strategy prole can be
transformed into a pure Nash equilibrium within at most
k
selsh steps (The-
orem 4.4).
•
There exists an algorithm, called
NashifyIdentical
, that transforms a given
strategy prole into a pure Nash equilibrium in
O(nlog n)
time using at most
n
selsh steps (Theorem 4.5). The algorithm does not increase makespan social
cost.
•
Combining the
PTAS
of Hochbaum and Shmoys [61] for scheduling
n
jobs
on
m
identical
machines
with
NashifyIdentical
yields a
PTAS
for computing
a pure Nash equilibrium with minimum makespan social cost (Theorem 4.6).
4.1.1.2 Price of Anarchy
Makespan Social Cost
For the case that social cost is dened as the expected maximum latency on a
link, we prove a comprehensive collection of bounds on the
pure
price of anarchy
for the case of unrestricted and restricted strategy sets.
For the case that strategy sets are
unrestricted
, we obtain the following:
•
We introduce a structural parameter
ρ
that species the relation between
the largest player trac and the link capacities. For the model of arbitrary
players and related links, we use
ρ
to prove an upper bound on the
pure
price of anarchy
(Theorem 4.10). This upper bound is tight up to an additive
constant (Theorem 4.16).
•
As a corollary to Theorem 4.10 we get that, for the model of arbitrary players
and related links, the pure price of anarchy is upper bounded by
Γ−1(m)
(Corollary 4.14). This upper bound is
asymptotically
tight (Theorem 4.17).
For the case that strategy sets are
restricted
, we prove:
•
For the model of identical players with restricted strategy sets and related
links, the pure price of anarchy is upper bounded by
Γ−1(n)+1
(Theorem 4.33).
This upper bound is tight up to an additive constant, if
n=m
(Theorem 4.32).
4.1 Introduction 25
•
For the model of arbitrary players with restricted strategy sets and identical
links, the pure price of anarchy is upper bounded by
Γ−1(m)
(Theorem 4.35).
This upper bound is tight up to an additive constant (Theorem 4.32).
•
For the model of arbitrary players with restricted strategy sets and related
links, the pure price of anarchy is upper bounded by
m
(Theorem 4.38) and
lower bounded by
m−1
(Theorem 4.37).
Polynomial Social Cost
For the case that social cost is dened as the expectation of the sum (over all links)
of a certain polynomial cost function of degree
d > 0
, we prove a comprehensive
collection of bounds on the price of anarchy. In particular, we show:
•
For the model of identical players and two identical links, the fully mixed Nash
equilibrium maximizes polynomial social cost (Theorem 4.42).
•
For the model of identical players and identical links, the fully mixed Nash
equilibrium maximizes polynomial social cost up to the factor
1 + 1
n−1d
(Theorem 4.44).
•
For the model of identical players and identical links, the price of anarchy is
upper bounded by
Bd
; here,
Bd
is the
Bell number
of order
d
. Our analysis rst
shows that
Bd
is an upper bound on the price of anarchy, if the polynomial cost
function is the
d
'th power (Theorem 4.48). As a corollary we get that the same
upper bound also holds for general polynomial cost functions (Corollary 4.49).
•
For the model of identical players and
two
identical links, the price of anarchy
is upper bounded by
2d−21 + 1
nd−1
, if the polynomial cost function is
the
d
'th power (Theorem 4.50). Moreover, this upper bound is
tight
for the
sub-case of two players. As a corollary we get that the same upper bound also
holds for general polynomial cost functions (Corollary 4.51).
4.1.2 Related Work
Koutsoupias and Papadimitriou [67] introduced and studied a model for selsh
routing on parallel links. They dened
makespan social cost
as their social cost
measure and showed the rst results on the
price of anarchy
.
The price of anarchy for makespan social cost, was further studied by Mavron-
icolas and Spirakis [74]. In this work, they also introduced and analyzed
fully
mixed Nash equilibria
. In the fully mixed Nash equilibrium, each player assigns
its trac to each link with strictly positive probability.
Tight bounds on the price of anarchy for makespan social cost were given
by Czumaj and Vöcking [23] and Koutsoupias et al. [66]. They showed that
the price of anarchy is
Θ(log m
log log m)
[23,66] for the model of identical links and
Θ(log m
log log log m)
[23] for the model of related links. Also for the model of related
links, but restricting to
pure
Nash equilibria, Czumaj and Vöcking [23] showed
two upper bounds of
Γ−1(m) + 1 = O(log m
log log m)
and
O(log(cmax
cmin ))
on the
pure
price of anarchy
.
26 4 Selsh Routing on Parallel Links
Independently of our work, Awerbuch et al. [6] also studied makespan social
cost for the case of restricted strategy sets. Awerbuch et al. [6] focused on the
model of arbitrary players and identical links, for which they proved that the
price of anarchy is
Olog m
log log m
for pure Nash equilibria and
Θlog m
log log log m
for
all (mixed) Nash equilibria. Suri et al. [95] studied a variant of the model of
parallel links with restricted strategy sets where the social cost is dened as the
total latency
. For this variant, Suri et al. [95] provided some
constant
bounds on
the price of anarchy. Elsässer et al. [25] studied a further restriction of the model
of parallel links with restricted strategy sets, called
interaction graphs,
where all
sets of allowed links for the players have size
2
. The results of Elsässer et al. [25]
for their model include bounds on price of anarchy for makespan social cost. In
particular, Elsässer et al. [25] proved that
Ωlog m
log log m
is still a lower bound on
price of anarchy for the case of identical players and identical links in the more
restricted model of interaction graphs.
Gairing et al. [42] and Lücking et al. [71] studied the
pure
price of anarchy for
polynomial social cost. For identical links, Gairing et al. [42] proved that the pure
price of anarchy is exactly
(2d−1)d
(d−1)(2d−2)d−1(d−1
d)d
, if the polynomial cost function
is the
d
'th power. For the special case of
d= 2
, this result was shown by Lücking
et al. [71]; here, the pure price of anarchy is
9
8
.
The
fully mixed Nash equilibrium conjecture
, which states that the fully mixed
Nash equilibrium has worst social cost among all Nash equilibria, was motivated
by some results from Mavronicolas and Spirakis [74] and explicitly formulated by
Gairing et al. [47]. The conjecture has been proved for several particular case by
Fotakis et al. [37], Gairing et al. [45,47] and Lücking et al. [71,72]. Fischer and
Vöcking [36] presented a counterexample to the fully mixed Nash equilibrium
conjecture for the case of identical links and makespan social cost.
It has been rst observed by Fotakis et al. [37] that for the model of related
links, Graham's LPT scheduling algorithm [56] can be used to compute a pure
Nash equilibrium in polynomial time. On the other hand, Fotakis et al. [37]
showed that the problem of computing a pure Nash equilibrium with minimum
(or maximum, respectively) makespan social cost is
NP
-hard. Fotakis et al. [37]
also showed that any sequence of selsh steps converges towards a pure Nash
equilibrium. Even-Dar et al. [26] studied the length of such sequences under
dierent policies to choose the deviating player, and Goldberg [54] considered
the expected length of such a sequence when a random policy is applied.
Selsh routing on parallel links is closely connected to multiprocessor schedul-
ing. Here, pure Nash equilibria and sequences of selsh steps translate to local
optima and sequences of local improvements. A schedule is said to be
jump op-
timal
if no job on a processor with maximum load can improve by moving to
another processor [93]. Obviously, the set of pure Nash equilibria is a subset of
the set of jump optimal schedules. Thus, for this model the strict upper bound
2−2/(m+ 1)
on the ratio between best and worst makespan of jump optimal
schedules [32,93] also holds for pure Nash equilibria. Algorithms for computing a
4.2 Identical Links 27
jump optimal schedule from any given schedule have been proposed in [14,32,93].
The fastest algorithm has been given by Schuurman and Vredeveld [93]. It always
moves the job with maximum weight from a makespan processor to a processor
with minimum load, using
O(n)
moves. However, in all algorithms the resulting
jump optimal schedule is not necessarily a Nash equilibrium.
Libman and Orda [68,69], Czumaj et al. [22] and Gairing et al. [45] con-
sidered selsh routing games on parallel links with more
general latency func-
tions
. Libman and Orda [68,69] allow for arbitrary increasing latency functions,
while Gairing et al. [45] restrict to convex (and increasing) latency functions.
Czumaj et al. [22] present a thorough study for the case of general continuous
non-decreasing latency functions, with emphasis on delay functions from queuing
theory.
Many results for routing games on parallel links have also been collected in
the surveys of Czumaj [21], Feldmann et al. [31] and Koutsoupias [65].
4.1.3 Organization
The rest of this chapter is organized as follows. Section 4.2 deals with the case
of identical links, whereas the results quoted in Section 4.3 hold for related links.
Section 4.4 studies the case of restricted strategy sets. Our results for polynomial
social cost are presented in Section 4.5. We conclude in Section 4.6.
4.2 Identical Links
In this section, we consider routing games on identical links. Here, we are in-
terested in the problem of computing a pure Nash equilibrium. Basically, two
dierent approaches can be found in the literature.
The rst approach is to directly compute a pure Nash equilibrium. Fotakis
et al. [37] showed that the LPT algorithm, rst explored by Graham [56], yields
some pure Nash equilibrium.
The second approach is to convert a given pure strategy prole into a pure
Nash equilibrium without increasing the social cost. This conversion process is
called
nashication
. Since selsh steps do not increase makespan social cost and
any sequence of selsh steps eventually reaches a pure Nash equilibrium, selsh
steps seem to be suitable for nashication.
However, our results in Section 4.2.1 and Section 4.2.2 will show, that we
can't use them uncoordinated. Section 4.2.1 deals with sequences of selsh steps,
whereas in Section 4.2.2 we present a nashication algorithm.
4.2.1 Sequences of Selsh Steps
In this section, we establish bounds on the maximum length of sequences of greedy
selsh steps. Recall, that in a greedy selsh step, the deviating player chooses its
best alternative. Afterwards, we consider the problem of deciding whether a given
28 4 Selsh Routing on Parallel Links
pure strategy prole can be transformed into a pure Nash equilibrium within a
given number of selsh steps.
As discussed above, performing greedy selsh steps will eventually convert
any pure strategy prole into a pure Nash equilibrium. However, this may take
exponential time, even for identical links, as shown in Theorem 4.2 and Theo-
rem 4.3.
The following lemma is crucial for proving the upper bound in Theorem 4.2.
Lemma 4.1.
Consider the model of arbitrary players and identical links. Then
a greedy selsh step of an unsatised player
i
with trac
wi
makes no player
k
with trac
wk≥wi
unsatised.
Proof.
We prove a more general result in Lemma 4.7.
Theorem 4.2.
Consider the model of arbitrary players and identical links. Then,
for any instance
hw, mi
, the length of a sequence of greedy selsh steps is at most
2n−1
.
Proof.
Without loss of generality assume
w1≥w2≥ ··· ≥ wn
. Let
1≤i≤n
.
We prove by induction on
i
that player
i
can make at most
2i−1
greedy selsh
steps.
Since
w1
is the largest trac, and because of Lemma 4.1, player
1
can make
at most one greedy selsh step. This proves the claim for
i= 1
. So assume
i≥2
.
Due to Lemma 4.1 player
i
can only become unsatised by a move of a player with
larger trac. By induction hypothesis, the number of greedy selsh steps made
by players
1, . . . , i −1
is at most
Pi−1
k=1 2k−1= 2i−1−1
. This shows that player
i
can become unsatised at most
2i−1−1
times. Since after a greedy selsh step
player
i
becomes satised and since player
i
can be unsatised at the beginning,
player
i
can make at most
2i−1
greedy selsh steps.
Summing up over all players, the total number of greedy selsh steps is at most
Pn
i=1 2i−1= 2n−1
. This completes the proof of the theorem.
A corresponding lower bound on the maximum length of a sequence of greedy
selsh steps was independently given by Feldmann et al. [31] and Even-Dar et al.
[26]. We include the latter, since it strictly dominates the former:
Theorem 4.3 (Even-Dar et al. [26]).
Consider the model of arbitrary players
and identical links. Then, there exists an instance and associated pure strategy
prole for which the maximum length of a sequence of greedy selsh steps is at
least
n
m−1m−1
2(m−1)! .
Instead of the maximum length one may ask about the minimum length of a
sequence of selsh steps. In particular, one may consider whether a given pure
strategy prole can be transformed into a pure Nash equilibrium with at most
k
selsh steps. We address this question with the following decision problem:
4.2 Identical Links 29
NASHIFY
INSTANCE:
A problem instance
hw,ci
, an associated pure strategy
prole
L
, and a positive integer
k
.
QUESTION:
Is there a sequence of at most
k
selsh steps that trans-
forms
L
into a pure Nash equilibrium?
If
k
is not part of the input, then the problem is called
k
-
NASHIFY
.
In order to prove that
NASHIFY
is
NP
-complete, we will employ a polynomial
time reduction from
PARTITION
.
PARTITION
already appeared in the original
list of 21
NP
-complete problems, presented by Karp [64]. In the notation of
Garey and Johnson [52],
PARTITION
is dened as follows:
PARTITION
INSTANCE:
A nite set
A
of items, a size
s(ai)∈N
for each item
ai∈A
,
i∈[|A|]
.
QUESTION:
Is there a subset
A0⊆A
such that
Pa∈A0s(a) =
Pa∈A\A0s(a)
?
We are now ready to establish
NP
-completeness for
NASHIFY
.
Theorem 4.4.
NASHIFY
is
NP
-complete, even for the case of two identical
links.
Proof.
Clearly,
NASHIFY
is in
NP
since it is solvable in polynomial-time by
a non-deterministic algorithm. We now prove
NP
-hardness by reduction from
PARTITION
, that is, we employ a polynomial-time transformation from
PARTI-
TION
to
NASHIFY
. Consider any arbitrary instance of
PARTITION
with
k≥2
items (an instance of partition with one items is a trivial
no
instance), and let
S=Pa∈As(a)
. From this instance construct an instance of
NASHIFY
as follows:
•
There are
n= 5k
players with weights
wi=s(ai)
if
i∈[k],
1
4k
if
k+ 1 ≤i≤5k.
•
There are
m= 2
identical links.
•
The pure strategy prole
L
is dened as follows: All players
i∈[3k]
are
assigned to link
1
and players
3k+ 1, . . . , 5k
are assigned to link
2
.
Clearly, this is a polynomial time transformation. We prove that this is a
transformation from
PARTITION
to
NASHIFY
.
(1.) The instance of
PARTITION
is positive:
Thus, there exists a subset
A0⊆A
such that
Pa∈A0s(a) = Pa∈A\A0s(a)
.
Since either
|A0| ≤ k
2
or
|A\A0| ≤ k
2
, assume, without loss of generality, that
30 4 Selsh Routing on Parallel Links
|A0| ≤ k
2
. Clearly, each player
i∈[3k]
assigned to link
1
is unsatised in
the constructed pure strategy prole
L
. Furthermore, transferring all players
that correspond to an element
a∈A0
from link
1
to link
2
(in any order) is a
sequence of at most
k
2< k
selsh steps. For the resulting strategy prole
L0
we have
Λ1(L0) = Λ2(L0) = S
2+1
2.
This implies that
L0
is a pure Nash equilibrium so that
NASHIFY
is positive.
(2.) The instance of
NASHIFY
is positive:
Thus, there exists a sequence of at most
k
selsh steps that transforms the
pure strategy prole
L
in the constructed instance of
NASHIFY
to a pure Nash
equilibrium
L0
. Assume that in
L0
players corresponding to a subset
A0⊆A
are assigned to link
j1
, players corresponding to the subset
A\A0⊆A
are
assigned to link
j2
, while the sums of trac of players with trac
1
4k
that
reside in link
j1
and link
j2
are
x
and
1−x
, respectively. Thus, the latencies
of the links are
Λj1(L0) = Pa∈A0s(a)+x
and
Λj2(L0) = Pa∈A\A0s(a)+1−x
.
Without loss of generality, assume, that
Pa∈A0s(a)≥Pa∈A\A0s(a)
.
We show that this implies
Pa∈A0s(a)−Pa∈A\A0s(a) = 0
. Assume otherwise
Pa∈A0s(a)−Pa∈A\A0s(a)>0
. Since the trac of players in
A
is integer, this
implies
Pa∈A0s(a)−Pa∈A\A0s(a)≥1
. Since
NASHIFY
is positive, we made
at most
k
selsh steps with the players having small trac; thus,
1
4≤x≤3
4
.
It follows that
Λj1(L0)−Λj2(L0) = X
a∈A0
s(a) + x−X
a∈A\A0
s(a)−1 + x
≥2x
≥1
2.
This implies that all remaining players with trac
1
4k≤1
8
on link
j1
are unsatised, a contradiction to the fact that
NASHIFY
is positive. So
Pa∈A0s(a)−Pa∈A\A0s(a) = 0
which implies that
PARTITION
is positive.
This completes our reduction.
We remark that
NASHIFY
is
NP
-complete in the strong sense (cf. [52, Section
4.2]) if
m
is part of the input. Thus, there is no pseudopolynomial-time algorithm
for
NASHIFY
(unless
P=NP
). In contrast, there is a natural pseudopolynomial-
time algorithm for
k
-
NASHIFY
, which exhaustively searches all sequences of
k
selsh steps; since a selsh step involves a (unsatised) player and a link for
a total of at most
mn
choices, this algorithm can be implemented to run in
Θ((mn)k)
time.
4.2.2 Nashication
We provide a polynomial-time algorithm to convert any pure strategy prole
into a pure Nash equilibrium with non-increased social cost. We call our algo-
rithm
NashifyIdentical
.
NashifyIdentical
solves
NASHIFY
when
n
selsh
4.2 Identical Links 31
steps are allowed. Together with the
PTAS
for scheduling
n
jobs on
m
identical
machines [61] this yields a
PTAS
for computing a best pure Nash equilibrium.
NashifyIdentical
(
L
)
Input:
A pure strategy prole
L
of
n
players with trac
w1,...,wn
.
Output:
A pure strategy prole
L0
that is a Nash equilibrium.
1: Sort the players' trac in non-increasing order so that
w1≥...≥wn
.
2:
for
i←1
to
n
do
3:
if
player
i
is unsatised
then
4: let player
i
perform a greedy selsh step;
5:
end if
6:
end for
7:
return
the resulting strategy prole
L0
Fig. 4.1.
The algorithm
NashifyIdentical
The algorithm
NashifyIdentical
sorts the players' trac in non-increasing
order so that
w1≥. . . ≥wn
. Then the algorithm examines the players in order
of non-increasing trac. For each player
i
we let player
i
perform a greedy selsh
step, if
i
is unsatised.
Theorem 4.5.
Given an instance
hw, mi
and an associated pure strategy prole
L=hl1, . . . , lni
, algorithm
NashifyIdentical
(
L
) computes a pure Nash equi-
librium
L0
with social cost
SCMSP(w, m, L0)≤SCMSP(w, m, L)
using at most
n
greedy selsh steps and
O(nlog n)
time.
Proof.
Clearly,
SCMSP(w, m, L0)≤SCMSP(w, m, L)
, since selsh steps do not
increase social cost. Furthermore, after every iteration the player that changed
its strategy is satised and stays satised in subsequent iterations by Lemma 4.1.
Thus
L0
is a Nash equilibrium.
The running time of algorithm
NashifyIdentical
is
O(nlog n)
for sorting
the
n
player by their trac,
O(mlog m)
for constructing a heap holding all link
latencies in the pure strategy prole
L
, and
O(log m)
for updating the heap
in each of the
n
iterations of the algorithm. Thus, the total running time is
O(nlog n+mlog m+nlog m)
. The interesting case is when
m≤n
(since oth-
erwise, a single player can be assigned to each link, achieving an optimal Nash
equilibrium). Thus, in the interesting case, the total running time of
NashifyI-
dentical
is
O(nlog n)
.
Since it is possible to compute a pure Nash equilibrium in polynomial time,
one may want to go one step further and ask, whether a pure Nash equilibrium
with
minimum
makespan social cost can also be computed in polynomial time.
Fotakis et al. [37] showed that this problem is
NP
-complete. The next logical step
is to ask for an approximation algorithm to perform this task. Here, the strong
connection between multiprocessor scheduling and routing on parallel links proves
useful.
32 4 Selsh Routing on Parallel Links
Since
NashifyIdentical
does not increase makespan social cost, we can
combine any approximation algorithm for the corresponding scheduling prob-
lem with
NashifyIdentical
. Hochbaum and Shmoys [61] presented a
PTAS
for scheduling
n
jobs
on
m
identical
machines
. Running this
PTAS
on an in-
stance
hw, mi
yields a pure strategy prole
L
such that
SCMSP(w, m, L)≤
(1 + ε)OPTMSP(w, m)
. On the other hand, applying the algorithm
Nashify-
Identical
on
L
yields a Nash equilibrium
L0
such that
SCMSP(w, m, L0)≤
SCMSP(w, m, L)
. Thus,
SCMSP(w, m, L0)≤(1+ε)OPTMSP(w, m)
. It follows that:
Theorem 4.6.
There exists a
PTAS
for computing a pure Nash equilibrium with
minimum makespan social cost, in the model of identical links.
4.3 Related Links
In this section, we consider routing games on parallel
related links
. In Section 4.3.1
we show that results from Section 4.2.2 can be generalized to the model of related
links. Section 4.3.2 holds our results that are related to the price of anarchy for
makespan social cost.
4.3.1 Nashication
We now consider the problem of computing a pure Nash equilibrium for the model
of related links. Again, Graham's LPT algorithm [56] can be used to compute
such a pure Nash equilibrium directly [37]. For related links, the makespan social
cost of the Nash equilibrium computed by LPT approximates the makespan social
cost of a pure Nash equilibrium with minimum social cost by a factor between
1.52
and
1.67
[40].
In this section we are interested in a nashication algorithm, that is, given
a pure strategy prole, we want to compute a pure Nash equilibrium with non-
increased makespan social cost. Selsh steps can also be used to compute a pure
Nash equilibrium, since selsh steps do not increase makespan social cost and
every sequence of selsh steps eventually reaches a pure Nash equilibrium. How-
ever, it is unknown whether selsh steps can be used to implement nashication
in polynomial time. Feldmann et al. [30] chose a dierent approach not only based
on selsh steps. Their algorithm relies on the following crucial observation:
Lemma 4.7.
Consider the model of arbitrary players and related links. Then, for
any pure strategy prole, a greedy selsh step of an unsatised player
i∈[n]
with
weight
wi
from a link
j∈[m]
to a link
k∈[m]
with
cj≤ck
makes no satised
player
s∈[n]
with weight
ws≥wi
unsatised.
Proof.
Let
L
and
L0
be the pure strategy proles before and after the greedy
selsh step of player
i
. By way of contradiction assume that some player
s
with
trac
ws≥wi
becomes unsatised due to this selsh step, and let player
s
be
4.3 Related Links 33
assigned to link
q
. Since only the loads on link
j
and
k
change due to the greedy
selsh step of player
i
we have to show that player
s
cannot improve by moving
to link
j
or if
q=k
that
s
does not become unsatised due to the arrival of
player
i
. We proceed by case study:
•
Assume rst,
q6=k
. As player
s
is satised in
L
,
δk(L) + ws
ck≥δq(L)
cq
.
Player
i
improves by moving to link
k
, thus,
δj(L)
cj
>δk(L) + wi
ck
=δk(L0)
ck
.
It follows
δj(L0) + ws
cj
=δj(L)−wi+ws
cj
>δk(L) + wi
ck
+ws−wi
cj
=δk(L) + ws
ck−ws−wi
ck
+ws−wi
cj
≥δq(L)
cq
+ (ws−wi)1
cj−1
ck
≥δq(L0)
cq
.
The last inequality holds since
ck≥cj
,
ws≥wi
and
δq(L0) = δq(L)
. Thus, in
L0
, player
s
cannot improve by moving to link
j
.
•
Now assume that
q=k
. Since player
i
performs a greedy selsh step from link
j
to link
k
, for all links
r∈[m]
,
δk(L) + wi
ck≤δr(L) + wi
cr
.
Because of
δj(L)−wi+ws
cj≥δj(L)
cj>δk(L)+wi
ck
, player
s
cannot improve by moving
to link
j
. Since player
i
performed a greedy selsh step, for all links
r∈
[m]\{j}
, we have,
δk(L0)
ck
=δk(L) + wi
ck
≤δr(L) + wi
cr
≤δr(L) + ws
cr
=δr(L0) + ws
cr
and therefore player
s
cannot improve by moving to any link
r6=j
.
34 4 Selsh Routing on Parallel Links
The algorithm of Feldmann et al. [30], call it
NashifyRelated
, works in two
phases. In the rst phase, given an instance
hw,ci
and an associated pure strategy
prole
L
, it lls up links with small capacities with players with small trac as
close to
SCMSP(w,c,L)
as possible (but without exceeding
SCMSP(w,c,L)
), and
it collects all these players in a set
U
. In the second phase, the algorithm performs
greedy selsh steps for unsatised players in
U
in non-increasing order of their
trac. Lemma 4.7 allows to show that this procedure results in a pure Nash
equilibrium. Implementing the algorithm in a proper way, we get:
Theorem 4.8 (Feldmann et al. [30]).
Consider the model of arbitrary play-
ers and related links. Then for any instance
hw,ci
and associated pure strategy
prole
L
, a Nash equilibrium
L0
can be computed from
L
with
SCMSP(w,c,L0)≤
SCMSP(w,c,L)
using at most
O(m2n)
time.
Hochbaum and Shmoys [62] presented a
PTAS
for scheduling
n
jobs on
m
related machines. Using the same arguments as in Section 4.2.2, we can combine
this
PTAS
with
NashifyRelated
to get a
PTAS
for computing a pure Nash
equilibrium with minimum makespan social cost.
Theorem 4.9 (Feldmann et al. [30]).
There exists a
PTAS
for computing a
pure Nash equilibrium with minimum makespan social cost, in the model of related
links.
4.3.2 Price of Anarchy
In this section we state results on the price of anarchy and the individual price
of anarchy for the case of related links and makespan social cost.
In this scenario, the rst results on the price of anarchy were give by Koutsou-
pias and Papadimitriou [67]. For the special case of
2
links, they showed that the
price of anarchy is the
golden ratio
. For the general case, Czumaj and Vöcking [23]
proved that the price of anarchy is
Θ(log m
log log log m)
. To show this asymptotically
tight upper bound, they rst provided upper bounds on the maximum expected
latency
Λ(P)
on a link in a mixed Nash equilibrium
P
. A Cherno bound then
gives the upper bound on the price of anarchy. The upper bounds on
Λ(P)
, given
by Czumaj and Vöcking [23], depend on the number of links
m
and the fraction
of the largest and the smallest link capacity. However, not only the capacities,
but the relation between the players' trac and the capacities determine the
(individual) price of anarchy.
To take this relation into account, we introduce a structural parameter
ρ
. We
denote
M1={j∈[m]|w1≤cj·OPTMSP(w,c)}.
Using
M1
, we dene
ρ=Pj∈M1cj
C.
(4.1)
4.3 Related Links 35
In other words,
ρ
is the ratio between the sum of link capacities of links to which
the largest trac can be assigned causing latency at most
OPTMSP(w,c)
and the
sum of all link capacities.
With the help of
ρ
we are able to prove an upper bound of
Γ−1(1
ρ)
on the
individual price of anarchy (Theorem 4.10). Since
w1
c1≤OPTMSP(w,c)
and
C=Pj∈[m]cj≤m·c1
, it follows that
ρ≥1
m
. Using this, we can upper
bound the maximum expected latency on a link in a mixed Nash equilibrium
by
Λ(P)≤Γ−1(m)·OPTMSP(w,c)
(Corollary 4.12), which slightly improves
the best known upper bound of
(Γ−1(m) + 1) ·OPTMSP(w,c)
by Czumaj and
Vöcking [23] and thus leads to an improvement of the upper bound on the
price of anarchy. Furthermore, it follows that the individual coordination ra-
tio is upper bounded by
Γ−1(m)
(Corollary 4.13). For pure Nash equilibria
L
, we
have
SCMSP(w,c,L) = IC(w,c,L)
. It follows that the pure price of anarchy for
makespan social cost is upper bounded by
Γ−1(m)
(Corollary 4.14).
We close this section with two lower bounds on the individual price of anarchy.
These lower bounds show that the upper bound from Theorem 4.10 is tight up
to an additive constant for
all
m
(Theorem 4.16), whereas the upper bound from
Corollary 4.13 is only tight for
large
m
(Theorem 4.17).
Theorem 4.10.
Consider the model of arbitrary players and related links. Then
for any instance
hw,ci
and associated Nash equilibrium
P
,
IC(w,c,P)
OPTMSP(w,c)<
3
2+q1
ρ−3
4
if
1
3≤ρ≤1,
2 + 3
q1
ρ−2
if
1
37 ≤ρ < 1
3,
Γ−11
ρ
if
ρ < 1
37 .
Proof.
Consider an arbitrary instance
hw,ci
with associated mixed Nash equi-
librium
P
such that
IC(w,c,P) = k·OPTMSP(w,c)
for some
k∈R+
. Furthermore, let
L
be a pure strategy prole with optimum
makespan social cost; thus,
SCMSP(w,c,L) = OPTMSP(w,c)
. Note that there
always exists such a pure strategy prole. We proceed as follows. In part (1.) and
(2.) we prove a lower bound on the total amount of trac that is necessary for
the Nash equilibrium
P
. In part (3.) we then use this lower bound to prove an
upper bound on
k
for each of the three cases. We continue with the details of the
formal proof.
(1.) Let
j1
be the maximum index of a link in
M1
, that is,
M1= [j1]
. Let
i1∈[n]
be a player and let
s1∈ M1
be a link with
pi1s1>0
and
λi1s1(P) = δ−i1
s1(P) + wi1
cs1
=IC(w,c,P).
Since
P
is a Nash equilibrium, we have for all links
j∈ M1
,
36 4 Selsh Routing on Parallel Links
IC(w,c,P) = δ−i1
s1(P) + wi1
cs1
≤δ−i1
j(P) + wi1
cj
≤δ−i1
j(P) + w1
cj
.
(4.2)
Furthermore, by denition of
M1
, we have for all links
j∈ M1
,
w1
cj≤OPTMSP(w,c).
(4.3)
This implies that
w1≤cj·OPTMSP(w,c)
for all
j∈ M1
, and thus
k·OPTMSP(w,c)≤IC(w,c,P)
(4.2)
≤δ−i1
j(P) + w1
cj
(4.3)
≤δ−i1
j(P) + cj·OPTMSP(w,c)
cj
=δ−i1
j(P)
cj
+OPTMSP(w,c),
or equivalently
δ−i1
j(P)≥(k−1) ·cj·OPTMSP(w,c).
Therefore, for all
j∈ M1
,
δj(P) = δ−i1
j(P) + pi1j·wi1
≥δ−i1
j(P)
≥(k−1) ·cj·OPTMSP(w,c).
(4.4)
Summing up all expected loads
δj(P)
on links in
M1
, the total expected
trac of links in
M1
is
X
j∈M1
δj(P)(4.4)
≥(k−1) ·X
j∈M1
cj·OPTMSP(w,c)
(4.1)
= (k−1) ·ρ·C·OPTMSP(w,c).
(4.5)
(2.) We prove an inductive claim:
Lemma 4.11.
For all integers
l
with
2≤l≤ dke−1
, there is a set of links
Ml= [jl]\[jl−1]
such that
4.3 Related Links 37
(a)the total capacity of links in
Ml
is at least:
X
j∈Ml
cj≥ρ·C·(k−2) ·
l−1
Y
j=2
(k−j),
(b)the expected load on each link
j
in
Ml
is at least:
δj(P)≥(k−l)·cj·OPTMSP(w,c),
(c)the total expected load on links in
Ml
is at least:
X
j∈Ml
δj(P)≥ρ·C·(k−2) ·
l
Y
j=2
(k−j)·OPTMSP(w,c),
(d)the dierence between the total expected load on links in
M1∪···∪Ml= [jl]
in the mixed Nash equilibrium
P
and the maximum total expected load on
the same links in the optimum strategy prole
L
is at least:
X
j∈[jl]
δj(P)−X
j∈[jl]
δj(L)≥ρ·C·(k−2) ·
l
Y
j=2
(k−j)·OPTMSP(w,c).
Proof.
We will rst show that the claim holds for
l= 2
. Let
wi2
be the small-
est trac of a player
i2
who chooses a link in
M1
with positive probability,
and let
s2∈ M1
be a link in
M1
with
pi2s2>0
. In the pure strategy prole
L
(with optimum social cost) at most
X
j∈M1
δj(L)≤OPTMSP(w,c)·X
j∈M1
cj
(4.1)
=ρ·C·OPTMSP(w,c)
(4.6)
total load can be assigned to links in
M1
. Therefore, in
L
the remaining
expected load which is greater or equal to
X
j∈M1
δj(P)−X
j∈M1
δj(L)(4.6)
≥X
j∈M1
δj(P)−ρ·C·OPTMSP(w,c)
(4.5)
≥ρ·C·(k−2) ·OPTMSP(w,c)
(4.7)
is assigned to links not in
M1
. This implies that there exists a set of links
M2= [j2]\[j1]
,
j2
minimal, with total capacity at least
X
j∈M2
cj≥Pj∈M1δj(P)−Pj∈M1δj(L)
OPTMSP(w,c)
(4.7)
≥ρ·C·(k−2),
(4.8)
38 4 Selsh Routing on Parallel Links
proving (a). Moreover, since
P
is a Nash equilibrium, for all links
j∈ M2
,
(k−1) ·OPTMSP(w,c)(4.4)
≤δs2(P)
cs2
≤δ−i2
s2(P) + wi2
cs2
≤δ−i2
j(P) + wi2
cj
.
(4.9)
By construction of
M2
there exists a player assigned in
P
with positive prob-
ability to a link in
M1
which is assigned to a link
j∈[m]\[j2−1]
in the
optimum strategy prole
L
. This player has trac at least
wi2
. Thus, for all
j∈ M2
,
wi2
cj≤OPTMSP(w,c).
(4.10)
So, for all
j∈ M2
,
(k−1) ·OPTMSP(w,c)(4.9)
≤δ−i2
j(P) + wi2
cj
(4.10)
≤δ−i2
j(P)
cj
+OPTMSP(w,c),
or equivalently
δ−i2
j(P)≥(k−2) ·cj·OPTMSP(w,c).
Therefore, for all
j∈ M2
,
δj(P) = δ−i2
j(P) + pi2j·wi2
≥δ−i2
j(P)
≥(k−2) ·cj·OPTMSP(w,c),
(4.11)
proving (b). Summing up all expected trac
δj(P)
on links in
M2
, the total
expected trac of links in
M2
is
X
j∈M2
δj(P)(4.11)
≥(k−2) ·OPTMSP(w,c)·X
j∈M2
cj
(4.8)
≥(k−2) ·OPTMSP(w,c)·ρ·C·(k−2)
=ρ·C·(k−2)2·OPTMSP(w,c).
proving (c). In the optimum strategy prole
L
at most expected trac
4.3 Related Links 39
X
j∈[j2]
δj(L)≤OPTMSP(w,c)·X
j∈[j2]
cj
(4.12)
can be assigned to links in
M1∪M2= [j2]
. So the remaining expected trac
on links in
M1∪M2
which has to be assigned to other links in the optimal
Nash equilibrium is at least
X
j∈[j2]
δj(P)−X
j∈[j2]
δj(L)
(4.12)
≥X
j∈M1
δj(P) + X
j∈M2
δj(P)−OPTMSP(w,c)·X
j∈[j2]
cj
(4.4)(4.11)
≥
(k−1) ·X
j∈M1
cj+ (k−2) ·X
j∈M2
cj−X
j∈[j2]
cj
·OPTMSP(w,c)
=
(k−2) ·X
j∈M1
cj+ (k−3) ·X
j∈M2
cj
·OPTMSP(w,c)
(4.1)(4.8)
≥((k−2) ·ρ·C+ (k−3) ·(k−2) ·ρ·C)·OPTMSP(w,c)
=ρ·C·(k−2)2·OPTMSP(w,c),
proving (d). This completes the proof of the claim for
l= 2
.
Now, assume inductively that for some integer
l≥3
the claim holds for all
integers not exceeding
(l−1)
. We will prove the claim for
l
.
Let
wil
be the smallest trac of a player
il
who assigns its trac to a link in
M1∪···∪Ml−1
with positive probability, and let
sl∈ M1∪···∪Ml−1
be
a link with
pilsl>0
. By induction hypothesis we have
X
j∈[jl−1]
δj(P)−X
j∈[jl−1]
δj(L)
≥ρ·C·(k−2) ·
l−1
Y
j=2
(k−j)·OPTMSP(w,c).
(4.13)
This implies that there exists a set of links
Ml= [jl]\[jl−1]
,
jl
minimal, with
total capacity at least
X
j∈Ml
cj≥Pj∈[jl−1]δj(P)−Pj∈[jl−1]δj(L)
OPTMSP(w,c)
(4.13)
≥ρ·C·(k−2) ·
l−1
Y
j=2
(k−j),
(4.14)
proving (a). Moreover, since
P
is a Nash equilibrium, for all links
j∈ Ml
,
40 4 Selsh Routing on Parallel Links
(k−(l−1)) ·OPTMSP(w,c)≤δsl(P)
csl
≤δ−il
sl(P) + wil
csl
≤δ−il
j(P) + wil
cj
.
(4.15)
By construction of
Ml
, there exists a player assigned in
P
with positive
probability to a link in
[jl−1]
which is assigned to a link
j∈[m]\[jl−1]
in
the optimum strategy prole
L
. This player has trac at least
wil
. Thus, for
all
j∈ Ml
,
wil
cj≤OPTMSP(w,c).
(4.16)
So, for all
j∈ Ml
,
(k−(l−1)) ·OPTMSP(w,c)(4.15)
≤δ−il
j(P) + wil
cj
(4.16)
≤δ−il
j(P)
cj
+OPTMSP(w,c),
or equivalently
δ−il
j(P)≥(k−l)·cj·OPTMSP(w,c).
Therefore, for all
j∈ Ml
,
δj(P) = δ−il
j(P) + pilj·wil
≥δ−il
j(P)
≥(k−l)·cj·OPTMSP(w,c),
(4.17)
proving (b). Summing up all expected trac
δj(P)
on links in
Ml
, the total
expected trac of links in
Ml
is
X
j∈Ml
δj(P)(4.17)
≥(k−l)·OPTMSP(w,c)·X
j∈Ml
cj
(4.14)
≥ρ·C·(k−2) ·
l
Y
j=2
(k−j)·OPTMSP(w,c),
proving (c). In the optimum strategy prole
L
at most trac
X
j∈[jl]
δj(L)≤OPTMSP(w,c)·X
j∈[jl]
cj
(4.18)
4.3 Related Links 41
can be assigned to links in
M1∪···∪Ml= [jl]
. So the remaining expected
trac on links in
M1∪···∪Ml
which has to be assigned to other links in
the optimal solution is at least
X
j∈[jl]
δj(P)−X
j∈[jl]
δj(L)
=X
j∈[jl−1]
δj(P)−X
j∈[jl−1]
δj(L) + X
j∈Ml
δj(P)−X
j∈Ml
δj(L)
(4.17)(4.18)
≥X
j∈[jl−1]
δj(P)−X
j∈[jl−1]
δj(L)
+(k−l)·X
j∈Ml
cj·OPTMSP(w,c)−X
j∈Ml
cj·OPTMSP(w,c)
(4.13)
≥ρ·C·(k−2) ·
l−1
Y
j=2
(k−j)·OPTMSP(w,c)
+(k−l−1) ·X
j∈Ml
cj·OPTMSP(w,c)
(4.14)
≥ρ·C·(k−2) ·
l−1
Y
j=2
(k−j)·OPTMSP(w,c)
+(k−l−1) ·ρ·C·(k−2) ·
l−1
Y
j=2
(k−j)·OPTMSP(w,c)
= (k−l)·ρ·C·(k−2) ·
l−1
Y
j=2
(k−j)·OPTMSP(w,c)
=ρ·C·(k−2) ·
l
Y
j=2
(k−j)·OPTMSP(w,c),
proving (d). This completes the proof of the inductive claim.
(3.) Summing up the lower bounds on the expected loads over all links we get
the lower bound
Pj∈[m]δj(P)< W
on the total trac
W
that is necessary
for a Nash equilibrium
P
with
IC(w,c,P) = k
. Note that the strict inequality
follows from the fact that we have at least one player with expected latency
k
. Using this lower bound, we now prove the upper bounds for the three
cases of the theorem by showing that a larger upper bound implies
W
C>
OPTMSP(w,c)
, a contradiction. Note that
Γ−1(1
ρ)
is also an upper bound on
the ratio for
ρ≥1
37
. However, in the ranges
1
3≤ρ≤1
and
1
37 ≤ρ < 1
3
the
given upper bounds are better. Now consider the three cases of the theorem:
(I)
1
3≤ρ≤1
: Assume
k≥3
2+q1
ρ−3
4
. This implies
k≥2
in the given range
of
ρ
. Then
42 4 Selsh Routing on Parallel Links
W > X
j∈M1
δj(P) + X
j∈M2
δj(P)
≥ρ·C·OPTMSP(w,c)·(k−1) + (k−2)2
=ρ·C·OPTMSP(w,c)·k2−3·k+ 3
≥ρ·C·OPTMSP(w,c)
· 3
2+r1
ρ−3
42
−3·3
2+r1
ρ−3
4+ 3!
=ρ·C·OPTMSP(w,c)
·9
4+ 3 ·r1
ρ−3
4+1
ρ−3
4−9
2−3·r1
ρ−3
4+ 3
=ρ·C·OPTMSP(w,c)·1
ρ
=C·OPTMSP(w,c).
(II)
1
37 ≤ρ < 1
3
: Assume
k≥2 + 3
q1
ρ−2
. This implies
k > 3
in the given
range of
ρ
. Then,
W > X
j∈M1
δj(P) + X
j∈M2
δj(P) + X
j∈M3
δj(P)
≥ρ·C·OPTMSP(w,c)·(k−1) + (k−2)2+ (k−2)2(k−3)
=ρ·C·OPTMSP(w,c)·k−1+(k−2)3
k>3
> ρ ·C·OPTMSP(w,c)·2+(k−2)3
≥ρ·C·OPTMSP(w,c)·2 + 1
ρ−2
=C·OPTMSP(w,c).
(III)
ρ < 1
37
: Assume
k≥Γ−1(1
ρ)
. Using the facts that
Γ(x+ 1) = x·Γ(x)
for
all
x∈R
and
Γ(x)≤x
for all
1≤x≤3
, we get
W > X
j∈Mbkc−2
δj(P) + X
j∈Mbkc−1
δj(P)
≥ρ·C·OPTMSP(w,c)·(k−2) ·
bkc−2
Y
j=2
(k−j) + bkc−1
Y
j=2
(k−j)
> ρ ·C·OPTMSP(w,c)·(k−2) ·
bkc−1
Y
j=3
(k−j) + bkc−1
Y
j=2
(k−j)
≥ρ·C·OPTMSP(w,c)·(k−2) ·(Γ(k−2) + Γ(k−1))
=ρ·C·OPTMSP(w,c)·(k−2) ·(Γ(k−2) + (k−2) ·Γ(k−2))
=ρ·C·OPTMSP(w,c)·(k−2) ·((k−1) ·Γ(k−2))
4.3 Related Links 43
=ρ·C·OPTMSP(w,c)·Γ(k)
≥ρ·C·OPTMSP(w,c)·ΓΓ−11
ρ
=C·OPTMSP(w,c).
In each of the cases we have
W
C>OPTMSP(w,c)
. This is a contradiction to
the fact that
W
C≤OPTMSP(w,c)
.
This completes the proof of the theorem.
Since
w1
c1≤OPTMSP(w,c)
, we have
ρ≥c1
C≥1
m
. Furthermore,
IC(w,c,P)≥
Λ(P)
holds for every strategy prole
P
. Thus, from Theorem 4.10 we get the
following corollaries:
Corollary 4.12.
Consider the model of arbitrary players and related links. Then
for any instance
hw,ci
and associated Nash equilibrium
P
, the maximum expected
latency
Λ(P)
is bounded from above by
Λ(P)≤Γ−11
ρ·OPTMSP(w,c)≤Γ−1(m)·OPTMSP(w,c).
Corollary 4.13.
Consider the model of arbitrary players and related links. Then
for any instance
hw,ci
and associated Nash equilibrium
P
,
IC(w,c,P)
OPTMSP(w,c)≤Γ−1(m).
Corollary 4.14.
Consider the model of arbitrary players and related links. Then
for any instance
hw,ci
and associated Nash equilibrium
P
,
pPoAMSP ≤Γ−1(m).
We now introduce a pure strategy prole in Example 4.3.1 for which we show
in Lemma 4.15 that it is a pure Nash equilibrium with certain properties. The
pure Nash equilibrium will be used in Theorem 4.16 and Theorem 4.17 to prove
that the upper bounds of
Γ−1(1
ρ)
and
Γ−1(m)
are tight.
Example 4.3.1
Let
k∈N
, and consider the following instance
hw,ci
with as-
sociated pure strategy prole
L
.
•
There are
k
dierent classes of players:
Class
U1
:
|U1|=k
players with trac
2k−1
Class
Ui
:
|Ui|= 2i−1·(k−1) Qj=1,...,i−1(k−j)
players with trac
2k−i
for all
2≤i≤k
.
•
There are
k+ 1
dierent classes of links:
Class
P0
: One link with capacity
2k−1
.
Class
P1
:
|P1|=|U1|−1
links with capacity
2k−1
.
Class
Pi
:
|Pi|=|Ui|
links with capacity
2k−i
for all
2≤i≤k
.
44 4 Selsh Routing on Parallel Links
k
P1
P0
k−1
k−2
P2Pk−1
2 2
1
Pk
1 1
k−2 k−2
2
2
2
2
k−1
k−1
k−2 2
2
2 2 2
k−1 k−1 k−1
2
2 2
2
k−3 k−3 1
2 2
k−2
k−2
k−2
k−3 k−3
1
11
Fig. 4.2.
Instance and associated strategy prole
L
from Example 4.3.1
•
Consider the following strategy prole
L
:
Class
P0
: All players in
U1
are assigned to this link.
Class
Pi
: On each link in
Pi
there are
2(k−i)
players from
Ui+1
, respec-
tively, for all
1≤i≤k−1
.
Class
Pk
: The links from
Pk
remain empty.
Lemma 4.15.
Consider the instance
hw,ci
and associated pure strategy prole
L
given in Example 4.3.1. Then
(a)
OPTMSP(w,c) = 1
, and
(b) the strategy prole
L
is a pure Nash equilibrium with
SCMSP(w,c,L) = k
.
Proof.
We will prove (a) and (b) separately.
(a) In an optimum strategy prole the player with largest trac must be assigned
to some link. Thus,
OPTMSP(w,c)≥w1
c1
= 1.
Now, consider the pure strategy prole
L0
that assigns one player from
U1
to
each link in
P0∪P1
and one player from
Ui
to each link in
Pi
for all
2≤i≤k
.
Then for all links
j∈[m]
, we have
δj(L0)
cj= 1
, so that
OPTMSP(w,c)≤SCMSP(w,c,L0) = max
j∈[m]
δj(L0)
cj
= 1.
It follows that
OPTMSP(w,c) = 1
.
(b) Clearly,
SCMSP(w,c,L) = k
. We now show, that
L
is a Nash equilibrium. In
the pure strategy prole
L
the latency of each link
j∈Pi
where
i∈[k]0
is
Λj(L) = δj(L)
cj
=k−i.
(4.19)
4.3 Related Links 45
By way of contradiction, assume that the pure strategy prole
L
is not a
Nash equilibrium. Then, there exists some player
s
with trac
ws
that is not
satised in
L
. Assume that player
s
can improve by moving from some link
j1∈Pi1
to some other link
j2∈Pi2
. Thus,
δj1(L)
cj1
>δj2(L) + ws
cj2
.
(4.20)
By (4.19) it must hold that
i1< i2
. We proceed by case analysis:
First assume
i1= 0
: Then
ws= 2k−1
and
k(4.19)
=δj1(L)
cj1
(4.20)
>δj2(L) + ws
cj2
(4.19)
= (k−i2) + 2k−1
2k−i2
= (k−i2)+2i2−1
i2≥1
≥(k−i2) + i2
=k,
a contradiction.
Now assume
i1≥1
: Then
ws= 2k−i1−1
and
k−i1
(4.19)
=δj1(L)
cj1
(4.20)
>δj2(L) + ws
cj2
(4.19)
= (k−i2) + 2k−i1−1
2k−i2
= (k−i2)+2i2−i1−1
i2−i1≥1
≥(k−i2) + i2−i1
=k−i1,
a contradiction. It follows that
L
is a pure Nash equilibrium.
This completes the proof of the lemma.
We are now ready to prove two lower bounds on the individual price of anarchy.
Our lower bounds hold for the case of pure Nash equilibria.
Theorem 4.16.
For each
k∈N
there exists an instance
hw,ci
and an associated
pure Nash equilibrium
L
with
k=IC(w,c,L)
OPTMSP(w,c)≥Γ−11
ρ−1.
46 4 Selsh Routing on Parallel Links
Proof.
Consider the instance
hw,ci
from Example 4.3.1 with associated pure
strategy prole
L
. By Lemma 4.15,
L
is a pure Nash equilibrium for
hw,ci
,
OPTMSP(w,c) = 1
and
IC(w,c,L) = SCMSP(w,c,L) = k
. We now prove that
Γ−1(1
ρ)−1
is a lower bound on
k
. We have
ρ=|P0∪P1|·2k−1
|P0∪P1|·2k−1+Pi∈[2,k]|Pi|·2k−i.
This implies
1
ρ=1
|P0∪P1|·2k−1·
|P0∪P1|·2k−1+X
i∈[2,k]|Pi|·2k−i
=1
k·2k−1·
k·2k−1+X
i∈[2,k]
2i−1·(k−1) Y
j∈[i−1]
(k−j)
·2k−i
=1
k·2k−1·
k·2k−1+ 2k−1·(k−1) X
i∈[2,k]=2 Y
j∈[i−1]
(k−j)
<1 + X
i∈[2,k]Y
j∈[i−1]
(k−j)
<1+(k−1) ·(k−1)!
= 1 + k!−(k−1)!
k≥1
≤k!
=Γ(k+ 1).
This yields
k≥Γ−11
ρ−1
, which completes the proof of the theorem.
Theorem 4.17.
For each
k∈N
there exists an instance
hw,ci
and an associated
pure Nash equilibrium
L
with
k=IC(w,c,L)
OPTMSP(w,c)≥Γ−1(m)·(1 + o(1)).
Proof.
Consider the instance
hw,ci
from Example 4.3.1 with associated pure
strategy prole
L
. By Lemma 4.15,
L
is a pure Nash equilibrium for
hw,ci
,
OPTMSP(w,c) = 1
and
IC(w,c,L) = SCMSP(w,c,L) = k
. Moreover, we have
m=X
i∈[k]0|Pi|=k+ (k−1) ·X
i∈[2,k]
2i−1·Y
j∈[i−1]
(k−j)
=k+ (k−1) ·2k−1·(k−1)! ·
1 + X
i∈[2,k−1]
1
2k−i·1
(k−i)!
≤k+ (k−1) ·2k·(k−1)!
≤2k·k!
≤α·kk,
4.4 Restricted Strategy Sets 47
for a constant
α∈R+
. We dene
r=α·kk.
Since
log(r) = k·log(k) + log(α)
=k·log(k)·(1 + o(1))
and
log(log(r)) = log(k) + log(log(k)) + o(1)
= log(k)·(1 + o(1)),
this implies
Γ−1(m)≤Γ−1(r)
=log(r)
log(log(r)) ·(1 + o(1))
=k·(1 + o(1)).
This completes the proof of the claim.
4.4 Restricted Strategy Sets
For all our results so far we have assumed that the strategy sets of the players
are unrestricted. We will now drop this assumption and consider routing games
on parallel links with
restricted strategy sets
. Here, each player
i∈[n]
is only
allowed to assigned its trac to links in its strategy set
Si
, where
Si⊆[m]
. In
Section 4.4.1 we study the problem of computing a pure Nash equilibrium, while
Section 4.4.2 deals with the pure price of anarchy.
4.4.1 Computation of Pure Nash Equilibria
For the model arbitrary players with restricted strategy sets and related links it
is not known whether a pure Nash equilibrium can be computed in polynomial
time. However, for the case of
identical
players, a result of Milchtaich [77] implies
the existence of a polynomial-time algorithm to perform this task.
Furthermore, for the model of arbitrary players with restricted strategy sets
and identical links, Gairing et al. [41] presented a polynomial-time algorithm,
called
Nashify-Restricted
, to compute a pure Nash equilibrium.
In the following, we will present
Nashify-Restricted
, which combines ideas
from blocking ows and the generic
Preflow-Push
algorithm [3]. Here, we
assume that all players' trac is integer. In Section 4.4.1.1, we will rst show,
how to represent a pure strategy prole by a residual network. Afterwards, in
Section 4.4.1.2, we introduce a blocking ow algorithm, called
Unsplittable-
Blocking-Flow
. In Section 4.4.1.3, we show how
Unsplittable-Blocking-
Flow
can be used to convert a given strategy prole into a pure Nash equilibrium
with non-increased makespan social cost.
48 4 Selsh Routing on Parallel Links
4.4.1.1 Residual Network Representation
We introduce a residual network
GL
representing a pure strategy prole
L
.
Denition 4.18.
Given a pure strategy prole
L= (`1, . . . , `n)
, we dene a di-
rected bipartite graph
GL= (V, EL)
, where
V=M∪U
such that each link
is represented by a node in
M
and each player is represented by a node in
U
.
Furthermore,
EL=E1
L∪E2
L
with
E1
L={(u, v) : u∈M, v ∈U, u =`v}
, and
E2
L={(u, v) : u∈U, v ∈M, v ∈Su\{`u}}.
For an arbitrary integer
w
, we use the graph
GL
from Denition 4.18 to dene
a graph
GL(w)
where
V
remains the same, but from
EL
we now only consider
edges
EL(w) = EL\{(u, v) : u∈U, v ∈V, wu> w}
. This implies that players
u
with
wu> w
remain assigned to their links. We use
GL
for
GL(w)
whenever
w
is clear from context.
4.4.1.2 Unsplittable Blocking Flow
We now introduce a blocking ow algorithm, called
Unsplittable-Blocking-
Flow
, which will be extensively used by our Nashication algorithm in Sec-
tion 4.4.1.3.
Unsplittable-Blocking-Flow
combines ideas from blocking
ows with the idea of pushing players without splitting them.
To control our blocking ow algorithm we use two integer parameters
a
and
w
. Here,
w
will be used to refer to a certain trac size, and
a
will be determined
by binary search. For every integer
a
and trac size
w
we partition the set of
links
M
into three subsets:
M−={j∈M|δj(L)≤a}
M0={j∈M|a+ 1 ≤δj(L)≤a+w}
M+={j∈M|δj(L)≥a+w+ 1}
In this setting, we do not have a dedicated source or sink. However, at each time
nodes in
M+
and
M−
can be interpreted as source and sink nodes, respectively.
Note, that those sets change over time.
Roughly speaking, algorithm
Unsplittable-Blocking-Flow
shifts players
so that the latencies of links from
M−
are never decreased, the latencies of links
from
M+
are never increased, and links from
M0
remain in
M0
. Our algorithm
is controlled by a height function
h:V→N0
with
h(j) =
dist
GL(j, M−)
for all
j∈V
. We call an edge
(u, v)
admissible
, if
h(u) = h(v) + 1
. In an
admissible
path
, all edges are admissible. For each node
j∈V
with
0< h(j)<∞
, dene
Suc(j)
to be the set of successors of node
j
; this is the set of nodes to which
j
has an admissible edge, so that
Suc(j) = {i∈V: (j, i)∈EL
and
h(j) = h(i)+1}.
4.4 Restricted Strategy Sets 49
Note that
Suc(j)
also denes the set of admissible edges leaving
j
. Let
suc(j)
be
the rst node in a list implementation of the set
Suc(j)
. We proceed to dene:
Denition 4.19.
A link
j∈M
with
0< h(j)<∞
is called helpful if
δj(L)≥
a+1+wsuc(j)
.
Lemma 4.20 (Gairing et al. [41]).
Let
v0
be a helpful link of minimum height.
Then there exists a sequence
v0, . . . , vr
, where
v2i∈M
for all
0≤i≤r/2
and
v2i+1 ∈U
for all
0≤i < r/2
such that:
(1)
(vi, vi+1)∈EL
and
h(vi) = h(vi+1)+1
.
(2)
δv0(L)≥a+1+wsuc(v0)
.
(3)
a+ 1 ≤δv2i(L) + wsuc(v2i−2)−wsuc(v2i)≤a+w, ∀0< i < r/2
.
(4)
δvr(L) + wsuc(vr−2)≤a+w
.
We are now ready to present the algorithm
Unsplittable-Blocking-Flow
.
The algorithm is depicted in Figure 4.3. Initially, the height function
h
is com-
puted as the distance in
GL
of each node to the set
M−
of nodes. Then, the
algorithm proceeds in phases. In each phase rst the minimum height
d=h(v)
of a node
v∈M+
is computed. Inside each phase, we do not update the height
function, but we successively choose a helpful link
v
of minimum height and we
push players along the helpful path induced by
v
and adjust the pure strategy
prole accordingly. In order to update
GL
we have to change the direction of two
arcs for each player push. The phase ends when there exists no further admissible
path from an node
v∈M+
with
h(v) = d
to some node in
M−
. Before the new
phase starts, we recompute
h
, and we check whether we need to start a new phase
or not.
Unsplittable-Blocking-Flow
stops when either
M−=∅
or for all
v∈M+
we have
h(v) = ∞
.
Unsplittable-Blocking-Flow
(
L, a, w
)
Input:
pure strategy prole
L
positive integers
a, w
Output:
pure strategy prole
L0
1: compute
h
;
2:
while
M−6=∅
and
∃v∈M+:h(v)<∞
do
3:
d←minv∈M+(h(v))
;
4:
while
∃
admissible path from
v∈M+, h(v) = d
to
M−
do
5: choose helpful link
v
of minimum height;
6: push players along helpful path dened by
v
;
7: update
L, GL
;
8:
end while
9: recompute
h
;
10:
end while
11:
return
L
;
Fig. 4.3.
Unsplittable-Blocking-Flow
Gairing et al. [41] showed that a call to
Unsplittable-Blocking-Flow
does not increase the load on any link in
M+
, does not decrease the load on any
50 4 Selsh Routing on Parallel Links
link in
M−
, and that links in
M0
remain in
M0
(Lemma 4.21). This implies that
Unsplittable-Blocking-Flow
does not increase the maximum load and not
decreases the minimum load on a link (Corollary 4.22). Furthermore, they showed
properties on the pure strategy prole computed by
Unsplittable-Blocking-
Flow
(Lemma 4.23) and the running time of
Unsplittable-Blocking-Flow
(Theorem 4.24).
Lemma 4.21 (Gairing et al. [41]).
Let
L0
be the pure strategy prole computed
by
Unsplittable-Blocking-Flow
(
L, a, w
). Then,
(1)
δj(L0)≥δj(L)
for each link
j∈M−(L)
.
(2)
a+ 1 ≤δj(L0)≤a+w
for each link
j∈M0(L)
.
(3)
δj(L0)≤δj(L)
for each node
j∈M+(L)
.
Corollary 4.22 (Gairing et al. [41]).
Let
L0
be the pure strategy prole com-
puted by
Unsplittable-Blocking-Flow
(
L, a, w
). Then,
max
j∈[m]δj(L0)≤max
j∈[m]δj(L)
, and
min
j∈[m]δj(L0)≥min
j∈[m]δj(L).
Lemma 4.23 (Gairing et al. [41]).
For the pure strategy prole
L0=(`0
1, . . . , `0
n)
computed by
Unsplittable-Blocking-Flow
(
L, a, w
) one of the following con-
ditions holds:
(1)
M−(L0) = ∅
.
(2)
M+(L0) = ∅
.
(3) There exists some set of links
B⊂[m]
such that
a)
δj(L0)≥a+ 1
for all
j∈B
, and
b)
δj(L0)≤a+w
for all
j∈[m]\B
, and
c)
`0
i∈B⇒Si⊆B
for all
i∈[n]
with
wi≤w
.
Theorem 4.24 (Gairing et al. [41]).
Unsplittable-Blocking-Flow
can
be implemented to run in
O(mA)
time, where
A=Pi∈[n]|Si|
.
4.4.1.3 Nashication
We now describe how
Unsplittable-Blocking-Flow
can be used to convert
any pure strategy prole into a pure Nash equilibrium with non-increased social
cost.
Our Nashication algorithm, called
Nashify-Restricted
, rst nds a pure
strategy prole satisfying all players with trac
w1
by recursively applying
Unsplittable-Blocking-Flow
. In this recursive procedure, called
Recur-
siveUBF
, we make extensive use of Lemma 4.23.
We then x the pure strategy prole of all players with trac
w1
and pro-
ceed with the next smaller trac while making sure that all xed players stay
satised. To make sure that all xed players stay satised, we introduce lower
4.4 Restricted Strategy Sets 51
and upper bounds on the load of the links, such that the load of each link
is always in its bounds, the lower bound only increases and the upper bound
only decreases. This is done until all players are satised. In order to achieve
this,
Nashify-Restricted
makes extensive use of algorithm
Unsplittable-
Blocking-Flow
.
We now proceed with a detailed description of our Nashication algorithm.
In the following, we denote
w=wi
for some player
i∈[n]
.
RecursiveUBF.
We rst turn our attention to
RecursiveUBF
, which is de-
picted in Figure 4.4. If
l≤δj(L)≤u+w
for all links
j∈B
prior to a call
RecursiveUBF
(
B, L(B),[l, u], w
)
Input:
A set of links
B
, a pure strategy prole
L(B)
, an interval
[l, u]
and a trac size
w
.
Output:
A a pure strategy prole
L0(B)
.
1:
a← d(l+u)/2e
;
2:
if
a=u
then
3:
return
L(B)
4:
end if
5:
L0(B)←
Unsplittable-Blocking-Flow
(L(B), a, w)
;
6:
if
M−(L0) = ∅
and
M+(L0)6=∅
then
7:
L(B)←
RecursiveUBF
(B, L(B),[a, u], w)
;
8:
else if
M−(L0)6=∅
and
M+(L0) = ∅
then
9:
L(B)←
RecursiveUBF
(B, L(B),[l, a], w)
;
10:
else if
M−(L0)6=∅
and
M+(L0)6=∅
then
11: split
B
(according to Lemma 4.23 (3)) into sets
B0
and
B0
;
12:
L(B0)←
RecursiveUBF
(B0,L(B0),[a, u], w)
;
13:
L(B0)←
RecursiveUBF
(B0,L(B0),[l, a], w)
;
14:
L(B)←L(B0)∪L(B0)
;
15:
end if
16:
return
L(B)
;
Fig. 4.4.
RecursiveUBF
to
RecursiveUBF
(
B, L(B),[l, u], w
), then it computes a pure strategy prole,
where no player with trac
w
that is assigned to some link in
B
can improve by
moving to some other link in
B
. By a series of calls to
Unsplittable-Blocking-
Flow
(
L(B), a, w
) we compute a pure strategy prole where
M−
and
M+
are
either both empty or both non-empty. Parameter
a
is chosen by binary search
a∈[l, u], a ∈N
, as follows: If
Unsplittable-Blocking-Flow
returns a pure
strategy prole with
M−=∅
and
M+6=∅
, then we increase
a
. On the other
hand, if
Unsplittable-Blocking-Flow
returns a pure strategy prole with
M−6=∅
and
M+=∅
, then we decrease
a
.
If after the binary search,
M−=∅
and
M+=∅
, then we have computed
a pure strategy prole where all players with trac at least
w
are satised. If
neither
M−=∅
nor
M+=∅
it follows that condition (3) from Lemma 4.23
holds. Dene
B0
as the set of links still reachable from
M+
and let
B0
be the
complement of
B0
in
B
. In this case we split our instance into two parts. One
part with all links in
B0
and all players that are currently assigned to a link in
52 4 Selsh Routing on Parallel Links
B0
, the other part holds the complement. Whenever
B
is split into
B0
and
B0
,
condition (3) from Lemma 4.23 implies that no player
v
with
wv≤w
, assigned
to a link in
B0
, has a link from
B0
in its strategy set.
We recursively proceed with the binary search on
a
in both parts of the in-
stance. For the part that corresponds to
B0
, we increase
a
, while in the other
part we decrease
a
. The recursive splitting of
B
(line 11) denes a partition of
the links into sets
B1, . . . , Bp
. At the end, all parts
B1, . . . , Bp
are put together
to form
L(B)
.
For each
Bk, k ∈[p]
, dene a lower bound
Low(Bk)
on the load of all links
from
Bk
as the last value for
a
after the binary search on
a
in
Bk
. This implies:
Lemma 4.25 (Gairing et al. [41]).
If
l≤δj(L)≤u+w
for all
j∈B
prior
to a call to
RecursiveUBF
, then
RecursiveUBF
(
B, L(B),[l, u], w
) returns a
pure strategy prole
L0(B)
of players in
B
, a partition of
B
into
p
sets
B1, . . . , Bp
for some
p
, and (implicit) numbers
Low(Bk)
for
k∈[p]
, such that:
(1)
u≥Low(B1)> . . . > Low(Bp)≥l
for all
k∈[p]
.
(2)
Low(Bk)≤δj≤Low(Bk) + w
for all
j∈Bk
and for all
k∈[p]
.
(3) No player
u
with
wu≤w
assigned to a link in
Bk
has a link from
B`
in its
strategy set, if
` > k
.
By the postconditions of Lemma 4.25 all players with trac
w
are satised in
the pure strategy prole computed by
RecursiveUBF
. In order to keep these
players satised, we have to ensure that in further computations the lower bounds
only increase and the upper bounds only decrease. We denote the upper bound
by
Up(Bk)
for all links from
Bk
, and in coincidence with (2) we set
Up(Bk) =
Low(Bk) + w
.
Nashify-Restricted.
We are ready to present algorithm
Nashify-Restricted
that converts any given pure strategy prole
L
into a pure Nash equilibrium
L0
with non-increased social cost. Let
ew1> . . . > ewr
be all dierent player traf-
c from
w1, . . . , wn
. The idea is to compute a sequence of pure strategy proles
L0, . . . , Lr
such that
L0=L
,
Lr=L0
and such that for all pure strategy proles
Li
with
1≤i≤r
, all players
u
with
wu≥ewi
are satised. We call the compu-
tation of
Li
from
Li−1
stage
i
. The aim in stage
i
is to compute a pure strategy
prole
Li
from
Li−1
such that in
Li
all players
u
with
wu≥ewi
are satised.
Figure 4.5 shows the high-level structure of our Nashication algorithm. It
rst uses the procedure
RecursiveUBF
to compute a pure strategy prole
L1
,
where all players with trac
ew1
are satised. Afterwards we iteratively satisfy
players with trac
ew2, . . . , ewr
making sure that players with larger trac remain
satised (lines 2-6). We do this by executing
Sweep
over the sets of active links.
In the following, we dene what we mean by sets of active links, and we describe
how a
Sweep
over these sets of active links is executed.
Lemma 4.25 implies that after stage 1, all players with trac
ew1
are satised.
Furthermore, the links are partitioned into
p1
sets
B1, . . . , Bp1
with
Up(Bk) =
Low(Bk) + ew1
for all
k∈[p1]
, and no player
u
with
wu≤ew1
, that is assigned to
a link from
Bk
can be assigned to a link from
B`
when
k < `
.
4.4 Restricted Strategy Sets 53
Nashify-Restricted
(
L0
)
.
stage 1:
1:
L1←
RecursiveUBF
(
[m],L0,[0,maxjδj(L0)],ew1
)
.
stages 2,..., r:
2:
for
i←2
to
r
do
3:
while
there are sets of active links
do
4: execute
Sweep
over the active links;
5:
end while
.Li
is the current pure strategy prole:
6:
end for
7:
return
Lr
Fig. 4.5.
Nashify-Restricted
We now describe stage
i > 1
(lines 2-6in Figure 4.5). The lower bound on the
load of a link only increases and the upper bound only decreases. This implies
that xed players remain satised. At the beginning of stage
i
, we have a pure
strategy prole
Li−1
, where the links are partitioned into
pi−1
sets
B1, . . . , Bpi−1
with
Up(Bk) = Low(Bk)+ ewi−1
, for all
k∈[pi−1]
, and no player
u
, that is assigned
to a link from
Bk
can be assigned to a link from
B`
when
k < `
.
During each stage
i
, we always maintain a pure strategy prole where the
links are partitioned into
q
sets
C1, . . . , Cq
for some
q
. They are ordered such
that
Up(Ck)>Up(Ck+1)
and
Low(Ck)≥Low(Ck+1)
for all
k
with
1≤k < q
.
... ...... wi−1
wi
x+1
C
x
Cy
C
x−1
Cy+1
C
CxCx+1 Cy
CxCx+1 Cy
~
~
Low( ) Low( )
Up( )
Up( )
Up( )
Low( )
Fig. 4.6.
Sets of active links in stage
i
at the beginning of a sweep
At the beginning of a
Sweep
, we have three classes of sets (see Figure 4.6):
•
Some sets of links
Ck,1≤k < x
, have not been considered yet and fulll
Up(Ck)−Low(Ck) = ewi−1
.
•
Moreover, some sets of links
Ck, q ≥k > y
, have been
done in stage
i
already
and fulll
Up(Ck)−Low(Ck) = ewi
.
•
Finally, we have sets
Cx, . . . , Cy
of
active
links, with
ewi<Up(Ck)−Low(Ck)≤
ewi−1
and
Low(Ck) = Low(Cy)
for all
k∈[x, y]
.
54 4 Selsh Routing on Parallel Links
Initially,
Cj=Bj
for all
1≤j≤pi−1
, the links from
Cpi−1
are active, and
the remaining links have not been considered. During a
Sweep
, the number of
partitions
q
may change. We will see in Lemma 4.28 that at the beginning of
each
Sweep
, the
sweep property
introduced below holds:
Denition 4.26 (Sweep Property during stage
i
).
(1) There is a partition of the links into
q
sets
C1, . . . , Cq
for some
q
with
Low(C1)≥. . . ≥Low(Cq)
and
Up(C1)> . . . > Up(Cq)
.
(2) If link
j∈Ck
, then
Low(Ck)≤δj≤Up(Ck)
.
(3) No player
u
with
wu≤ewi
that is assigned to a link in
Ck
has a link from
C`
in its strategy set
Su
, if
` > k
.
(4) There exist integers
x, y
with
1≤x≤y≤q
and
a)
Up(Ck)−Low(Ck) = ewi−1
for
1≤k < x
,
b)
Up(Ck)−Low(Ck) = ewi
for
y < k ≤q
, and
c)
ewi<Up(Ck)−Low(Ck)≤ewi−1
and
Low(Ck) = Low(Cy)
for all
x≤k≤
y
.
We use the denition of sweep property to dene active links.
Denition 4.27.
Let
x, y
be as in Denition 4.26. Then, a link
j∈Ck
,
x≤k≤
y
, is called
active
and a link
j∈Ck
,
y < k ≤p
, is called
done in stage
i
.
A
Sweep
is shown in Figure 4.7 and works on active links as follows: At the
beginning of
Sweep
, the sweep property holds. The aim of
Sweep
is to process
links in
Cy
such that they do not have to be considered again in this stage, or
to make all links in
Cx−1
active by increasing the lower bound of all active links
to
Low(Cx−1)
. In order to preserve the structure of our pure strategy prole, we
choose
a= min{Up(Cy)−ewi,Low(Cx−1)}
. We insert all sets into a list
L
such
that
L= [Cx, . . . , Cy]
. Then, as long as there are at least two sets in
L
, we do the
following: We extract the rst element, say
D1
, of
L
and apply
Unsplittable-
Blocking-Flow
to the sub-instance dened by the set
D1
.
Unsplittable-
Blocking-Flow
(
L(D1), a, ewi
) returns a pure strategy prole
L0
, where one of
the following conditions holds:
1.
M+(L0) = ∅
: In this case, all links in
D1
have load at most
a+ewi
, and
Corollary 4.22 implies that this property is preserved. Let
D2
be the next
element in
L
. Before the call,
Up(D1)>Up(D2)> a +ewi
was true. After
the call, the loads of all links in
D1
are bounded by
a+ewi
. So, by setting
Up(D1)←Up(D2)
, we get a new upper bound on the loads of the links in
D1
, and we fulll the requirement that upper bounds can be only decreased.
D1
and
D2
are merged, and the union of both sets is inserted into
L
. This
way, the number of sets in the list is decreased by
1
.
2.
M−(L0) = ∅
and
M+(L0)6=∅
: In this case, all links in
D1
have load at least
a
, and Corollary 4.22 implies that this property is preserved. Thus, we are
allowed to set
Low(D1)←a
. We are done with
D1
during this execution of
Sweep
.
4.4 Restricted Strategy Sets 55
Require:
L= [Cx,...,Cy]
is a list of the sets of
active
links
1:
a←min{Up(Cy)−ewi,Low(Cx−1)}
;
2:
while
|L| ≥ 2
do
3:
D1←ExtractFirst(L)
;
4:
L0←
Unsplittable-Blocking-Flow
(L(D1), a, ewi)
;
5:
if
M+(L0) = ∅
then
6:
D2←ExtractFirst(L)
;
7:
Up(D1)←Up(D2)
;
8:
D1←D1∪D2
;
Insert(D1,L)
;
9:
else if
M−(L0) = ∅
and
M+(L0)6=∅
then
10:
Low(D1)←a
and
output:
"links in
D1
are
done in this sweep
";
11:
else if
M−(L0)6=∅
and
M+(L0)6=∅
then
12: split
D1
(according to Lemma 4.23 (3)) into sets
D0
1
and
D0
1
;
13:
Low(D0
1)←a
and
output:
"links in
D0
1
are
done in this sweep
";
14:
D2←ExtractFirst(L)
;
15:
Up(D0
1)←Up(D2)
;
16:
D1←D0
1∪D2
;
Insert(D1,L)
;
17:
end if
18:
end while
.
Dierent handling of last set
19:
D1←ExtractFirst(L)
;
20:
if
a=Up(D1)−ewi
then
21:
RecursiveUBF
(
D1,L(D1),[Low(D1), a],ewi
) and
output:
"links in
D1
are
done in
stage
i
";
22:
else
23:
L0←
Unsplittable-Blocking-Flow
(L(D1), a, ewi)
;
24:
if
M−(L0) = ∅
then
25:
Low(D1)←a
and
output:
"links in
D1
are
done in this sweep
";
26:
else if
M−(L0)6=∅
and
M+(L0) = ∅
then
27:
Up(D1)←a+ewi
;
28:
RecursiveUBF
(
D1,L0(D1),[Low(D1), a],ewi
) and
output:
"links in
D1
are
done
in stage
i
";
29:
else if
M−(L0)6=∅
and
M+(L0)6=∅
then
30: split
D1
(according to Lemma 4.23 (3)) into sets
D0
1
and
D0
1
;
31:
Low(D0
1)←a
and
output:
"links in
D0
1
are
done in this sweep
";
32:
Up(D0
1)←a+ewi
;
33:
RecursiveUBF
(
D0
1,L0(D0
1),[Low(D0
1), a],ewi
) and
output:
"links in
D0
1
are
done
in stage
i
";
34:
end if
35:
end if
Fig. 4.7.
Sweep
over the sets of active links
3.
M−(L0)6=∅
and
M+(L0)6=∅
: In this case, we split
D1
according to condition
(3) from Lemma 4.23 into sets
D0
1
and
D0
1
. Condition (3c) implies, that no
player that is assigned to a link in
D0
1
can be assigned to a link in
D0
1
. Since
the load on each link in
D0
1
is at least
a
, we can set
Low(D0
1)←a
. The load
of each link in
D0
1
is at most
a+ewi
. Thus, since the upper bound of the next
element, say
D2
, in
L
is
Up(D2)> a +ewi
, we again can extract
D2
from
L
,
set
Up(D0
1)←Up(D2)
, merge
D0
1
and
D2
, and insert it in
L
. We are done
with
D0
1
during this execution of
Sweep
.
56 4 Selsh Routing on Parallel Links
So, in each case, the number of sets in list
L
is decreased by
1
. Now, we
consider the case that there is only one set, say
D1
, in
L
. This case has to be
handled dierently.
If
a=Up(D1)−ewi
, then we simply apply
RecursiveUBF
to the sub-instance
dened by
D1
in the interval
[Low(D1), a]
with trac size
ewi
. Otherwise, we ap-
ply
Unsplittable-Blocking-Flow
to the sub-instance dened by the set
D1
.
Unsplittable-Blocking-Flow
(
L(D1), a, ewi
) returns a pure strategy prole
L0
where one of the following conditions holds.
1.
M−(L0) = ∅
: Here, we set
Low(D1)←a
.
2.
M−(L0)6=∅
and
M+(L0) = ∅
: In this case, we set
Up(D1)←a+ewi
and
apply
RecursiveUBF
to the sub-instance dened by
D1
in the interval
[Low(D1), a]
with trac size
ewi
.
3.
M−(L0)6=∅
and
M+(L0)6=∅
: Here, we split
D1
according to condition (3)
from Lemma 4.23 into sets
D0
1
and
D0
1
. For
D0
1
we set
Low(D0
1)←a
and for
D0
1
we set
Up(D0
1)←a+ewi
and we apply
RecursiveUBF
to the sub-instance
dened by
D0
1
in the interval
[Low(D0
1), a]
with trac size
ewi
.
After each sweep, by renumbering the partitions, we get a new pure strategy
prole that again has the same structure as in Denition 4.26. This completes
the description of
Sweep
. Gairing et al. [41] proved:
Lemma 4.28 (Gairing et al. [41]).
The sweep property holds at the beginning
of each execution of
Sweep
. Moreover, in each execution, either a non-empty set
of links is added to the set of active links, or some non-empty set of links is done
in the current stage.
Lemma 4.29 (Gairing et al. [41]).
After stage
i
, every player
u
with trac
wu≥ewi
is satised.
Theorem 4.30 (Gairing et al. [41]).
Consider the model of arbitrary players
with restricted strategy sets and identical links. Given an instance
hw, mi
and an
associated pure strategy prole
L
,
Nashify-Restricted
(L)
computes a Nash
equilibrium with non-increased makespan social cost in polynomial time.
Remark 4.31.
The algorithm
Unsplittable-Blocking-Flow
has been proved
useful also for the problem of scheduling unrelated parallel machines with the
objective to minimize makespan. For this problem, we were able to provide a
combinatorial
2
-approximation algorithm [51], which is simpler and faster than
previously known algorithms. For the approximation algorithm, the procedure
Unsplittable-Blocking-Flow
is an essential element.
4.4.2 Price of Anarchy
In this section we present a comprehensive collection of bounds on the pure price
of anarchy for the model of restricted strategy sets and makespan social cost.
Independently of our work, Awerbuch et al. [6] also have studied makespan social
4.4 Restricted Strategy Sets 57
cost for the model of restricted strategy sets. They focused on the case of arbitrary
players and identical links, for which they proved that the pure price of anarchy
is
Θ(log m
log log m)
. Awerbuch et al. [6] also consider mixed Nash equilibria. Here, they
showed that the price of anarchy is
Θ(log m
log log log m)
.
We structure our results on the pure price of anarchy in this model as follows.
In Section 4.4.2.1 we prove a lower bound of
Γ−1(m)−2
that holds for the case of
identical players and identical links (Theorem 4.4.2.1). This lower bound serves
also as a lower bound for the more general cases of arbitrary players or related
links (or of both). Section 4.4.2.2 shows an upper bound of
Γ−1(n)+1
for the case
of identical players and related links (Theorem 4.33). Section 4.4.2.3 states that
for arbitrary players and identical links the pure price of anarchy is upper bounded
by
Θ(log m
log log m)
(Theorem 4.35). We stress that Theorem 4.35 is the only result
that can be found in (or even follows from) Awerbuch et al. [6]. Section 4.4.2.4
studies the general model of arbitrary players and related links. Here, we show
that the pure price of anarchy lies in between
m−1
and
m
(Theorem 4.37 and
Theorem 4.38). For our upper bounds in Section 4.4.2.2 and Section 4.4.2.3, we
use similar techniques as in [23].
4.4.2.1 Identical Players and Identical Links
We start by proving a lower bound on the pure price of anarchy. This lower bound
holds for the model of identical players with restricted strategy sets and identical
links.
Theorem 4.32.
Consider the model of identical players with restricted strategy
sets and identical links. Then,
pPoAMSP >Γ−1(m)−2 = Ωlog m
log log m.
Proof.
Consider an instance
hn, mi
with
n
identical players with restricted strat-
egy sets and
m
identical links. We construct the strategy sets of the players as
follows. Fix some suciently large integer
p
(to be determined later).
•
Partition the set of links into
p+ 1
disjoint subsets
M0,M1, . . . , Mp
with:
|M0|= 1
.
For each integer
l
, where
1≤l≤p
,
|Ml|= (p−1) ·Qj∈[l−1](p−j)
.
Note that since
|M0| ≤ |M1|< . . . < |Mp|
the partition implies that
m <
(p+1) ·|Mp|= (p+1)(p−1)(p−1)! <(p+1)! = Γ(p+2)
. So,
p > Γ−1(m)−2
.
•
Partition the set of players into
p
disjoint subsets
U0,U1, . . . , Up−1
with:
For each integer
k
, where
0≤k≤p−1
,
|Uk|= (p−k)·|Mk|
.
The strategy set of each player in
Uk
is
Mk∪Mk+1
.
We now construct a pure Nash equilibrium
L
and an optimal strategy prole
Q
such that
SCMSP(n, m, L) = p
and
SCMSP(n, m, Q) = 1
.
•
Construct a pure strategy prole
L
as follows.
58 4 Selsh Routing on Parallel Links
All
p
players from the set
U0
are assigned to the single link in
M0
.
For each integer
k
, where
1≤k≤p−1
,
p−k
players from
Uk
are assigned
to each link in
Mk
. (Note that no player is assigned to any link in
Mp
.)
By the construction of
L
, the latency on each link in the set
Ml
, where
0≤l≤p
, is
p−l
. Thus, for each integer
l
, where
0≤l≤p−1
, no player
assigned to a link in the set
Ml
can decrease its private cost by switching
either to a dierent link from the set
Ml
or to a link from the set
Ml+1
. So,
all players are satised in
L
and
L
is a Nash equilibrium with
SCMSP(n, m, L) = max
j∈[m]Λj(L)
= max
0≤l≤p(p−l)
=p .
•
Note that
|M0|+|M1|=p
and
|U0|=p
. Note also that for each integer
k
,
1≤k≤p−1
,
|Uk|= (p−k)·|Mk|
= (p−k) (p−1) ·Y
j∈[k−1]
(p−j)
= (p−1) ·Y
j∈[k]
(p−j)
=|Mk+1|.
So, it is possible to assign each player in
U0
to a distinct link in
M0∪M1
,
and to assign each player in
Uk
, where
1≤k≤p−1
, to a distinct link in
Mk+1
. Call
Q
the resulting pure strategy prole. Then,
SCMSP(n, m, Q)=1
and
Q
is optimal. So,
OPTMSP(n, m) = 1
.
It follows that
pPoAMSP ≥SCMSP(n, m, L)
OPTMSP(n, m)
=p
>Γ−1(m)−2
=Ωlog m
log log m,
as needed.
Theorem 4.32 implies that
Ωlog m
log log m
is a lower bound on the pure price of
anarchy for the more general cases of arbitrary players or related links (or of
both).
4.4.2.2 Identical Players and Related Links
We proceed with an upper bound on the pure price of anarchy for the model of
identical players with restricted strategy sets and related links.
4.4 Restricted Strategy Sets 59
Theorem 4.33.
Consider the model of identical players with restricted strategy
sets and related links. Then,
pPoAMSP ≤Γ−1(n) + 1 = Olog n
log log n.
Proof.
Consider any arbitrary instance
hn, ci
with an associated pure Nash equi-
librium
L
such that
k·OPTMSP(n, c)≤SCMSP(n, c,L)<(k+ 1) ·OPTMSP(n, c)
for some integer
k∈N
, and an optimal strategy prole
Q
. To prove an upper
bound on the price of anarchy, it suces to prove an upper bound on
k+ 1
. To
do so, we will prove a lower bound (as a function of
k
) on the number of players
that are necessary for such a Nash equilibrium
L
. We will then use this lower
bound to prove an upper bound of
Olog n
log log n
on
k+ 1
. We continue with the
details of the formal proof.
Consider now a link
j∈[m]
with
cj<1
OPTMSP(n,c)
. Note that in the optimal
strategy prole
Q
, no player is assigned to link
j
(since otherwise
1
cj≤Λj(Q)≤
OPTMSP(n, c)
, or
cj≥1
OPTMSP(n,c)
). If, in addition,
Λj(L)<SCMSP(n, c,L)
,
then link
j
can be eliminated (together with all players assigned to it in
L
)
with no change to
SCMSP(n, c,L)
and no increase to
OPTMSP(n, c)
. So, assume,
without loss of generality, that for each link
j∈[m]
, either
cj≥1
OPTMSP(n,c)
or
Λj(L) = SCMSP(n, c,L)
.
Dene
M0
as the set of links
j∈[m]
with latency
Λj(L)≥k·OPTMSP(n, c).
Clearly,
M06=∅
. By denition of latency, this implies that
X
j∈M0
δj(L)≥k·OPTMSP(n, c)·X
j∈M0
cj.
We prove an inductive claim:
Lemma 4.34.
For each
l∈[k−1]
, there is a set of links
Ml
with
Ml∩(M0∪. . . ∪Ml−1) = ∅
such that:
(1)
Pj∈Mlcj≥(k−1) ·Qj∈[l−1](k−j)·Pj∈M0cj
.
(2)
For each link
j∈ Ml
,
Λj(L)≥(k−l)·OPTMSP(n, c)
.
(3)
Pj∈Mlδj(L)≥(k−1) ·Qj∈[l](k−j)·OPTMSP(n, c)·Pj∈M0cj
.
(4)
There are at least
(k−1) ·Qj∈[l](k−j)·OPTMSP(n, c)·Pj∈M0cj
players
assigned by
L
to links in
M0∪. . . ∪ Ml
whose strategy sets include links
outside
M0∪. . . ∪Ml
.
Proof.
By (strong) induction on
l
. For the sake of shortening the proof, we merge
the proof for the basis case (where
l= 1
) into the proof for the induction step;
60 4 Selsh Routing on Parallel Links
thus, the case
l= 1
will be treated separately (where needed) along the proof of
the induction step.
Assume inductively that for some integer
l≥1
, the claim holds for all integers
not exceeding
(l−1)
. Notice that if
l= 1
, the induction hypothesis is empty. We
will prove the claim for
l
.
Assume rst that
l= 1
. Recall that
X
j∈M0
δj(L)≥k·OPTMSP(n, c)·X
j∈M0
cj.
In the optimal strategy prole
Q
,
Λj(Q)≤OPTMSP(n, c)
for each link
j∈
[m]
. By denition of latency, this implies that
Pj∈M0δj(Q)≤OPTMSP(n, c)·
Pj∈M0cj
. It follows that there are at least
k·OPTMSP(n, c)·X
j∈M0
cj−OPTMSP(n, c)·X
j∈M0
cj
= (k−1) ·OPTMSP(n, c)·X
j∈M0
cj
excess players
assigned by
L
to links in
M0
whose strategy sets include links
outside
M0
.
Assume now that
l > 1
. By induction hypothesis (condition
(4)
), there are at
least
(k−1) ·Qj∈[l−1](k−j)·OPTMSP(n, c)·Pj∈M0cj
excess players assigned
by
L
to links in
M0∪. . . ∪ Ml−1
whose strategy sets include links outside
M0∪. . . ∪Ml−1
.
Dene
Ml
as the set of all links outside
M0∪. . .∪Ml−1
that are included
in the strategy sets of such excess players; so,
Ml∩(M0∪. . .∪Ml−1) = ∅
.
Clearly, in
Q
, all these excess players are assigned to links in
Ml
, so that
X
j∈Ml
δj(Q)≥(k−1) ·Y
j∈[l−1]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj.
We now prove the four claimed properties for the set
Ml
.
•
Clearly,
X
j∈Ml
cj=X
j∈Ml
δj(Q)
Λj(Q)
≥Pj∈Mlδj(Q)
OPTMSP(n, c)
≥(k−1) ·Qj∈[l−1](k−j)·OPTMSP(n, c)·Pj∈M0cj
OPTMSP(n, c)
= (k−1) ·Y
j∈[l−1]
(k−j)·X
j∈M0
cj,
which proves
(1)
.
4.4 Restricted Strategy Sets 61
•
To prove
(2)
, consider any link
j∈ Ml
. Since
j6∈ M0
, it follows that
Λj(L)<
k·OPTMSP(n, c)
. Since
SCMSP(n, c)≥k·OPTMSP(n, c)
, this implies that
Λj(L)<SCMSP(n, c)
. Therefore,
cj≥1
OPTMSP(n,c)
.
If
l= 1
, there is some link
j0∈ M0
to which
L
assigns some excess player.
So, assume
l > 1
. Recall that in the optimal strategy prole
Q
,
Λj(Q)≤
OPTMSP(n, c)
for each link
j∈ Ml
. By denition of latency, this implies
that
Pj∈Ml−1δj(Q)≤OPTMSP(n, c)·Pj∈Ml−1cj
. By induction hypothesis
(condition
(3)
),
X
j∈Ml−1
δj(L)≥(k−1) ·Y
j∈[l−1]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj.
It follows that there is some excess player assigned to some link
j0∈ Ml−1
.
Since
L
is a Nash equilibrium, for each link
j∈ Ml
,
Λj0(L)≤Λj(L) + 1
cj
≤Λj(L) + OPTMSP(n, c).
Assume rst that
l= 1
. By denition of the set
M0
,
Λj0(L)≥k·OPTMSP(n, c).
It follows that
Λj(L)≥(k−1) ·OPTMSP(n, c),
and the proof of
(2)
for the basis case is now complete.
So, assume
l > 1
. By induction hypothesis (condition
(2)
),
Λj0(L)≥(k−(l−1)) ·OPTMSP(n, c)
= (k−l)·OPTMSP(n, c) + OPTMSP(n, c).
It follows that
Λj(L)≥(k−l)·OPTMSP(n, c),
and the proof of
(2)
is now complete.
•
To prove
(3)
, we use
(2)
and
(1)
to derive that
X
j∈Ml
δj(L) = X
j∈Ml
Λj(L)·cj
≥(k−l)·OPTMSP(n, c)·X
j∈Ml
cj
≥(k−l)·OPTMSP(n, c)·(k−1) ·Y
j∈[l−1]
(k−j)·X
j∈M0
cj
= (k−1) ·Y
j∈[l]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj,
as needed for proving
(3)
.
62 4 Selsh Routing on Parallel Links
•
Recall rst that in the optimal strategy prole
Q
,
Λj(Q)≤OPTMSP(n, c)
for
each link
j∈[m]
. By denition of latency, this implies that
Pj∈Mlδj(Q)≤
OPTMSP(n, c)·Pj∈Mlcj
.
Clearly, the number of players assigned by
L
to links in
M0∪. . . ∪Ml
whose
strategy sets include links outside
M0∪. . . Ml
is at least
X
r∈[0,l]X
j∈Mr
(δj(L)−δj(Q))
=X
r∈[0,l−1] X
j∈Mr
(δj(L)−δj(Q)) + X
j∈Ml
δj(L)−X
j∈Ml
δj(Q)
≥(k−1) ·Y
j∈[l−1]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj
+(k−l)·OPTMSP(n, c)·X
j∈Ml
cj−OPTMSP(n, c)·X
j∈Ml
cj
= (k−1) ·Y
j∈[l−1]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj
+(k−l−1) ·OPTMSP(n, c)·X
j∈Ml
cj
≥(k−1) ·Y
j∈[l−1]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj
+(k−l−1) ·OPTMSP(n, c)·(k−1) ·Y
j∈[l−1]
(k−j)·X
j∈M0
cj
= (1 + (k−l−1)) ·(k−1) ·Y
j∈[l−1]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj
= (k−1) ·Y
j∈[l]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj,
as needed for proving
(4)
.
The proof of the inductive claim is now complete.
We now prove an upper bound on
k+ 1
. Fix any link
j∈ M0
. Clearly,
Λj(L)≤SCMSP(n, c,L)<(k+ 1) ·OPTMSP(n, c)
. Recall that by denition of
M0
,
Λj(L)≥k·OPTMSP(n, c)>0
. This implies that
Λj(L)≥1
cj
. It follows that
1
cj<(k+ 1) ·OPTMSP(n, c)
. This implies that
OPTMSP(n, c)·X
j∈M0
cj>1
k+ 1 .
Assume, without loss of generality, that
k≥3
(otherwise
k+1 ∈O(1)
). Then,
by Lemma 4.34 (condition
(3)
),
4.4 Restricted Strategy Sets 63
n≥X
j∈Mk−1
δj(L) + X
j∈Mk−2
δj(L)
≥(k−1) ·Y
j∈[k−1]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj
+(k−1) ·Y
j∈[k−2]
(k−j)·OPTMSP(n, c)·X
j∈M0
cj
>2·(k−1) ·(k−1)! ·1
k+ 1
≥(k−1)!
=Γ(k).
Hence
k+ 1 <Γ−1(n)+1
=Olog n
log log n,
as needed.
We remark that Theorems 4.32 and 4.33 leave a gap between our bounds on the
pure price of anarchy for the case of identical players. Closing this gap remains
an interesting open problem.
4.4.2.3 Arbitrary Players and Identical Links
With a similar proof as in Theorem 4.33, we can prove an upper bound on the pure
price of anarchy for the model of arbitrary players with restricted strategy sets
and identical links. This upper bound matches asymptotically the lower bound
shown in Theorem 4.32.
Theorem 4.35.
Consider the model of arbitrary players with restricted strategy
sets and identical links. Then,
pPoAMSP =Γ−1(m) = Olog m
log log m.
Theorems 4.33 and 4.35 together imply:
Theorem 4.36.
Consider the model of identical players with restricted strategy
sets and identical links. Then,
pPoAMSP =Olog min{m, n}
log log min{m, n}.
We remark that in the interesting cases where
n≥m
, Theorems 4.32 and 4.36
provide asymptotically tight bounds on the pure price of anarchy for the case of
identical players and identical links.
64 4 Selsh Routing on Parallel Links
4.4.2.4 Arbitrary Players and Related Links
We now turn to the model of arbitrary players with restricted strategy sets and
related links. For this model, we provide almost matching upper and lower bounds
on the pure price of anarchy. We rst prove the lower bound:
Theorem 4.37.
Consider the model of arbitrary players with restricted strategy
sets and related links. Then,
pPoA ≥m−1
.
Proof.
Consider an instance
hw,ci
as follows:
•
For each link
j∈[m]
, the capacity
cj
is
cj=(m−1)!
(j−1)! .
•
There are
n=m−1
players; the weight of player
i∈[m−1]
is
wi=ci
.
Moreover, assume that for each player
i∈[m−1]
, the strategy set
Si
is
Si=
{i, i + 1}
.
•
Construct a pure strategy prole
L
as follows:
Each player
i∈[m−1]
is assigned to link
i+ 1
.
We will argue that all players are satised in
L
.
On the one hand, the private cost of each player
i∈[m−1] \{1}
is
PCi(L) = Λi+1(L)
=wi
ci+1
=i .
On the other hand, moving to the other link
i
in its strategy set would
lead to latency
δi+wi
ci
=wi−1+wi
ci
=ci−1+ci
ci
= (i−1) + 1
=i .
It follows that player
i∈[m−1] \{1}
is satised in
L
.
Consider now player
1
. Since
c1=c2
and there are no players assigned to
link
1
, player
1
cannot decrease its private cost by switching from link
2
to link
1
. So, player
1
is also satised in
L
.
It follows that
L
is a Nash equilibrium. Clearly,
SCMSP(w,c,L) = max
j∈[m]Λj(L)
=m−1.
4.4 Restricted Strategy Sets 65
•
Construct now a pure strategy prole
Q
as follows:
Each player
i∈[m−1]
is assigned to link
i
.
Clearly, for each link
j∈[m−1]
,
Λj(L) = wj
cj= 1
and
Λm(L) = 0
. So,
SCMSP(w,c,Q) = 1
. Thus,
OPTMSP(w,c)≤1
.
It follows that
pPoAMSP ≥SCMSP(w,c,L)
OPTMSP(w,c)
=m−1,
as needed.
We now prove the upper bound:
Theorem 4.38.
Consider the model of arbitrary players with restricted strategy
sets and related links. Then,
pPoA < m
.
Proof.
Consider any arbitrary instance
hw,ci
with an associated Nash equilib-
rium
L
such that
k·OPTMSP(w,c)≤SCMSP(w,c,L)<(k+ 1) ·OPTMSP(w,c)
for some integer
k∈N
, and an optimal strategy prole
Q
. To prove an upper
bound on the price of anarchy, it suces to prove an upper bound on
k+ 1
. To
do so, we will prove a lower bound (as a function of
k
) on the number of links
that are necessary for such a Nash equilibrium
L
. We will then use this lower
bound to prove an upper bound of
m
on
k+ 1
. We continue with the details of
the formal proof.
We prove an inductive claim:
Lemma 4.39.
For each integer
i∈[k]
, there exists a distinct link
li∈[m]
with
latency
Λli(L)≥(k−i+ 1) ·OPTMSP(w,c)
.
Proof.
By (strong) induction on
i
. For the basis case, let
i= 1
. Since
SCMSP(w,c,L)≥k·OPTMSP(w,c)
, there is a link
l1∈[m]
with latency
Λl1(L)≥k·OPTMSP(w,c)
, as needed.
Assume inductively that the claim holds for all integers not exceeding
(i−1)
where
i≥2
. We will prove the claim for
i
. By induction hypothesis, there exist
i−1
distinct links
l1, . . . , li−1
with
Λlj(L)≥(k−j+ 1) ·OPTMSP(w,c),
for each integer
j∈[i−1]
. Since
j≤i−1
and
i≤k
, it follows that
j≤k−1
.
So,
k−j+ 1 ≥2
. It follows that for each integer
j∈[i−1]
,
Λlj(L)>OPTMSP(w,c).
Since
OPTMSP(w,c) = SCMSP(w,c,Q)≥Λlj(Q)
for each integer
j∈[i−1]
, it
follows that for each integer
j∈[i−1]
,
Λlj(L)> Λlj(Q)
. So,
Pj∈[i−1] Λlj(L)>
66 4 Selsh Routing on Parallel Links
Pj∈[i−1] Λlj(Q)
. It follows that there is some player
i0
assigned by
L
to some
link in the set
{l1, . . . , li−1}
that is assigned by
Q
to some link
li6∈ {l1, . . . , li−1}
(otherwise,
Pj∈[i−1] Λlj(Q)≥Pj∈[i−1] Λlj(L)
). Thus,
li
is an allowed link for
player
i0
.
Since
L
is a Nash equilibrium, player
i0
has no incentive to switch from its
link
lj
, where
j∈[i−1]
, to link
li
. Since player
i0
is assigned to link
li
in
Q
, the
additional latency on link
li
in
L
due to player
i0
switching to link
li
is at most
the latency on link
li
in
Q
; since
Q
is optimal, this additional latency is at most
OPTMSP(w,c)
. It follows that
Λlj(L)≤Λli(L) + OPTMSP(w,c).
By induction hypothesis,
Λlj(L)≥(k−j+ 1) ·OPTMSP(w,c)
≥(k−(i−1) + 1) ·OPTMSP(w,c)
= (k−i+ 1) ·OPTMSP(w,c) + OPTMSP(w,c).
It follows that
Λli(L)≥(k−i+ 1) ·OPTMSP(w,c).
The proof of the inductive claim is now complete.
Lemma 4.39 implies that for
L
, there are
k
distinct links with latency
larger than
OPTMSP(w,c)
. Since
Pj∈[m]Λj(L) = Pj∈[m]Λj(Q)
and
Λj(Q)≤
OPTMSP(w,c)
for each link
j∈[m]
, it follows that there is some other link with
latency smaller than
OPTMSP(w,c)
. So,
k≤m−1
or
k+ 1 ≤m
, as needed.
4.5 Polynomial Social Cost
We now come back to the case of unrestricted strategy sets and study
polynomial
social cost
for routing games on parallel links. Throughout this section, we restrict
to the case of
identical links
. Recall, that polynomial social cost is dened with
the help of a certain polynomial cost function
πd(λ)
of degree
d
. Using a dierent
denition of social cost does not alter the denition of private cost nor the set
of (pure) Nash equilibria. Thus, the results on sequences of selsh steps from
Section 4.2.1 also apply to this model. Furthermore, Lücking [70] showed that
selsh steps do not increase polynomial social cost. This implies that
Nashify-
Identical
from Section 4.2.2 can be used to convert a given strategy prole into
a pure Nash equilibrium with non-increased polynomial social cost.
In this section, we are interested in the price of anarchy for polynomial social
cost. We start by proving a simple fact (Lemma 4.40), that will be instrumental
for reducing the polynomial price of anarchy for arbitrary polynomials (with non-
negative coecients) to the monomial price of anarchy. This result holds for arbi-
trary players. We then focus on the case of identical players. Here, we show that
4.5 Polynomial Social Cost 67
the monomial social cost of the fully mixed Nash equilibrium can be expressed
as a combinatorial sum of Stirling numbers of the second kind (Corollary 4.41).
With the help of Corollary 4.41, we show that fully mixed Nash equilibria max-
imize polynomial social cost, for the case of identical players and two identical
links (Theorem 4.42). Afterwards, we consider the case of identical players and
arbitrary many identical links. Here, we show that fully mixed Nash equilibria
maximize polynomial social cost up to a factor
1 + 1
n−1d
(Theorem 4.44). Re-
cently, it was shown that this factor is not necessary (Theorem 4.47).
Equipped with these results, we show that for the model of identical players
and identical links, the price of anarchy is upper bounded by
Bd
. Recall that
Bd
is the
Bell number
of order
d
. Our analysis rst shows that
Bd
is an upper
bound on the price of anarchy, if the polynomial cost function is the
d
'th power
(Theorem 4.48). As a corollary we get that the same upper bound also holds for
general polynomial cost functions (Corollary 4.49). We show in Theorem 4.50
and Corollary 4.51 that, for the special case of
2
links, both upper bounds reduce
from
Bd
to
2d−21 + 1
nd−1
.
We start by proving a simple fact that holds for arbitrary players.
Lemma 4.40 (From Polynomials to Monomials).
Consider the model of
arbitrary players and identical links. Fix an instance
hw, mi
with an associated
Nash equilibrium
P
. Then,
SCπd(λ)(w, m, P)
OPTπd(λ)(w, m)≤max
t∈[d]SCλt(w, m, P)
OPTλt(w, m).
Proof.
Our proof will use the expression of polynomial social cost as a linear
combination of monomial social costs (see Section 3.1.5.2). The proof will manip-
ulate sums of fractions while relying on the non-negativeness of the coecients
in the latency cost function. We continue with the details of the formal proof.
Let
Q
be an optimal strategy prole for the instance
hw, mi
. Then,
SCπd(λ)(w, m, P)
OPTπd(λ)(w, m)=SCπd(λ)(w, m, P)
SCπd(λ)(w, m, Q)
=a0+Pt∈[d]at·SCλt(w, m, P)
a0+Pt∈[d]at·SCλt(w, m, Q).
Since
Q
is an optimal strategy prole,
SCπd(λ)(w, m, P)≥SCπd(λ)(w, m, Q)
.
Since the coecients
at
are non-negative for all
t∈[d]
, it follows that
Pt∈[d]at·
SCλt(w, m, P)≥Pt∈[d]at·SCλt(w, m, Q)
. Since
a0≥0
, this implies that
a0+Pt∈[d]at·SCλt(w, m, P)
a0+Pt∈[d]at·SCλt(w, m, Q)≤Pt∈[d]at·SCλt(w, m, P)
Pt∈[d]at·SCλt(w, m, Q).
68 4 Selsh Routing on Parallel Links
Now dene, for each
t∈[d]
,
K(t, P,Q) = SCλt(w, m, P)
SCλt(w, m, Q).
Observe, that for each
t∈[d]
,
SCλt(w, m, Q)>0
; so
K(t, P,Q)
is well dened.
It follows that
Pt∈[d]at·SCλt(w, m, P)
Pt∈[d]at·SCλt(w, m, Q)
=Pt∈[d]at·SCλt(w, m, Q)·K(t, P,Q)
Pt∈[d]at·SCλt(w, m, Q)
≤maxt∈[d]{K(t, P,Q)}·Pt∈[d]at·SCλt(w, m, Q)
Pt∈[d]|at>0at·SCλt(w, m, Q)
= max
t∈[d]{K(t, P,Q)}
= max
t∈[d]SCλt(w, m, P)
SCλt(w, m, Q).
Since
SCλt(w, m, Q)≥OPTλt(w, m)
the claim follows.
4.5.1 Identical Players
In the following we restrict to the case of identical players. In this case, by De-
nition 2.1, polynomial social cost reduces to
SCπd(λ)(n, m, P) = X
j∈[m]X
A⊆[n] Y
i∈A
pij!
Y
i6∈A
(1 −pij)
πd(|A|)
=X
j∈[m]
BF(hp1j, . . . , pnji, πd(λ)) .
So, polynomial social cost is a sum of binomial functions, one for each link.
4.5.1.1 Fully Mixed Nash Equilibria
We proceed by providing a simple expression for the monomial social cost of
the fully mixed Nash equilibrium. Recall that in the case of identical links, all
probabilities are identical (and equal to
1
m
) for the fully mixed Nash equilibrium
F
. Hence, Proposition 2.2 implies now that the monomial social cost of the fully
mixed Nash equilibrium
F
is a combinatorial sum of Stirling numbers of the
second kind.
Corollary 4.41.
Consider the model of identical players and identical links. Fix
an instance
hn, mi
. Then,
SCλd(n, m, F) = mX
t∈[d]1
mt
·S(d, t)·nt.
4.5 Polynomial Social Cost 69
A lower bound on the monomial optimum for the case of identical players is
OPTλd(n, m)≥mn
md
if
n≥m
, while
OPTλd(n, m) = n
if
n<m
.
We are now ready to prove that fully mixed Nash equilibria maximize poly-
nomial social cost for the case of identical players and two identical links.
Theorem 4.42.
Consider the model of identical players and two identical links.
Fix an instance
hn, 2i
with associated Nash equilibrium
P
and fully mixed Nash
equilibrium
F
. Then,
SCπd(λ)(n, 2,P)≤SCπd(λ)(n, 2,F).
Proof.
Since polynomial social cost is a linear combination (with non-negative
coecients) of monomial social costs, it suces to prove the claim for a monomial
latency cost function
πd(λ) = λd
.
We partition the set of players
[n]
into three sets:
U1={i∈[n]|supporti(P) = {1}},
U2={i∈[n]|supporti(P) = {2}},
U12 ={i∈[n]|supporti(P) = {1,2}}.
Without loss of generality, assume that
|U1| ≤ |U2|
. Denote
u=|U1|
,
v=|U2|−u
and
r=|U12|
.
We will treat separately pure and non-pure Nash equilibria. In the second
case, we will distinguish between the subcase where there are no pure players
assigned to one of the two links (that is,
u= 0
), and the subcase where there are
pure players assigned to each of the two links (that is,
u > 0
). The second subcase
will be reduced to the rst. We now continue with the details of the formal proof.
We proceed by case analysis.
1. Assume rst that
P
is pure. Since
P
is a Nash equilibrium,
Λ1(P)≤Λ2(P)+1
and
Λ2(P)≤Λ1(P)+1
. So,
|Λ1(P)−Λ2(P)| ≤ 1
. Note that
SCλd(n, 2,P) =
(Λ1(P))d+(Λ2(P))d
and
Λ1(P)+Λ2(P) = n
. Hence,
SCλd(n, 2,P)
is minimum
when
|Λ1(P)−Λ2(P)| ≤ 1
. It follows that
P
is an optimal strategy prole,
so that
SCλd(n, 2,P)≤SCλd(n, 2,F)
, as needed.
2. Assume now that
P
is not pure, so that
r > 0
. There are two separate cases.
a) Assume rst that
u= 0
. So, no pure player is assigned to link
1
. We rst
prove that, in this case,
r > 1
. Assume, by way of contradiction, that
r= 1
, and consider the single mixed player
i0
. Then,
λi01(P) = 1
, while
λi02(P) = |U2|+ 1
. Since
P
is a Nash equilibrium,
λi01(P) = λi02(P)
. It
follows that
|U2|= 0
. This implies that
n=r= 1
, a contradiction. It
follows that
r > 1
. So,
r∈[n−1]\{1}
mixed players are assigned to both
links and
n−r
pure players are assigned to link
2
.
Consider any arbitrary player
i∈ U12
. Clearly,
λi1(P) = Λ1(P)−pi1+ 1
70 4 Selsh Routing on Parallel Links
and
λi2(P) = Λ2(P)−pi2+ 1
=Λ2(P)−(1 −pi1)+1.
Since
P
is a Nash equilibrium,
λi1(P) = λi2(P)
. This implies that
pi1=Λ1(P)−Λ2(P)+1
2.
So,
pi1
is independent of
i
. It follows that each mixed player chooses link
1 with probability
p1=pi1
and link
2
with probability
p2= 1 −p1
.
Hence,
Λ1(P) = rp1
and
Λ2(P) = (n−r) + r(1 −p1)
. We obtain that
p1=rp1−((n−r)+r(1−p1))+1
2
, from which we derive that
p1=1
2+n−r
2(r−1) ,
p2=1
2−n−r
2(r−1) .
Clearly,
Λ1(P) = rp1
and
Λ2(P) = n−Λ1(P)
. Denote
α=n
2
and
β=
n−r
2(r−1)
to derive that
Λ1(P) = α+β
and
Λ2(P) = α−β
. Then, the average
probabilities on links 1 and 2 are
ep1=Λ1(P)
r=α+β
r,
and
ep2=Λ2(P)
n=α−β
n,
respectively.
On one hand, Lemma 2.3 and Proposition 2.2 imply that
SCλd(n, 2,P) = BF
hp1, . . . , p1
|{z }
r
entries
i, λd
+BF
hp2, . . . , p2
| {z }
r
entries
,1, . . . , 1
| {z }
n−r
entries
i, λd
≤BF ep1, r, λd+BF ep2, n, λd
=X
t∈[d](ep1)t·S(d, t)·rt+X
t∈[d](ep2)t·S(d, t)·nt
=X
t∈[d]
S(d, t)(α+β)t·rt
rt+ (α−β)t·nt
nt.
4.5 Polynomial Social Cost 71
On the other hand, Lemma 4.41 and Proposition 2.2 imply that
SCλd(n, 2,F) = BF hf11, . . . , fn1i, λd+BF hf12, . . . , fn2i, λd
= 2 BF 1
2, n, λd
= 2 BF α
n, n, λd
= 2 X
t∈[d]α
nt·S(d, t)·nt
=X
t∈[d]
S(d, t)·2αt·nt
nt.
So, clearly,
SCλd(n, 2,F)−SCλd(n, 2,P) = X
t∈[d]
S(d, t)·∆(t),
where for each integer
t∈[d]
,
∆(t)=2αt·nt
nt−(α+β)t·rt
rt+ (α−β)t·nt
nt.
We prove:
Lemma 4.43.
For each integer
t≥1
,
∆(t)≥0
.
Proof.
By induction on
t
. For the basis case where
t= 1
, the claim holds
since
2α−((α+β)+(α−β)) = 0
. Assume inductively that the claim
holds for
(t−1)
, where
t≥2
. For the induction step, we will prove the
claim for
t
.
Note rst that by the denition of
α
and
β
and since
r≤n
,
(α+β)·r−(t−1)
r=(n−1)r
2(r−1) ·r−(t−1)
r
=n−1
2·r−(t−1)
r−1
≤n−1
2·n−(t−1)
n−1
=n
2·n−(t−1)
n
=α·n−(t−1)
n.
72 4 Selsh Routing on Parallel Links
We now use this fact to derive that
(α+β)t·rt
rt+ (α−β)t·nt
nt
= (α+β)·r−(t−1)
r·(α+β)t−1·rt−1
rt−1
+(α−β)·n−(t−1)
n·(α−β)t−1·n(t−1)
nt−1
≤α·n−(t−1)
n·(α+β)t−1·rt−1
rt−1
+(α−β)·n−(t−1)
n·(α−β)t−1·n(t−1)
nt−1
≤α·n−(t−1)
n·(α+β)t−1·rt−1
rt−1
+α·n−(t−1)
n·(α−β)t−1·n(t−1)
nt−1
≤α·n−(t−1)
n· (α+β)t−1·rt−1
rt−1+ (α−β)t−1·n(t−1)
nt−1!
≤α·n−(t−1)
n·2αt−1·nt−1
nt−1
= 2αt·nt
nt,
where we used the induction hypothesis in the last inequality. This com-
pletes the proof of the lemma.
Lemma 4.43 implies that
SCλd(n, 2,P)≤SCλd(n, 2,F)
. The proof for the
case
u= 0
that
SCπd(λ)(n, 2,P)≤SCπd(λ)(n, 2,F)
is now complete.
b) Assume now that
u > 0
.
Consider the mixed strategy prole
Q
for the instance
hn, 2i
, which assigns
u
pure players to each link (with probability 1) and
en=n−2u
mixed
players to each link with probability
1
2
. Clearly,
Q
is a Nash equilibrium.
Note that the average probability for each link is
u·1+(n−2u)·1
2
n=1
2
, which
is precisely the probability with which each player is assigned to a link in
the fully mixed Nash equilibrium
F
. Since polynomial social cost is a sum
of binomial functions and the function
πd(λ) = λd
is convex, Lemma 2.3
implies that
SCλd(n, 2,Q)≤SCλd(n, 2,F).
In the rest, we will prove that
SCλd(n, 2,P)≤SCλd(n, 2,Q)
, and this will
complete the proof.
Denote
e
F
the (unique) fully mixed Nash equilibrium associated with the
instance
hen, 2i
. On one hand,
SCλd(n, 2,Q) = SC(λ+u)d(en, 2,e
F).
4.5 Polynomial Social Cost 73
On the other hand,
SCλd(en, 2,P) = SC(λ+u)d(en, 2,e
P),
where
e
P
is a mixed strategy prole associated with the instance
hen, 2i
that assigns
v
pure players to link 2 and
r
mixed players to both links.
Since the function
(λ+u)d
is a linear combination of monomials in
λ
with non-negative coecients, we are reduced to Case (a). Hence, it fol-
lows that
SC(λ+u)d(en, 2,e
P)≤SC(λ+u)d(en, 2,e
F)
. Hence, this implies that
SCλd(n, 2,P)≤SCλd(n, 2,Q)
, as needed. The proof for the case
u > 0
that
SCπd(λ)(n, 2,P)≤SCπd(λ)(n, 2,F)
is now complete.
Since we examined all possible cases, the proof is now complete.
We now turn to the case of
m
identical links. We prove that the polynomial
social cost of any Nash equilibrium is upper bounded by
(1 + 1
n−1)d
times the
polynomial social cost of the fully mixed Nash equilibrium.
Theorem 4.44.
Consider the model of identical players and identical links. Fix
an instance
hn, mi
with associated Nash equilibrium
P
and fully mixed Nash equi-
librium
F
. Then,
SCπd(λ)(n, m, P)≤1 + 1
n−1d
·SCπd(λ)(n, m, F).
Proof.
We rst consider the case of the monomial latency cost function
πd(λ) =
λd
. We will later reduce the general case to this case.
Denote
α=n
m
. For each link
j∈[m]
, denote
rj=|{i∈[n] : pij >0}|
.
Assume, without loss of generality, that for each link
j∈[m]
,
rj≥1
. Clearly, the
average probability on link
j
is
Λj(P)
rj
. Dene
βj=|Λj(P)−α|
. Roughly speaking,
βj
is the excess expected latency on link
j
from the fair share
α
. Partition the
set of links
[m]
into
M1={j∈[m]|0< Λj(P)≤α},
M2={j∈[m]|Λj(P)> α}.
Clearly,
Λj(P) = α−βj
for
j∈ M1
and
Λj(P) = α+βj
for
j∈ M2
. Dene now
qj= mini∈[n]{pij |pij >0}
. Clearly,
qj≤Λj(P)
rj
.
Dene
β= maxj∈M1βj
. We prove a simple fact.
Lemma 4.45.
For each link
j∈ M2
with
rj≥2
,
βj≤α−rjβ
rj−1
.
Proof.
Fix a link
j∈ M2
with
rj≥2
and a player
i0∈ {i∈[n]|pij >0}
such
that
pi0j=qj
. Consider a link
j0∈ M1
such that
βj0=β
. Since
P
is a Nash
equilibrium,
λi0j(P)≤λi0j0(P)
. Clearly,
λi0j(P) = Λj(P)−qj+1 = α+βj−qj+1
.
Also,
λi0j0(P) = Λj0(P)−pi0j0+ 1 ≤Λj0(P) + 1 = α−β+ 1
. It follows that
α+βj−qj+ 1 ≤α−β+ 1
or
βj+β≤qj≤Λj(P)
rj=α+βj
rj
. This implies that
βj≤α−rjβ
rj−1
, as needed.
74 4 Selsh Routing on Parallel Links
On one hand, Lemma 2.3 and Proposition 2.2 imply that
SCλd(n, m, P)
=X
j∈[m]
BF hp1j, . . . , pnji, λd
=X
j∈M1
BF hp1j, . . . , pnji, λd+X
j∈M2
BF hp1j, . . . , pnji, λd
≤X
j∈M1
BF α−βj
rj
, rj, λd+X
j∈M2
BF α+βj
rj
, rjλd
=X
j∈M1X
t∈[d]α−βj
rjt
·S(d, t)·(rj)t+X
j∈M2X
t∈[d]α+βj
rjt
·S(d, t)·(rj)t
=X
t∈[d]
S(d, t)·
X
j∈M1α−βj
rjt
·(rj)t+X
j∈M2α+βj
rjt
·(rj)t
.
On the other hand Lemma 4.41 and Proposition 2.2 imply that
SCλd(n, m, F) = X
j∈[m]
BF hf1j, . . . , fnji, λd
=m·BF 1
m, n, λd
=X
t∈[d]
S(d, t)·m·αt·nt
nt.
For each integer
t∈[d]
dene the function
∆(t) = n
n−1tm·αt·nt
nt
−
X
j∈M1α−βj
rjt
·(rj)t+X
j∈M2α+βj
rjt
·(rj)t
.
We prove:
Lemma 4.46.
For each integer
t≥1
,
∆(t)≥0
.
Proof.
By induction on
t
. For the basis case, let
t= 1
. Then,
∆(t) = m·α−
X
j∈M1
(α−βj) + X
j∈M2
(α+βj)
=m·α−
X
j∈M1
Λj(P) + X
j∈M2
Λj(P)
=m·α−n
= 0 ,
4.5 Polynomial Social Cost 75
as needed.
Assume inductively that the claim holds for
(t−1)
, for some integer
t≥2
.
For the induction step, we will prove the claim for
t
.
Since
(rj)t= 0
for each
j∈ M1∪M2
with
rj< t
, it follows that
X
j∈M1α−βj
rjt
(rj)t+X
j∈M2α+βj
rjt
(rj)t
=X
j∈M1:rj≥tα−βj
rjt
(rj)t+X
j∈M2:rj≥tα+βj
rjt
(rj)t
=X
j∈M1:rj≥t
(α−βj)rj−(t−1)
rjα−βj
rjt−1
(rj)t−1
+X
j∈M2:rj≥t
(α+βj)rj−(t−1)
rjα+βj
rjt−1
(rj)t−1.
Note that for each link
j∈[m]
, the fraction
rj−(t−1)
rj
is monotonically increasing
in
rj
(since
t≥2
). Since for each
j∈[m]
,
rj≤n
and
βj≥0
, it follows that
X
j∈M1α−βj
rjt
(rj)t+X
j∈M2α+βj
rjt
(rj)t
≤X
j∈M1:rj≥t
αn−(t−1)
nα−βj
rjt−1
(rj)t−1
+X
j∈M2:rj≥t
(α+βj)rj−(t−1)
rjα+βj
rjt−1
(rj)t−1.
(4.21)
We proceed by case analysis.
1. Assume that for each link
j∈ M2
with
rj≥t
,
βj≤n−rj
m(rj−1)
. The implies
that for each link
j∈ M2
with
rj≥t
,
α+βj≤(n−1)rj
m(rj−1)
. Then, with (4.21),
X
j∈M1α−βj
rjt
(rj)t+X
j∈M2α+βj
rjt
(rj)t
≤X
j∈M1:rj≥t
αn−(t−1)
nα−βj
rjt−1
(rj)t−1
+X
j∈M2:rj≥t
(n−1)rj
m(rj−1)
rj−(t−1)
rjα+βj
rjt−1
(rj)t−1
76 4 Selsh Routing on Parallel Links
=X
j∈M1:rj≥t
αn−(t−1)
nα−βj
rjt−1
(rj)t−1
+X
j∈M2:rj≥t
n−1
m
rj−(t−1)
rj−1α+βj
rjt−1
(rj)t−1
≤X
j∈M1:rj≥t
αn−(t−1)
nα−βj
rjt−1
(rj)t−1
+X
j∈M2:rj≥t
n−1
m
n−(t−1)
n−1α+βj
rjt−1
(rj)t−1
=X
j∈M1:rj≥t
αn−(t−1)
nα−βj
rjt−1
(rj)t−1
+X
j∈M2:rj≥t
αn−(t−1)
nα+βj
rjt−1
(rj)t−1
=αn−(t−1)
n·
·
X
j∈M1:rj≥tα−βj
rjt−1
(rj)t−1+X
j∈M2:rj≥tα+βj
rjt−1
(rj)t−1
=αn−(t−1)
n
X
j∈M1α−βj
rjt−1
(rj)t−1+X
j∈M2α+βj
rjt−1
(rj)t−1
≤αn−(t−1)
nn
n−1t−1m·αt−1·nt−1
nt−1
<n
n−1tmαtnt
nt,
where the induction hypothesis was used for the last inequality. This implies
that
∆(t)≥0
, as needed.
2. Assume now that there exists a link
j∈ M2
with
rj≥t
such that
βj>
n−rj
m(rj−1)
. By Lemma 4.45, for each link
j∈ M2
with
rj≥t≥2
,
(α+βj)·rj−(t−1)
rj≤α+α−rjβ
rj−1·rj−(t−1)
rj
β≥0
≤αrj
rj−1·rj−(t−1)
rj
=α·rj−(t−1)
rj−1
≤α·n−(t−1)
n−1.
4.5 Polynomial Social Cost 77
Thus, with (4.21),
X
j∈M1α−βj
rjt
(rj)t+X
j∈M2α+βj
rjt
(rj)t
≤X
j∈M1:rj≥t
αn−(t−1)
nα−βj
rjt−1
(rj)t−1
+X
j∈M2:rj≥t
αn−(t−1)
n−1α+βj
rjt−1
(rj)t−1
≤n
n−1αn−(t−1)
n·
·
X
j∈M1:rj≥tα−βj
rjt−1
(rj)t−1+X
j∈M2:rj≥tα+βj
rjt−1
(rj)t−1
≤n
n−1αn−(t−1)
nn
n−1t−1m·αt−1nt−1
nt−1
=n
n−1tmαtnt
nt,
where the induction hypothesis was used for the last inequality. This implies
that
∆(t)≥0
, as needed.
This completes the proof of the Lemma 4.46.
Lemma 4.46 implies that
SCλd(n, m, P)≤1 + 1
n−1d·SCλd(n, m, F)
. Hence,
SCπd(λ)(n, m, P) = X
0≤t≤d
at·SCλt(n, m, P)
≤X
0≤t≤d
at·1 + 1
n−1t
·SCλt(n, m, F)
≤1 + 1
n−1d
·X
0≤t≤d
at·SCλt(n, m, F)
=1 + 1
n−1d
·SCπd(λ)(n, m, F),
as needed. The proof of Theorem 4.44 is now complete.
We remark that the proof of Theorem 4.44 follows the proof of Theorem 4.42.
However, it is more complicated in dealing with an arbitrary number of links.
In the process of publishing the corresponding paper in a journal, an unknown
referee contributed a substantial improvement to Theorem 4.44, showing that the
factor
1 + 1
n−1d
is
not
necessary.
78 4 Selsh Routing on Parallel Links
Theorem 4.47 (Gairing et al. [43]).
Consider the model of identical players
and identical links. Fix an instance
hn, mi
with associated Nash equilibrium
P
and fully mixed Nash equilibrium
F
. Then,
SCπd(λ)(n, m, P)≤SCπd(λ)(n, m, F).
4.5.1.2 The Monomial and Polynomial Prices of Anarchy.
We are now ready to prove our upper bounds on the price of anarchy for monomial
and polynomial social cost.
Identical Players and Identical Links
We rst consider the model of identical players and identical links. We use The-
orem 4.47 to prove upper bounds on the price of anarchy for monomial and
polynomial social cost (Theorem 4.48 and Corollary 4.49).
Theorem 4.48.
Consider the model of identical players and identical links.
Then,
PoAλd≤Bd.
Proof.
Fix any instance
hn, mi
with an associated fully mixed Nash equilibrium
F
. By Proposition 2.2,
SCλd(n, m, F) = X
j∈[m]
BF hf1j, . . . , fnji, λd
=m·BF hf1j, . . . , fnji, λd
=m·X
t∈[d]1
mt
·S(d, t)·nt
≤m·X
t∈[d]1
mt
·S(d, t)·nt.
We now proceed by case analysis.
1. Assume rst that
n≥m
. Recall that in this case,
OPTλd(w, m)≥m·n
md
.
Hence,
SCλd(n, m, F)
OPTλd(n, m)≤1
m·m
nd·m·X
t∈[d]1
mt
·S(d, t)·nt
=X
t∈[d]m
nd−t·S(d, t)
≤X
t∈[d]
S(d, t)
=Bd.
4.5 Polynomial Social Cost 79
2. Assume now that
n<m
. Recall that, in this case,
OPTλd(n, m)≥n
. Hence,
SCλd(n, m, F)
OPTλd(n, m)≤1
nm·X
t∈[d]1
mt
·S(d, t)·nt
=X
t∈[d]n
mt−1·S(d, t)
≤X
t∈[d]
S(d, t)
=Bd.
So, in all cases,
SCλd(n,m,F)
OPTλd(n,m)≤Bd
. Theorem 4.47 implies now the claim.
Note that the upper bound on the monomial price of anarchy established in
Theorem 4.48 approaches
Bd
as
n
approaches innity. By Lemma 4.40 and since
the Bell numbers are increasing in their order, Theorem 4.48 immediately implies:
Corollary 4.49.
Consider the model of identical players and identical links.
Then,
PoAπd(λ)≤Bd.
Identical Players and Two Identical Links
We now turn to the model of identical players and
two
identical links. Again,
we use Theorem 4.47 to prove (tight) upper bounds on the price of anarchy for
monomial and polynomial social cost (Theorem 4.50 and Corollary 4.51).
Theorem 4.50.
Consider the model of identical players and two identical links.
Then,
PoAλd≤2d−2 1 + 1
nd−1!.
This bound is tight for
n= 2
.
Proof.
We start with the upper bound. Fix any instance
hn, 2i
with an associated
Nash equilibrium
P
. On the one hand, by Theorem 4.47,
80 4 Selsh Routing on Parallel Links
SCλd(n, 2,P)≤SCλd(n, 2,F)
= 2 ·BF(1
2, n, λd)
= 2 ·X
t∈[d]1
2t
·S(d, t)·nt
≤2·
1
2·S(d, 1) ·n+1
4·X
2≤i≤d
S(d, i)ni
= 2 ·1
2·S(d, 1) ·n+1
4·(nd−S(d, 1) ·n)
= 2 ·n
4+nd
4.
On the other hand,
OPTλd(n, 2) ≥2·n
2d.
It follows that
SCλd(n, 2,F)
OPTλd(n, 2) ≤2
nd
·n
4+nd
4
= 2d−2 1 + 1
nd−1!,
as needed. To prove that the upper bound is tight for
n= 2
, note that it becomes
2d−2+1
2
. We continue to prove that this is also a lower bound for
n= 2
. Fix an
instance
h2,2i
. Then,
OPTλd(2,2) = 2 ,
while
SCλd(2,2,F)=2·X
t∈[d]1
2t
·S(d, t)·2t
= 2 ·1
2·S(d, 1) ·2 + 1
4·S(d, 2) ·2·1
= 2 ·S(d, 1) + 1
2·S(d, 2)
= 2 ·1 + 1
2·(2d−1−1)
= 2 ·2d−2+1
2.
It follows that
PoAλd≥2d−2+1
2
as needed.
By Lemma 4.40, Theorem 4.50 immediately implies:
4.6 Conclusion and Discussion 81
Corollary 4.51.
Consider the model of identical players and two identical links.
Then,
PoAπd(λ)≤2d−2 1 + 1
nd−1!.
This bound is tight for
n= 2
.
Proof.
We will show that for all integer
t∈[2, d]
the upper bound on
PoAλt
is
larger than the upper bound on
PoAλt−1
. Clearly,
2t−2 1 + 1
nt−1!−2t−3 1 + 1
nt−2!
= 2t−3 2 + 2
n1
nt−2!−2t−3 1 + 1
nt−2!
= 2t−3 1−1−2
n1
nt−2!
≥2t−31−1−2
n
=2t−2
n
>0,
as needed. Tightness, for
n= 2
, follows from the tightness of Theorem 4.50
4.6 Conclusion and Discussion
In this chapter, we have studied routing games on parallel links. For this setting,
we have provided many results concerning the computational complexity of pure
Nash equilibria. Moreover, we proved an extensive collection of results related to
the price of anarchy in various sub-models. Although, routing games on parallel
links have received a lot of attention, many problems are still tantalizing open.
We only state some of them.
•
For the model of identical links, our nashication algorithm
NashifyIdenti-
cal
is based only on selsh steps. Is it also possible to provide a polynomial-
time nashication algorithm, for the model of
related
links, that solely depends
on selsh steps?
•
We have described a polynomial-time algorithm to compute a pure Nash equi-
librium for the model of restricted strategy sets and identical links. Is it possi-
ble to provide such an algorithm for the model of restricted strategy sets and
related
links?
•
Our bounds on the price of anarchy for polynomial social cost are all for the
model of identical players. Proving such bounds for
arbitrary
players remains
a challenging open problem.
5
Weighted Congestion Games
5.1 Introduction
In this chapter, we present strong results on the price of anarchy for
congestion
games
and
weighted congestion games
. Such games have been formally introduced
in Section 3.2. In a congestion game, there is a set of
resources
and the strat-
egy set of each player is a subset of the power set of the resources. Thus, a pure
strategy might consist of multiple resources. This stands in contrast to the games
studied in Chapter 4where each pure strategy consists of a single resource (link).
For each resource, there is a
latency function
which describes the latency of this
resource. In this chapter, we allow for polynomial latency functions with max-
imum degree
d
and non-negative coecients. Each player aims to minimize its
private cost
which is dened as the (expected) sum of the latencies of its chosen
resources. For (unweighted) congestion games the latency of a resource only de-
pends on the number of players sharing this resource. In a
weighted
congestion
game, players have weights and thus dierent inuence on the congestion of the
resources. Weighted congestion games provide us with a general framework for
modeling any kind of non-cooperative resource sharing problem. A typical re-
source sharing problem is that of routing. In a routing game the strategy sets of
the players correspond to paths in a network. Routing games where the demand
of the players cannot be split among multiple paths are also called
(weighted)
network congestion games
.
5.1.1 Summary of Results
In this chapter, we prove
exact
bounds on the price of anarchy for unweighted and
weighted congestion games with polynomial latency functions. We use the total
latency as social cost measure. This improves on results by Awerbuch et al. [5]
and Christodoulou and Koutsoupias [17], where non-matching upper and lower
bounds are given.
We now describe our ndings in more detail.
•
For
unweighted congestion games
we show that the price of anarchy (
PoATL
)
is exactly
84 5 Weighted Congestion Games
PoATL =(k+ 1)2d+1 −kd+1(k+ 2)d
(k+ 1)d+1 −(k+ 2)d+ (k+ 1)d−kd+1 ,
where
k=bΦdc
and
Φd
is a natural generalization of the golden ratio to
larger dimensions such that
Φd
is the solution to
(Φd+ 1)d=Φd+1
d
. Prior
to this paper the best known upper and lower bounds were shown to be of
the form
dd(1−o(1))
[17]. However, the term
o(1)
still hides a gap between the
upper and the lower bound.
•
For
weighted congestion games
we show that the price of anarchy (
PoATL
) is
exactly
PoATL =Φd+1
d.
This result closes the gap between the so far best upper and lower bounds of
O(2ddd+1)
and
Ω(dd/2)
from [5].
We show that the above values on the price of anarchy also hold for the subclasses
of unweighted and weighted
network
congestion games.
For our upper bounds we use a similar analysis as in [17]. The core of our
analysis is to determine parameters
c1
and
c2
such that
y·f(x+ 1) ≤c1·x·f(x) + c2·y·f(y)
(5.1)
for all polynomial latency functions
f
of maximum degree
d
and for all reals
x, y ≥
0
. For the case of unweighted demands it suces to show (5.1) for all integers
x, y
.
In order to prove their upper bound Christodoulou and Koutsoupias [17] looked
at (5.1) with
c1=1
2
and gave an asymptotic estimate for
c2
. In our analysis
we optimize both parameters
c1, c2
. This optimization process requires new ideas
and is non-trivial.
unweighted
PoATL
weighted
PoATL
d Φd
Our exact result Upper Bound [17] Lower bound [17] Our exact result Lower bound [5]
1 1.618
2.5
2.5 2.5
2.618
2.618
2 2.148
9.583
10 (2.5)
9.909
(2.618)
3 2.630
41.54
47 (2.5)
47.82
5
4 3.080
267.6
269 21.33
277.0
15
5 3.506
1,514
2,154 42.67
1,858
52
6 3.915
12,345
15,187 85.33
14,099
203
7 4.309
98,734
169,247 170.7
118,926
877
8 4.692
802,603
1,451,906 14,762
1,101,126
4,140
9 5.064
10,540,286
20,241,038 44,287
11,079,429
21,147
10 5.427
88,562,706
202,153,442 132,860
120,180,803
115,975
Table 5.1.
Comparison of our results to [17] and [5]
Table 5.1 shows a numerical comparison of our bounds with the previous
results of Awerbuch et al. [5] and Christodoulou and Koutsoupias [17].
For
d≥2
, the table only gives the respective lower bounds that are given in
the cited works (before any estimates are applied). Values in parentheses denote
cases in which the bound for linear functions is better than the general case.
5.1 Introduction 85
In [5, Theorem 4.3], a construction scheme for networks is described with price
of anarchy approximating
1
eP∞
k=1 kd
k!
which yields the
d
-th Bell number. In [17,
Theorem 10], a network with price of anarchy
(N−1)d+2
N
is given, with
N
being
the largest integer for which
(N−1)d+2 ≤Nd
holds.
The column with the upper bound from [17] is computed by using (5.1) with
c1=1
2
and optimizing
c2
with help of our analysis. Thus, the column shows the
best possible bounds that can be shown with
c1=1
2
.
5.1.2 Related Work
The papers most closely related to our work are those of Awerbuch et al. [5]
and Christodoulou and Koutsoupias [17,16]. For (unweighted) congestion games
and social cost dened as average private cost (which in this case is the same as
total latency) it has been shown that the price of anarchy of pure Nash equilibria
is
5
2
for linear latency functions and
dΘ(d)
for polynomial latency functions of
maximum degree
d
[5,17]. The bound of
5
2
for linear latency function also holds
for the (mixed) price of anarchy [16]. For
weighted
congestion games and social
cost dened as the total latency, the (mixed) price of anarchy is
3+√5
2
in case of
linear latency functions and
dΘ(d)
in case of polynomial latency functions [5].
Since the routing games on parallel links are a special class of weighted conges-
tion games, all results concerning the price of anarchy that are either described
in Section 4.1.2 or presented throughout Chapter 4belong to the related work
here. Of particular interest is the paper of Lücking et al. [71], where they studied
the total latency (they call it quadratic social cost) for routing games on paral-
lel links. For this model, Lücking et al. [71] showed that the price of anarchy is
exactly
4
3
for the case of identical players and related links and
9
8
for the case of
arbitrary players and identical links.
The class of
congestion games
has been introduced by Rosenthal [88] and
extensively studied afterwards (see e.g. [1,2,27,38,39,77,78,96]). In Rosen-
thal's model the strategy of each player is a subset of resources. Resource utility
functions can be arbitrary but they only depend on the number of players shar-
ing the same resource. Rosenthal showed that such games always admit a pure
Nash equilibrium using a potential function. Monderer and Shapley [78] and later
Voorneveld et al. [96] characterized games that possess a potential function as
potential games and showed their relation to congestion games. Milchtaich [77]
considers weighted congestion games with player-specic payo functions and
shows that these games do not admit a pure Nash equilibrium in general. Acker-
mann et al. [2] further studied the existence of pure Nash equilibria in weighted
congestion games and (unweighted) congestion games with player-specic payo
functions. Fotakis et al. [38,39] considered the price of anarchy for symmetric
weighted network congestion games in layered networks [38] and for symmetric
(unweighted) network congestion games in general networks [39]. In both cases
they dened social cost as expected maximum latency. The complexity of com-
puting a pure Nash equilibrium has been studied by Fabrikant et al. [27]. On
86 5 Weighted Congestion Games
the one hand, they proved that for any symmetric network congestion game, a
pure Nash equilibrium can be constructed in polynomial time, by computing the
optimum of Rosenthal's potential function. This is accomplished through a nice
reduction to an instance of the min-cost ow problem. On the other hand, Fab-
rikant et al. [27] showed that the problem of computing a pure Nash equilibrium
becomes PLS-complete, if we allow for asymmetric or non-network congestion
games. Ackermann et al. [1] further studied the complexity of computing a pure
Nash equilibrium in congestion games. In particular, they showed that a pure
Nash equilibrium can be computed in polynomial time, if the strategy sets of the
players have a certain property. For a survey on weighted congestion games we
refer to [48].
Inspired by the arisen interest in the price of anarchy Roughgarden and Tardos
[92] re-investigated the Wardrop model and used the
total latency
as a social cost
measure. In this context the price of anarchy was shown to be
4
3
for linear latency
functions [92] and
Θ(d
log d)
[89] for polynomial latency functions of maximum
degree
d
. An overview on results for this model can be found in the recent book
of Roughgarden [90].
5.1.3 Organization
The rest of this chapter is organized as follows. We present our exact bounds
on the price of anarchy for unweighted congestion games in Section 5.2 and for
weighted congestion games in Section 5.3. We conclude in Section 5.4 with a
discussion on our results.
5.2 Price of Anarchy for Unweighted Congestion Games
In this section, we prove the exact value for the price of anarchy of
unweighted
congestion games with polynomial latency functions. We start by showing the
upper bound in Section 5.2.1. In Section 5.2.2 we provide a matching lower bound.
5.2.1 Upper Bound
Before we can state our upper bound on the price of anarchy for unweighted con-
gestion games, we introduce two technical lemmas (Lemma 5.1 and Lemma 5.2).
These lemmas are crucial for determining
c1
and
c2
in (5.1) and thus for proving
the upper bound in Theorem 5.3.
Lemma 5.1.
Let
0≤c < 1
and
d∈N0
then
max
x∈N0,y∈N(x+ 1
yd
−c·x
yd+1)= max
x∈N0n(x+ 1)d−c·xd+1o.
5.2 Price of Anarchy for Unweighted Congestion Games 87
Proof.
Let
g(x, y, c) = x+ 1
yd
−c·x
yd+1
.
We will show that for all
x∈N0
,
y∈N
there exists
ˆx∈N0
such that
g(ˆx, 1, c)≥g(x, y, c)∀0≤c < 1.
Let
x∈N0
,
y∈N
be arbitrary non-negative integers. If
y≥x+ 1
then we can
choose
ˆx= 0
to see that
g(0,1, c) = 1 ≥g(x, y, c)
. So in the following we may
assume that
y≤x
.
Let
ˆx
be the smallest integer such that
g(ˆx, 1,0) ≥g(x, y, 0)
, that is
ˆx=x+ 1 −y
y.
To complete the proof, we will show that also
g(ˆx, 1,1) ≥g(x, y, 1)
, or equiva-
lently
(ˆx+ 1)d−ˆxd+1 ≥x+ 1
yd
−x
yd+1
⇔x+ 1
yd
−x+ 1 −y
yd+1
≥x+ 1
yd
−x
yd+1
⇐x
yd+1
≥x+ 1 −y
yd+1
⇔x
y≥x+ 1 −y
y
To see that the last inequality holds, recall that
x≥y
. Thus we can express
x
as
x=b1·y+b2
where
b1
and
b2
are integers with
b1≥1
and
0≤b2< y
. Then
the last inequality reduces to
b1+b2
y≥b1
, which is fullled. This completes the
proof of the lemma.
Lemma 5.2.
Let
d∈N
and
Fd={g(d)
r:R→R|g(d)
r(x) = (r+ 1)d−x·rd+1, r ∈R≥0}
be an innite set of linear functions. Furthermore, let
γ(s, t)
for
s, t ∈R≥0
and
s6=t
denote the intersection abscissa of
g(d)
s
and
g(d)
t
. Then it holds for any
s, t, u ∈R≥0
with
s<t<u
that
γ(s, t)> γ(s, u)
and
γ(u, s)> γ(u, t)
.
Proof.
We rst show that
γ(v, v +δ)
is strictly decreasing in
v∈R≥0
, for any
δ∈R>0
. Afterwards we will show that this implies the lemma.
For some
v∈R≥0
, consider now the two linear functions
g(d)
v
,
g(d)
v+δ
from
Fd
.
They intersect at
γ(v, v +δ) = (v+1+δ)d−(v+ 1)d
(v+δ)d+1 −vd+1 .
88 5 Weighted Congestion Games
Computing the rst derivative in
v
of
γ(v, v +δ)
yields
∂γ(v, v +δ)
∂v =d(v+1+δ)d−1−(v+ 1)d−1·(v+δ)d+1 −vd+1
((v+ 1)d+1 −vd+1)2
−(d+ 1) (v+1+δ)d−(v+ 1)d·(v+δ)d−vd
((v+ 1)d+1 −vd+1)2
<d(v+1+δ)d−1−(v+ 1)d−1·(v+δ)d+1 −vd+1
((v+ 1)d+1 −vd+1)2
−d(v+1+δ)d−(v+ 1)d·(v+δ)d−vd
((v+ 1)d+1 −vd+1)2
=d·"(1 + δ)(v+1+δ)d−1vd+ (1 −δ)(v+ 1)d−1(v+δ)d
((v+ 1)d+1 −vd+1)2
−(v+1+δ)d−1(v+δ)d+ (v+ 1)d−1vd
((v+ 1)d+1 −vd+1)2#
We now show (by induction over
d
) that
(1 + δ)(v+1+δ)d−1vd+ (1 −δ)(v+ 1)d−1(v+δ)d
−(v+1+δ)d−1(v+δ)d−(v+ 1)d−1vd<0
(5.2)
and thus
γ(v, v +δ)
is strictly decreasing in
v
:
Clearly, (5.2) holds for
d= 1
as
(1 + δ)v+ (1 −δ)(v+δ)−(v+δ)−v=−δ2<0.
It also holds for
v= 0
as
(1 −δ)δd−(1 + δ)d−1δd<0.
Thus, we only consider
v > 0
in the following. Assume that our induction hy-
pothesis (5.2) holds for a natural
d
. We then multiply with
(v+1+δ)v
and
get:
(1 + δ)(v+1+δ)dvd+1 + (1 −δ)(v+ 1)d−1(v+δ)d(v+1+δ)v
−(v+1+δ)d(v+δ)d
| {z }
=A
v−(v+ 1)d−1vd+1
| {z }
=B
(v+1+δ)<0
⇔(1 + δ)(v+1+δ)dvd+1 + (1 −δ)(v+ 1)d−1(v+δ)d(v+1+δ)v
| {z }
=C
−δB +δA −(v+1+δ)d(v+δ)d+1 −(v+ 1)dvd+1 <0.
Thus, if we dene
D= (1 −δ)(v+ 1)d(v+δ)d+1
, proving the inductive step
d→d+ 1
reduces to showing that
C−δB +δA ≥D
5.2 Price of Anarchy for Unweighted Congestion Games 89
or equivalently
C−δB +δA −D≥0.
Now,
C−δB +δA −D
= (1 −δ)(v+ 1)d−1(v+δ)d(v+1+δ)v−δB +δA −D
= (1 −δ)(v+ 1)d−1(v+δ)d(v+ 1)v−δB +δA
+δ(1 −δ)(v+ 1)d−1(v+δ)dv−D
= (1 −δ)(v+ 1)d(v+δ)d(v+δ)−δB +δA
+δ(1 −δ)(v+ 1)d−1(v+δ)dv−δ(1 −δ)(v+ 1)d(v+δ)d−D
=D−δB +δA −δ(1 −δ)(v+ 1)d−1(v+δ)d−D
=δ−B+A−(1 −δ)(v+ 1)d−1(v+δ)d
> δ −B+A−(v+ 1)d−1(v+δ)d
> δ −(v+ 1)d−1(v+δ)dv+A−(v+ 1)d−1(v+δ)d
=δ−(v+ 1)d(v+δ)d+ (v+1+δ)d(v+δ)d
>0.
This last inequality obviously holds for any
δ > 0
, thus
γ(v, v +δ)
is strictly
decreasing in
v∈R≥0
, for any
δ∈R>0
.
It follows that
γ(v+k·δ, v + (k+ 1) ·δ)
strictly decreases as
k∈Z
,
k≥ −v
δ
,
becomes larger. We separately consider functions
g(d)
v+k·δ
for integers
k > 0
and
k < 0
:
•
We rst turn to the case where
k > 0
. Then the intersection of
g(d)
v+k·δ
and
g(d)
v+(k+1)·δ
must lie above
g(d)
v
. This can easily be seen due to the fact that
g(d)
v+k·δ(x)> g(d)
v(x)
holds for any
x < γ(v, v +δ)
and any
k∈N
. Now recall
that the slope of
g(d)
v+k·δ
is
−(v+k·δ)d+1
, which is decreasing as
k
becomes
larger. Since the aforementioned intersection lies above
g(d)
v
, we have that
γ(v, v + (k+ 1) ·δ)< γ(v, v +k·δ)
.
•
Now let
−v
δ≤k < 0
. For clarity set
j=−k
and consider the intersection of
g(d)
v−j·δ
and
g(d)
v−(j+1)·δ
which still lies above
g(d)
v
. Obviously,
g(d)
v−j·δ(x)> g(d)
v(x)
holds for any
x > γ(v, v +δ)
and any
j∈N
. With corresponding arguments
as in the rst case we get that
γ(v, v −j·δ)< γ(v, v −(j+ 1) ·δ)
.
Hence,
γ(v, v +k·δ)
is increasing in any
k∈Z\{0}
,
k≥ −v
δ
, and the lemma
holds for any
s<t<u
where
(t−s)
and
(u−s)
are rational by choosing
δ
as the reciprocal of a common denominator of
(t−s)
and
(u−s)
. Note that this
is also a denominator of
(u−t)
. Finally, as
x7→ γ(v, v +x)
is continuous in any
x∈[−v, ∞)\{0}
, the lemma follows.
90 5 Weighted Congestion Games
We are now ready to prove our upper bound on the price of anarchy for un-
weighted congestion games with polynomial latency functions.
Theorem 5.3.
For unweighted congestion games with polynomial latency func-
tions of maximum degree
d
and non-negative coecients, we have
PoATL ≤(k+ 1)2d+1 −kd+1(k+ 2)d
(k+ 1)d+1 −(k+ 2)d+ (k+ 1)d−kd+1
where
k=bΦdc
.
Proof.
Let
P= (P1, ..., Pn)
be a (mixed) Nash equilibrium and let
Q=
(Q1, ..., Qn)
be a pure strategy prole with optimum social cost. Since
P
is a
Nash equilibrium, player
i∈[n]
cannot improve by switching from strategy
Pi
to strategy
Qi
. Thus,
PCi(P) = X
s∈S
p(s)X
e∈si
fe(δe(s))
≤PCi(P−i, Qi)
=X
s0∈S−i
p(s0)
X
e∈Qi∩si
fe(δe(s)) + X
e∈Qi\si
fe(δe(s) + 1)
≤X
s0∈S−i
p(s0)X
e∈Qi
fe(δe(s) + 1)
=X
si∈Si
p(i, si)X
s0∈S−i
p(s0)X
e∈Qi
fe(δe(s) + 1)
=X
s∈S
p(s)X
e∈Qi
fe(δe(s) + 1).
Summing up over all players
i∈[n]
yields
SCTL(Γ, P) = X
i∈[n]X
s∈S
p(s)X
e∈si
fe(δe(s))
≤X
i∈[n]X
s∈S
p(s)X
e∈Qi
fe(δe(s) + 1)
=X
s∈S
p(s)X
e∈E
δe(Q)·fe(δe(s) + 1).
Now,
δe(Q)
and
δe(s)
are both integer, since
Q
and
s
are both pure strategy
proles. Thus, by choosing
c1, c2
such that
y·f(x+ 1) ≤c1·x·f(x) + c2·y·f(y)
(5.3)
for all polynomials
f
with maximum degree
d
and non-negative coecients and
for all
x, y ∈N0
, we get
5.2 Price of Anarchy for Unweighted Congestion Games 91
SCTL(Γ, P)≤X
s∈S
p(s)X
e∈E
[c1·δe(s)·fe(δe(s))+c2·δe(Q)·fe(δe(Q))]
=c1·SCTL(Γ, P) + c2·SCTL(Γ, Q).
With
c1<1
it follows that
SCTL(Γ,P)
SCTL(Γ,Q)≤c2
1−c1.
Since
P
is an arbitrary (mixed)
Nash equilibrium we get
PoATL ≤c2
1−c1
.
(5.4)
In fact,
c1
and
c2
depend on the maximum degree
d
, however, for the sake of
readability we omit this dependence in our notation.
We will now show how to determine constants
c1
and
c2
such that Inequality
(5.3) holds and such that the resulting upper bound of
c2
1−c1
is minimal. To do
so, we will rst show, that it suces to consider Inequality (5.3) with
y= 1
and
f(x) = xd
.
Since
f
is a polynomial of maximum degree
d
with non-negative coecients,
it is sucient to determine
c1
and
c2
that fulll (5.3) for
f(x) = xr
for all integers
0≤r≤d
.
So let
f(x) = xr
for some
0≤r≤d
. In this case (5.3) reduces to
y·(x+ 1)r≤c1·xr+1 +c2·yr+1.
(5.5)
For any given constant
0≤c1<1
let
c2(r, c1)
be the minimum value for
c2
such
that (5.5) holds, that is
c2(r, c1) = max
x∈N0,y∈Ny(x+ 1)r−c1·xr+1
yr+1
= max
x∈N0,y∈N(x+ 1
yr
−c1·x
yr+1).
Note that (5.5) holds for any
c2
when
y= 0
. By Lemma 5.1 we have
c2(r, c1) = max
x∈N0(x+ 1)r−c1·xr+1.
(5.6)
Now,
c2(r, c1)
is the maximum of innitely many linear functions in
c1
; one for
each
x∈N0
. Denote
Fr
as the (innite) set of linear functions dening
c2(r, c1)
.
Thus,
Fr={g(r)
x: (0,1) →R|g(r)
x(c1) = (x+ 1)r−c1·xr+1, x ∈N0}.
For the partial derivative of any function
(x, r, c1)7→ g(r)
x(c1)
we get
∂((x+ 1)r−c1·xr+1)
∂r = (x+ 1)r·ln(x+ 1) −c1·xr+1 ·ln(x)
>ln(x+ 1) (x+ 1)r−c1·xr+1
≥0,
92 5 Weighted Congestion Games
for
(x+ 1)r−c1·xr+1 ≥0
, that is, for the positive range of the chosen function
from
Fr
. Thus, the positive range of
(x+ 1)d−c1·xd+1
dominates the positive
range of
(x+ 1)r−c1·xr+1
for all
0≤r≤d
. Since
c2(r, c1)>0
for all
0≤r≤d
,
it follows that
c2(d, c1)≥c2(r, c1)
, for all
0≤r≤d
. Thus, without loss of
generality, we may assume that
f(x) = xd
.
For
s, t ∈R≥0
and
s6=t
dene
γ(s, t)
as the intersection abscissa of
g(d)
s
and
g(d)
t
(as in Lemma 5.2). Now consider the intersection of the two functions
g(d)
v
and
g(d)
v+1
from
Fd
for some
v∈N
. We show that this intersection lies above all
other functions from
Fd
.
•
First consider any function
g(d)
z
with
z > v + 1
. We have
g(d)
z(0) > g(d)
v+1(0) >
g(d)
v(0)
. Furthermore, by Lemma 5.2 we get
γ(v, z)< γ(v, v + 1)
. It follows
that
g(d)
v(γ(v, v + 1)) > g(d)
z(γ(v, v + 1))
.
•
Now consider any function
g(d)
z
with
z < v
. We have
g(d)
v+1(0) > g(d)
v(0) >
g(d)
z(0)
. Furthermore, by Lemma 5.2 we get
γ(v, z)> γ(v, v + 1)
. Again, it
follows that
g(d)
v(γ(v, v + 1)) > g(d)
z(γ(v, v + 1))
.
Thus, all intersections of two consecutive linear functions from
Fd
lie on
c2(d, c1)
.
The structure of function
c2(d, c1)
is illustrated in Figure 5.1.
c (d)
1
c (d)
2
21
c (d, c )
0 0 1
x=0
x=1
x=2
x=3
x=4 x=5
Fig. 5.1.
The function
c2(d, c1)
By (5.4), any point that lies on
c2(d, c1)
gives an upper bound on
PoATL
.
Let
k
be the largest integer such that
(k+ 1)d≥kd+1
, that is
k=bΦdc
. Then
(k+ 2)d<(k+ 1)d+1
. Choose
c1
and
c2
at the intersection of the two lines
from
Fd
with
x=k
and
x=k+ 1
, that is
c2= (k+ 1)d−c1·kd+1
and
c2= (k+ 2)d−c1·(k+ 1)d+1
. Thus,
c1=(k+ 2)d−(k+ 1)d
(k+ 1)d+1 −kd+1
and
c2=(k+ 1)2d+1 −(k+ 2)d·kd+1
(k+ 1)d+1 −kd+1 .
5.2 Price of Anarchy for Unweighted Congestion Games 93
Note that by the choice of
k
we have
0< c1<1
.
It follows that
PoATL ≤(k+ 1)2d+1 −kd+1(k+ 2)d
(k+ 1)d+1 −(k+ 2)d+ (k+ 1)d−kd+1 .
This completes the proof of the theorem.
5.2.2 Lower Bound
In Theorem 5.4 we give a matching lower bound which also holds for unweighted
network congestion games (Corollary 5.5).
Theorem 5.4.
For unweighted congestion games with polynomial latency func-
tions of maximum degree
d
and non-negative coecients, we have
PoATL ≥(k+ 1)2d+1 −kd+1(k+ 2)d
(k+ 1)d+1 −(k+ 2)d+ (k+ 1)d−kd+1 ,
where
k=bΦdc
.
Proof.
Given the maximum degree
d∈N
for the polynomial latency functions,
we construct a congestion game for
n≥k+ 2
players and
|E|= 2n
facilities.
We divide the set
E
into two subsets
E1={g1, . . . , gn}
and
E2={h1, . . . , hn}
.
Each player
i
has two pure strategies,
Pi={gi+1, . . . , gi+k, hi+1, . . . , hi+k+1}
and
Qi={gi, hi}
where
gj=gj−n
and
hj=hj−n
for
j > n
. I.e.
Si={Qi, Pi}.
Each of the facilities in
E1
share the latency function
x7→ axd
for an
a∈R>0
(yet to be determined) whereas the facilities in
E2
have latency
x7→ xd
.
Obviously, the optimal allocation
Q
is for every player
i
to choose
Qi
. Now we
determine a value for
a
such that the allocation
P= (P1, . . . , Pn)
becomes a Nash
equilibrium, i.e. each player
i
is satised with
P
, that is
PCi(P)≤PCi(P−i, Qi)
for all
i∈[n]
, or equivalently
k·a·kd+ (k+ 1) ·(k+ 1)d≤a·(k+ 1)d+ (k+ 2)d.
Resolving to the coecient
a
gives
a≥(k+ 1)d+1 −(k+ 2)d
(k+ 1)d−kd+1 >0.
(5.7)
Because
(k+ 1)d6=kd+1
due to either
k+ 1
or
k
being odd and the other being
even,
a
is well dened and positive. Now since for any player
i
the private costs
are
PCi(Q) = a+ 1
and
PCi(P) = a·kd+1 + (k+ 1)d+1
, it follows that
SCTL(Γ, P)
SCTL(Γ, Q)=Pi∈[n]PCi(P)
Pi∈[n]PCi(Q)=a·kd+1 + (k+ 1)d+1
a+ 1 .
(5.8)
94 5 Weighted Congestion Games
Provided that
(k+ 1)d≥kd+1
, it is not hard to see that (5.8) is monotonically
decreasing in
a
. Thus, we assume equality in (5.7), which then gives
PoATL ≥SCTL(Γ, P)
SCTL(Γ, Q)=(k+ 1)2d+1 −kd+1(k+ 2)d
(k+ 1)d+1 −(k+ 2)d+ (k+ 1)d−kd+1 .
This completes the proof of the theorem.
We close this section by showing that the just shown lower bound also holds for
unweighted
network
congestion games.
Corollary 5.5.
The lower bound in Theorem 5.4 on
PoATL
also holds for un-
weighted network congestion games.
Proof.
Instances of the congestion game in Theorem 5.4 can be characterized by
two parameters: the maximum degree
d
of the latency functions and the number
of players
n≥ bΦdc+ 2
. The number of edges is then given by
2n
.
2
1
4
g2
h3
g3
h1
g1
h2
2
43
3
g4
1
h4
v1v2
v4
v3
fe(x)
v1v2
fe(x)
(a) (b)
Fig. 5.2.
Network congestion game for
d= 2
and 4 players
Figure 5.2 (a) shows an example of the network congestion game for quadratic
latency functions (i.e.
d= 2
) and for
n= 4
players. Unlabeled edges have
fe(x) =
0
as their latency function. We say that these edges are
free
. All other edges have
the associated latency function as in Theorem 5.4. In the following we outline
the general construction scheme.
The network corresponding to an instance characterized by
(d, n)
can be con-
structed as follows:
There is a circle of
2n
undirected edges
g1, h1, g2, h2, . . . , gn, hn
. Each undi-
rected edge
(v1, v2)
has to be replaced by the construction shown in Figure 5.2 (b).
This insures that no matter in which direction a player uses edge
(v1, v2)
it pro-
duces load on the directed edge
(v3, v4)
.
Now, every player
i
has its own origin node outside the circle which is
indicated by a gray background in the example. This node has an edge to the
connecting node of
gi
and
hi−1
. The destination node of each player
i
is the
5.3 Price of Anarchy for Weighted Congestion Games 95
node between
hi
and
gi+1
, represented by a thick outline in the gure. To also
allow for the strategy
Pi
for each player
i
(which essentially goes the other way
round inside the circle), we nally add one more edge from
i
's origin node to the
connecting node of
hi+k+1
and
gi+k+1
. As before, let
k=bΦdc
.
5.3 Price of Anarchy for Weighted Congestion Games
In this section, we prove the exact value for the price of anarchy of weighted
congestion games with polynomial latency functions.
We prove the upper bound in Theorem 5.6. In Theorem 5.7 we give a match-
ing lower bound which also holds for weighted network congestion games (Corol-
lary 5.9). Corollary 5.10 shows the impact of player weights to the price of anar-
chy.
5.3.1 Upper Bound
Theorem 5.6.
For weighted congestion games with polynomial latency functions
of maximum degree
d
and non-negative coecients we have
PoATL ≤Φd+1
d
.
Proof.
Let
P= (P1, . . . , Pn)
be a (mixed) Nash equilibrium and let
Q=
(Q1, . . . , Qn)
be a pure strategy prole with optimum total latency. We rst
note that due to the Nash inequalities it holds that
PCi(P) = X
s∈S
p(s)X
e∈si
fe(δe(s))
≤X
s∈S
p(s)X
e∈Qi
fe(δe(s) + wi)
≤X
s∈S
p(s)X
e∈Qi
fe(δe(s) + δe(Q)).
This gives the following upper bound for the total latency:
SCTL(Γ, P) =
n
X
i=1
wi·PCi(P)
≤
n
X
i=1
wiX
s∈S
p(s)X
e∈Qi
fe(δe(s) + δe(Q))
=X
s∈S
p(s)X
e∈E
δe(Q)·fe(δe(s) + δe(Q)).
Similarly to Theorem 5.3, by choosing
c1, c2∈R
such that
y·f(x+y)≤c1·x·f(x) + c2·y·f(y)
(5.9)
for all polynomials
f
with maximum degree
d
and non-negative coecients and
for all
x, y ∈R≥0
, we get
96 5 Weighted Congestion Games
SCTL(Γ, P)≤X
s∈S
p(s)X
e∈Ehc1·δe(s)·fe(δe(s)) + c2·δe(Q)·fe(δe(Q))i
=c1·SCTL(Γ, P) + c2·SCTL(Γ, Q).
Note here that (5.9) varies from (5.3) of Theorem 5.3 and hence the former values
for
c1
and
c2
in Theorem 5.3 cannot simply be reused.
For (5.9) to hold for all
x, y ∈R≥0
, both
c1
and
c2
must be non-negative.
With
0≤c1<1
it follows that
SCTL(Γ,P)
SCTL(Γ,Q)≤c2
1−c1
. Since
P
is an arbitrary Nash
equilibrium we get
PoATL ≤c2
1−c1
.
Now let
r∈[d]0
. Because of the equivalences
∀x, y ∈R≥0:y·(x+y)r≤c1·xr+1 +c2·yr+1
⇔∀x∈R≥0, y ∈R>0:x
y+ 1r
≤c1·x
yr+1
+c2
⇔∀x∈R≥0: (x+ 1)r≤c1·xr+1 +c2
it is sucient to choose
c2
depending on
c1∈(0,1)
and
r∈[d]0
as
c2≥c2(r, c1) = max
x∈R≥0{(x+ 1)r−c1·xr+1},
in order to fulll (5.9) for every monomial
f
of degree
r
. With the same argument
as for (5.6) in Theorem 5.3 it is sucient to consider only the monomial of the
largest degree
d
, so that (5.9) will then hold for any polynomial of maximum
degree
d
with positive coecients.
Let again
Fd
denote the innite set of linear functions dening
c2(d, c1)
, i.e.
Fd={g(d)
x: (0,1) →R|g(d)
x(c1) = (x+ 1)d−c1·xd+1, x ∈R≥0}.
Keep
d
xed and like in Lemma 5.2 dene
γ(s, t)
for
s, t ∈R≥0
and
s6=t
as the intersection abscissa of
g(d)
s
and
g(d)
t
. From Lemma 5.2, we know that
x7→ γ(v, x)
is both continuous and strictly decreasing on
(0, v)
and then again
on
(v, ∞)
, for any
v∈R>0
. We are interested in the limit of
x7→ γ(v, x)
at
x=v
and get
lim
ε→0γ(v, v +ε) = lim
ε→0
(v+1+ε)d−(v+ 1)d
(v+ε)d+1 −vd+1 = lim
ε→0Pd−1
i=0 d
i·(v+ 1)i·εd−i
Pd
i=0 d+1
i·vi·εd+1−i
which yields (by canceling one
ε
)
lim
ε→0γ(v, v +ε) = d·(v+ 1)d−1
(d+ 1) ·vd.
(5.10)
Note here that this limit exists regardless of the direction in which
ε
approaches
0. Therefore, we make the natural extension of dening
γ(v, v)
to be the limit
from (5.10). We observe that this extension makes
x7→ γ(v, x)
strictly decreasing
and continuous in all of its domain
R>0
.
5.3 Price of Anarchy for Weighted Congestion Games 97
Now consider the intersection of the two functions
g(d)
v
and
g(d)
v+ε
from
Fd
for
ε→0
and for some
v∈R>0
. For
ε→0
these functions intersect at
(γ(v, v), g(d)
v(γ(v, v)))
. We show that this intersection lies above all other func-
tions from
Fd
.
•
First consider any function
g(d)
z
with
z > v
. We have
g(d)
z(0) > g(d)
v+ε(0) >
g(d)
v(0)
for
0< ε < z −v
. Furthermore, by Lemma 5.2 we get
γ(v, z)<
γ(v, v +ε)< γ(v, v)
. It follows that
g(d)
v(γ(v, v)) > g(d)
z(γ(v, v))
.
•
Now consider any function
g(d)
z
with
z < v
. We have
g(d)
v+ε(0) > g(d)
v(0) >
g(d)
z(0)
. Furthermore, by Lemma 5.2 we get
γ(v, z)> γ(v, v)> γ(v, v +ε)
.
Again, it follows that
g(d)
v(γ(v, v)) > g(d)
z(γ(v, v))
.
Thus,
g(d)
v(γ(v, v)) = maxx∈R≥0g(d)
x(γ(v, v))
for all
v∈R>0
.
Therefore, we can express
c2(d, c1)
as
c2(d, c1) = max
x∈R≥0{g(d)
x(c1)}=g(d)
s(c1)
(5.11)
where
s∈R≥0
solely depends on
c1
and
d
and is dened by
γ(s, s) = c1
. That
means
g(d)
s
and
g(d)
s+ε
, for
ε→0
, have
c1
as their intersection abscissa. We choose
c1=γ(Φd, Φd)
=d·(Φd+ 1)d−1
(d+ 1) ·Φd
d
=d·Φd+1
d
(d+ 1) ·(Φd+ 1) ·Φd
d
=d·Φd
(d+ 1)(Φd+ 1) ∈(0,1).
With (5.11), this yields
c2(d, c1) = g(d)
Φd(c1) = Φd+1
d·(1−c1)
. Thus, we can choose
c2=Φd+1
d·(1 −c1)
and get
PoATL ≤c2
1−c1
=Φd+1
d·(1 −c1)
1−c1
=Φd+1
d.
This completes the proof of the theorem.
5.3.2 Lower Bound
Theorem 5.7.
For weighted congestion games with polynomial latency functions
of maximum degree
d
and non-negative coecients, we have
PoATL ≥Φd+1
d
.
Proof.
Given the maximum degree
d∈N
for the polynomial latency functions,
set
k≥max{d
bd/2c,2}
. Note, that
d
bd/2c= maxj∈[d]0d
j
. We construct a con-
gestion game for
n= (d+ 1) ·k
players and
|E|=n
facilities.
98 5 Weighted Congestion Games
We divide the set
E
into
d+ 1
partitions:
For
i∈[d]0
, let
Ei={gi,1, . . . , gi,k}
, with each
gi,j
sharing the latency function
x7→ ai·xd
. The values of the coecients
ai
will be determined later. For simplicity
of notation, set
gi,j =gi,j−k
for
j > k
in the following.
Similarly, we partition the set of players
[n]
:
For
i∈[d]0
, let
Ni={ui,1, . . . , ui,k}
. The weight of each player in set
Ni
is
Φi
d
,
so
wui,j =Φi
d
for all
i∈[d]0
,
j∈[k]
.
Now, for every set
Ni
, each player
ui,j ∈Ni
has exactly two strategies:
Qui,j ={gi,j}
and
Pui,j =({gd,j+1, . . . , gd,j+(d
i), gi−1,j}
for
i= 1
to
d,
{gd,j+1}
for
i= 0.
Now let
Q= (Q1, . . . , Qn)
and
P= (P1, . . . , Pn)
be strategy proles. The facili-
ties in each set
Ei
then have the following loads for
Q
and
P
:
load on every facility
e∈Ei
i δe(Q)δe(P)
d Φd
dPd
l=0 d
lΦl
d= (Φd+ 1)d=Φd+1
d
0
to
d−1Φi
dΦi+1
d
For
P
to become a Nash Equilibrium, we need to fulll the following Nash in-
equalities for each set
Ni
of players:
i
Nash inequality to fulll
1 to
dPCui,j (P) = d
i·ad·(Φd+1
d)d+ai−1·(Φi
d)d
≤ai·(Φi+1
d+Φi
d)d=PCui,j (P−ui,j , Qui,j )
0
PCu0,j (P) = ad·(Φd+1
d)d≤a0·(Φd+ 1)d=PCu0,j (P−u0,j , Qu0,j )
Replacing
≤
by = yields a homogeneous system of linear equations, i.e. the
system
Bd·a= 0
where
Bd
is the following
(d+ 1) ×(d+ 1)
matrix:
Bd=
−Φd2+d+1
d+Φd2+d
dΦd2
d0··· ··· 0
d
d−1Φd2+d
d−Φd2+1
d
....
.
.
.
.
.
0
...
.
.
..
.
........
.
.
d
iΦd2+d
d0··· 0−Φid+d+1
dΦid
d0··· 0
.
.
..
.
.
0
.......
.
.
.
.
....
0
.
.
..
.
..
.
....
Φd
d
Φd2+d
d0··· 0··· 0−Φd+1
d
(5.12)
5.3 Price of Anarchy for Weighted Congestion Games 99
and
a= (ad. . . a0)t
. Obviously, a solution to this system fullls the initial Nash
inequalities. Note, that
(Φi+1
d+Φi
d)d= (Φi
d)d·(Φd+ 1)d=Φid+d+1
d.
Claim 5.8.
The
(d+ 1) ×(d+ 1)
matrix
Bd
from (5.12) has rank
d
.
Proof.
We use the well known fact from linear algebra that if a matrix
C
results
from another matrix
D
by adding a multiple of one row (or column) to another
row (or column, respectively) then
rank(C) = rank(D)
.
Now consider the matrix
Cd
that results from adding row
j
multiplied by the
factor
Φ−1
d
to row
j−1
, sequentially done for
j=d+ 1, d, . . . , 2
. Obviously,
Cd
is a lower triangular matrix with nonzero elements only in the rst column and
on the principal diagonal.
For the top left element of
Cd
we get
−Φd2+d+1
d+
d
X
j=0 d
jΦd2+j
d=Φd2
d· −Φd+1
d+
d
X
j=0 d
jΦj
d
| {z }
(Φd+1)d
!= 0.
Since all elements on the principal diagonal of
Cd
with the just shown
exception of the rst one are nonzero, it is easy to see that
Cd
(and thus also
Bd
) has rank
d
.
By the above claim it follows that the column vectors of
Bd
are linearly de-
pendent and thus there are with degree of freedom 1 innitely many linear
combinations of them yielding 0. In other words,
Bd·a= 0
has a one-dimensional
solution space.
We now show (by induction over
i
) that all coecients
ai
,
i∈[d]0
must
have the same sign and thus we can alway nd a valid solution. From the last
equality, for
i= 0
, we have that
ad
and
a0
must have the same sign. Now for
i= 1, . . . , d −1
, it follows that
ai
must have the same sign as
ai−1
and
ad
, for
(Φd+1
d)d
,
(Φi
d)d
, and
(Φi+1
d+Φi
d)d
are all positive.
Choosing
a6= 0
with all components being positive, all coecients of the
latency functions are positive. We get,
PoATL ≥SCTL(Γ, P)
SCTL(Γ, Q)=k·Pd
i=0 ai(Φi+1
d)d+1
k·Pd
i=0 ai(Φi
d)d+1 =Φd+1
d.
We proceed to show that just given lower bound also holds for weighted
network
congestion games.
Corollary 5.9.
The lower bound in Theorem 5.7 on
PoATL
also holds for weighted
network congestion games.
100 5 Weighted Congestion Games
Proof.
Each instance of the congestion game in Theorem 5.7 essentially can be
characterized by two parameters: The maximum degree
d
of the latency functions
and the number of facilities
k≥max{d
bd/2c,2}
in each class
Ei
, where
i∈[d]0
.
Remember that the number of facilities as well as the number of players is
given by
(d+ 1) ·k
.
Confer Figure 5.3 for an example in the case of quadratic latency functions
(i.e.
d= 2
) and
k= 2
. For their respective players, gray nodes denote origins,
whereas nodes with a thick outline represent destinations. Note that for the sake
of clarity not all edges are shown, as will be explained later. Edges without a
label have
fe(x) = 0
as their latency function. Again, we call these edges
free
edges. All other edges have the associated latency function as in Theorem 5.7. In
the following, we will outline the general construction scheme.
u2,1 u1,1 u0,1
u2,2 u1,2 u0,2
u2,1
u1,2
u0,1
u2,2
u1,1
u0,2
g2,2
g2,1
g0,2
g1,1
g0,1
g1,2
Fig. 5.3.
Network congestion game for
d= 2
and
k= 2
The network corresponding to an instance characterized by
(d, k)
can be con-
structed as follows:
There is a circle of
2·k
edges where every other edge represents a resource
gd,1, gd,2, . . . , gd,k
. All remaining edges in the circle are free edges. Furthermore,
every player
ui,j
has its own origin node which has a single free edge to
gd,j+1
.
Consequently, circle edge
gd,j+(d
i)
connects to a free edge which then in turn
connects to edge
gi−1,j
. (In case
i= 0
, the latter simply is another free edge.)
From there, there is another free edge to the destination node of player
ui,j
. Note
that, thus far, the graph has exactly one acyclic path for each player, i.e. for each
origin-destination pair. Each of these paths represents that player's unfavorable
strategy which has been denoted as
Pui,j
in Theorem 5.7.
One can now add two more free edges, for each player
ui,j
, that allow
ui,j
to
also use its optimal strategy
Qui,j
: From
ui,j
's origin node add a free link to
gi,j
,
and from
gi,j
add a free link to
ui,j
's destination node. We call the rst type of
links
A-Links
, the latter
B-Links
. Note that in Figure 5.3, A- and B-Links are
5.4 Conclusion and Discussion 101
only shown for player
u1,1
. (The gure is complete otherwise.) The thick gray
path denotes player
u1,1
's strategy in the system optimum, whereas the hatched
path indicates its strategy in the worst-case Nash equilibrium.
A-Links obviously cannot create shortcuts for
other
players as origin nodes
only have outgoing edges. Similarly, destination nodes only have incoming edges
and therefore B-Links cannot create shortcuts for other players, either. Eventu-
ally, neither A- nor B-Links can create a shortcut for the same player's other
strategy
Pu∗,∗
as they do not share any nodes, except for the origin and destina-
tion nodes.
Note, however, that B-Links do create additional paths: In Figure 5.3, for
instance, player
u1,1
now has the further option of using a path consisting of ve
edges: three free ones,
g2,2
, and
g1,1
. Nevertheless, all such additional paths are
supersets of the player's optimal strategy and thus neither change the system
optimum nor the worst-case Nash equilibrium.
We close this section by studying the impact of weights to the price of anarchy.
Corollary 5.10.
The exact price of anarchy for
unweighted congestion games
PoATL =(k+ 1)2d+1 −kd+1(k+ 2)d
(k+ 1)d+1 −(k+ 2)d+ (k+ 1)d−kd+1 ,
where
k=bΦdc
, is bounded by
bΦdcd+1 ≤PoATL ≤Φd+1
d
.
Proof.
The upper bound obviously is a direct consequence of Theorem 5.6. For
the lower bound, dene
A= (k+ 1)d−kd+1
and
B= (k+ 1)d+1 −(k+ 2)d
. Note
that
A, B > 0
by choice of
k
. Then,
(k+ 1)2d+1 −kd+1(k+ 2)d
(k+ 1)d+1 −(k+ 2)d+ (k+ 1)d−kd+1 =(A+kd+1)(k+ 1)d+1 −kd+1(k+ 2)d
A+B
=kd+1 ·A·k+1
kd+1 +B
A+B≥ bΦdcd+1.
This completes the proof of the lower bound.
5.4 Conclusion and Discussion
In this chapter we closed the problem of obtaining the exact value of the price of
anarchy for (weighted) congestion games with polynomial latency functions. We
assumed polynomial latency functions of maximum degree
d
with non-negative
coecients. We considered the cases of unweighted and weighted players. The
given results improve on the two recent STOC papers of Awerbuch et al. [5]
and Christodoulou and Koutsoupias [17]. Our bounds on the price of anarchy
depend on a new combinatorial sequence
Φd
, which is a generalization of the
golden ratio to higher dimensions. We believe that this sequence is of independent
combinatorial interest.
102 5 Weighted Congestion Games
The key to our results was an improved analysis of an optimization process
that was also considered by Christodoulou and Koutsoupias [17]. Our improved
analysis uses some new and non-trivial ideas.
The techniques used for proving our upper bounds are also practical for other
models. In particular, by applying them, we were able to prove (tight) upper
bounds on the price of anarchy for weighted congestion games and Wardrop
games, both with player-specic linear latency functions [50]. Moreover, we de-
ployed them to show upper bounds on the price of anarchy for Wardrop games
with certain polynomial latency functions [24]. In both works [24,50] we also
used the total latency as our social cost measure.
6
Bayesian Routing Games
6.1 Introduction
In recent years, motivated by non-cooperative systems like the Internet, com-
bining ideas from Game Theory and Theoretical Computer Science has become
more and more attractive. In many of these large-scale, non-cooperative systems,
users have only incomplete information about the system for several reasons. In
his honored work, Harsanyi [58] introduced an elegant approach to studying non-
cooperative games with
incomplete information
, where the players are uncertain
about some parameters. To model such games, Harsanyi introduced the
Harsanyi
transformation
, which converts a (strategic) game with incomplete information
to a strategic game where players have dierent
types
. The type of a player rep-
resents its private information that is not common knowledge to all players. In
the resulting
Bayesian game
, each player's uncertainty about each other's type
is described by a probability distribution over all possible
type proles
. Using
this probability distribution, players make their decisions according to
Bayesian
Decision Theory
[12]. In Bayesian Decision Theory probabilities are used as a
measure of the degree of belief a person has in some proposition.
In this chapter, we introduce a particular selsh routing game with incomplete
information that we call
Bayesian routing game
. These games have been formally
introduced in Section 3.3. Bayesian routing games are a generalization to the
routing games on parallel links studied in Chapter 4. In a Bayesian routing game,
each of
n
selsh
players
wishes to assign its
trac
to one of
m
parallel
links
. Each
link has a certain
capacity
, which species the rate at which the link processes
trac. In the case of
identical links
, all links have equal capacity. Link capacities
vary arbitrary, in the case of
related links
. The
latency
of a link is the total
trac on the link divided by the capacity of the link. Players do not know each
other's trac. Following Harsanyi's approach, we introduce for each player a
set of possible
types
. We assume that all
type sets
are nite. Each type of a
player corresponds to some trac. Furthermore, we assume that there is a joint
probability distribution
Ψ
, called
type distribution
, over the set of all possible type
realizations. In general,
Ψ
can be arbitrary; however, sometimes we assume
Ψ
to
104 6 Bayesian Routing Games
be
independent
in that case,
Ψ
is expressed as the product of
n
independent
probability distributions, one for each player type set.
In a
pure strategy
, a player chooses for each of its types a particular link. So,
a pure strategy is a function from the type set of a player to the set of links. In a
mixed strategy
, a player uses a probability distribution over all its possible pure
strategies. A
strategy prole
species a strategy for each of the players. Users
choose strategies in order to minimize their
private cost
, which is dened as the
expected latency experienced by the player. Note that due to the Bayesian model,
the private cost in a pure strategy prole is given by the expectation over the
type distribution
Ψ
. For mixed strategy proles, the expectation is taken over
both the type distribution
Ψ
and the mixed strategies of the players.
The players neither cooperate with each other nor adhere to a global objective
function, the so called
social cost
[67]. A stable state in which no player has an
incentive to unilaterally change its strategy is called a
Bayesian Nash equilibrium
.
In our study, we distinguish between
pure
and
mixed
Bayesian Nash equilibria.
Of special interest to our work are
fully mixed
Bayesian Nash equilibria, where
each player assigns strictly positive probability to each of its pure strategies.
If each player has only a single type, so that players are completely informed
about each other's trac, then we are in the setting of the routing games on
parallel links (with complete information) that we studied in Chapter 4. In the
following, we call them
complete information routing games
, in order to empha-
size the connection to Bayesian routing games. In this setting, Bayesian Nash
equilibria become Nash equilibria.
As before, we use the
price of anarchy
[86], as a measure of the maximum
performance degradation due to the selsh behavior of the players. The price of
anarchy can be dened with respect to dierent social cost measures.
As a generalization of our Bayesian routing games we also introduce
weighted
Bayesian congestion games
. Weighted Bayesian congestion games generalize the
weighted congestion games (as studied in Chapter 5) by incorporating incomplete
information about the players' trac. So, in a weighted Bayesian congestion
game, the strategy set of each player is a subset of the power set of given resources.
Weighted Bayesian congestion games provide us with a general framework for
modeling any kind of non-cooperative resource sharing problem where the players
do not know each other's trac.
6.1.1 Summary of Results
Due to the new dimension that the incomplete information introduces to the
routing game, the analysis of the Bayesian routing game requires new techniques.
In this chapter, we introduce such techniques and we present a comprehensive
collection of results for the Bayesian routing game. We partition our results into
three major parts:
6.1 Introduction 105
6.1.1.1 Existence and Computational Complexity of Pure Bayesian
Nash Equilibria
We dene a new potential function that we use to prove that every weighted
Bayesian congestion game possesses a pure Bayesian Nash equilibrium (The-
orem 6.1). Observe that this existence result applies for the class of
weighted
Bayesian congestion games
.
For the case of Bayesian routing games, identical links and independent type
distributions, a pure Bayesian Nash equilibrium can be computed in polynomial
time (Theorem 6.2). This computation is based on Graham's LPT scheduling al-
gorithm [56]. For the case of related links and independent type distribution, and
also for the case of identical links and arbitrary type distribution, the complexity
of computing a pure Bayesian Nash equilibrium remains open.
6.1.1.2 Properties of Fully Mixed Bayesian Nash Equilibria
We show that for the case of identical links, the private cost of each player is
maximized in a fully mixed Bayesian Nash equilibrium (Theorem 6.7). This also
implies that a player has the same private cost in any fully mixed Bayesian
Nash equilibrium. We dene a certain fully mixed Bayesian Nash equilibrium
that always exists. We show that, in general, there might exist more than one
fully mixed Bayesian Nash equilibrium, and we study their structural proper-
ties (Theorem 6.9). Finally, we determine the dimension of the space of fully
mixed Bayesian Nash equilibria for the case of independent type distributions
(Theorem 6.10).
6.1.1.3 Bounds on the Price of Anarchy
We close this chapter with bounds on the price of anarchy for three dierent
social cost measures and for the case of identical links.
•
The
makespan social cost
, which is dened by the expected maximum latency
on a link, is a social cost measure that expresses the social welfare of the sys-
tem. Here, we are able to show lower and upper bounds on the price of anarchy
for dierent special cases (Theorem 6.12, Theorem 6.15 and Theorem 6.16).
The exact price of anarchy for this social cost measure remains open, even for
the case of identical links.
•
A social cost measure that describes average player welfare is the
sum of
private costs
. In this setting, it follows that for the case of identical links,
each fully mixed Bayesian Nash equilibrium has maximum social cost (The-
orem 6.17). Using this fact, we prove an upper bound of
m+n−1
m
on the price
of anarchy for the case of identical links (Theorem 6.18). We prove that
this bound is asymptotically tight, already for complete information routing
games.
106 6 Bayesian Routing Games
•
We also study social cost as
maximum of private costs
. For identical links, we
show asymptotically tight upper bounds on the price of anarchy of
m+n−1
m
for
Bayesian routing games and of
2−1
m
for complete information routing games
(Theorem 6.20).
6.1.2 Related Work
Bayesian routing games and weighted Bayesian congestion games generalize the
games studied in Chapter 4and Chapter 5. So, many results that are already
described in Section 4.1.2 and Section 5.1.2 are also of interest here. To keep
this chapter self-contained, we again include those results that are most closely
related.
Rosenthal [88] introduced the class of
congestion games
and showed that they
always possess a pure Nash equilibrium. Fotakis et al. [38] considered
weighted
congestion games
and proved the existence of a pure Nash equilibria, for the case
where resources have linear latency functions. They also showed that a pure Nash
equilibrium might not exist for weighted congestion games with general latency
functions.
Harsanyi developed in his pioneering work [58,59] a framework for studying
competitive situations where the players have incomplete information. For an
introduction to these so-called Bayesian games, we refer to Mas-Colell et al. [73]
and Myerson [79]. Facchini et al. [28] considered Bayesian congestion models with
players of identical weight, which have incomplete information about each other's
preferences. Beier et al. [9] focused on a service provider congestion game with
incomplete information.
Complete information routing games on parallel links were introduced by
Koutsoupias and Papadimitriou [67]. Graham's LPT scheduling algorithm [56]
computes a pure Nash equilibrium in this setting [37].
Mavronicolas and Spirakis [74] introduced the notion of
fully mixed Nash equi-
libria
to complete information routing games. They showed that, in case of exis-
tence, the fully mixed Nash equilibrium is unique. For the case of identical links,
Gairing et al. [47] showed that fully mixed Nash equilibria maximize private costs.
For complete information routing games and makespan social cost, there exist
tight bounds on the price of anarchy. These bounds are
Θ(log m
log log m)
for identical
links [23,66] and
Θ(log m
log log log m)
for related links [23]. For complete information
routing games and social cost dened as the sum of private cost, Berenbrink et al.
[11] gave a lower bound of
n
5
on the price of anarchy. Restricting to pure Nash
equilibria they also showed an upper bound that solely depends on the players'
trac.
Subsequently to our work Georgiou et al. [53] introduced a routing game with
incomplete information where the players have complete information about each
other's trac but only incomplete information about the latency functions in the
network.
6.2 Pure Bayesian Nash Equilibria 107
6.1.3 Organization
The rest of this chapter is organized as follows. Pure Bayesian Nash equilibria
are studied in Section 6.2. Some interesting structural properties of fully mixed
Bayesian Nash equilibria are treated in Section 6.3. Section 6.4 studies the price
of anarchy. We conclude, in Section 6.5, with a summary of our results and some
open problems.
6.2 Pure Bayesian Nash Equilibria
In this section we study the existence and the computational complexity of pure
Bayesian Nash equilibria.
We rst show that there is always a pure Bayesian Nash equilibrium in any
Bayesian routing game. In fact, we show this result for the more general class
of weighted Bayesian congestion games (Theorem 6.1). Then, we present a poly-
nomial time algorithm, called
PureBayesian
, that computes a pure Bayesian
Nash equilibrium for a Bayesian routing game with identical links and indepen-
dent type distribution (Theorem 6.2). Finally, we show that
PureBayesian
cannot be used to compute a pure Bayesian Nash equilibrium for Bayesian rout-
ing games with
related
links (Proposition 6.3) or with
correlated
type distribution
(Proposition 6.4).
6.2.1 Existence
We start by proving existence of pure Bayesian Nash equilibria.
Theorem 6.1.
Every weighted Bayesian congestion game
Γ
with linear latency
functions has a pure Bayesian Nash equilibrium.
Proof.
Given a pure strategy prole
σ= (σ1, . . . , σn)
, dene the function
Φ(σ) = X
i∈[n]X
t∈TiX
e∈σi(t)
Ψ(i, t)·w(t)·ge(δ−i
e(σ,(Ψ|ti=t))+w(t))+ge(w(t)).
We will prove that any unilateral strategy change of a type agent that decreases
its private cost also decreases the value of the function
Φ
.
Given a pure strategy prole
σ
, dene for every player
r∈[n]
and type
t∈Tr
,
Φ(r,t)(σ) = X
e∈σr(t)
Ψ(r, t)·w(t)·ge(δ−r
e(σ,(Ψ|tr=t))+w(t)) + ge(w(t));
and for every resource
e∈[m]
and player
r∈[n]
,
Φ−r
e(σ)
=X
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
Ψ(i, t)·w(t)·ge(δ−i
e(σ,(Ψ|ti=t))+w(t))+ge(w(t)).
108 6 Bayesian Routing Games
Observe rst that
X
t∈Tr
Φ(r,t)(σ) + X
e∈[m]
Φ−r
e(σ)
=X
t∈TrX
e∈σr(t)
Ψ(r, t)·w(t)·ge(δ−r
e(σ,(Ψ|tr=t))+w(t)) + ge(w(t))
+X
e∈[m]X
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
Ψ(i, t)·w(t)·ge(δ−i
e(σ,(Ψ|ti=t))+w(t))+ge(w(t))
=X
t∈TrX
e∈σr(t)
Ψ(r, t)·w(t)·ge(δ−r
e(σ,(Ψ|tr=t))+w(t)) + ge(w(t))
+X
i∈[n]\{r}X
t∈TiX
e∈σi(t)
Ψ(i, t)·w(t)·ge(δ−i
e(σ,(Ψ|ti=t))+w(t))+ge(w(t))
=X
i∈[n]X
t∈TiX
e∈σi(t)
Ψ(i, t)·w(t)·ge(δ−i
e(σ,(Ψ|ti=t))+w(t)) + ge(w(t))
=Φ(σ).
Consider a unilateral strategy change of type agent
(r, ˆ
t)
from the set of
resources
σr(ˆ
t)∈Sr
to the set of resources
σ0
r(ˆ
t)∈Sr
. Set
σ0
r(t) = σr(t)
for all
t∈Tr\ {ˆ
t}
and dene
σ0= (σ1, . . . , σr−1, σ0
r, σr+1, . . . σn)
as the
pure strategy prole resulting from
σ
after this strategy change. Assume that
v(r,ˆ
t)(σ0,Ψ)< v(r,ˆ
t)(σ,Ψ)
, that is, the private cost of type agent
(r, ˆ
t)
decreases.
Thus,
v(r,ˆ
t)(σ0,Ψ)−v(r,ˆ
t)(σ,Ψ)
=X
e∈σ0
r(ˆ
t)
geδ−r
e(σ0,(Ψ|tr=ˆ
t)) + w(ˆ
t)−X
e∈σr(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)
<0.
Moreover,
•Φ(r,t)(σ) = Φ(r,t)(σ0)
for all type agents
(r, t)
where
t∈Tr\{ˆ
t}
, and
•Φ−r
e(σ) = Φ−r
e(σ0)
for all resources
e
that are neither in
σr(ˆ
t)
nor in
σ0
r(ˆ
t)
or
that are in both
σr(ˆ
t)
as well as in
σ0
r(ˆ
t)
, that is,
e∈[m]\(σr(ˆ
t)∪σ0
r(ˆ
t))∪
σr(ˆ
t)∩σ0
r(ˆ
t)
. Observe that these are the resources where the load does not
change.
Now, consider the change
∆(Φ)
to the function
Φ
due to this strategy change of
type agent
(r, ˆ
t)
. Clearly,
6.2 Pure Bayesian Nash Equilibria 109
∆(Φ)
=Φ(σ0)−Φ(σ)
=X
t∈TrΦ(r,t)(σ0)−Φ(r,t)(σ))+X
e∈[m]Φ−r
e(σ0)−Φ−r
e(σ)
=Φ(r,ˆ
t)(σ0)−Φ(r,ˆ
t)(σ))
+X
e∈σ0
r(ˆ
t)\σr(ˆ
t)Φ−r
e(σ0)−Φ−r
e(σ)+X
e∈σr(ˆ
t)\σ0
r(ˆ
t)Φ−r
e(σ0)−Φ−r
e(σ)
=∆1(Φ) + ∆2(Φ) + ∆3(Φ),
where
∆1(Φ) = Φ(r,ˆ
t)(σ0)−Φ(r,ˆ
t)(σ),
∆2(Φ) = X
e∈σ0
r(ˆ
t)\σr(ˆ
t)Φ−r
e(σ0)−Φ−r
e(σ),
and
∆3(Φ) = X
e∈σr(ˆ
t)\σ0
r(ˆ
t)Φ−r
e(σ0)−Φ−r
e(σ).
Clearly,
∆1(Φ)
=Ψ(r, ˆ
t)·w(ˆ
t)·
X
e∈σ0
r(ˆ
t)geδ−r
e(σ0,(Ψ|tr=ˆ
t))+w(ˆ
t)+ge(w(ˆ
t))
−X
e∈σr(ˆ
t)geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)+ge(w(ˆ
t))
=Ψ(r, ˆ
t)·w(ˆ
t)·
X
e∈σ0
r(ˆ
t)\σr(ˆ
t)geδ−r
e(σ0,(Ψ|tr=ˆ
t))+w(ˆ
t)+ge(w(ˆ
t))
−X
e∈σr(ˆ
t)\σ0
r(ˆ
t)geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)+ge(w(ˆ
t))
=Ψ(r, ˆ
t)·w(ˆ
t)·
X
e∈σ0
r(ˆ
t)\σr(ˆ
t)geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)+ge(w(ˆ
t))
−X
e∈σr(ˆ
t)\σ0
r(ˆ
t)geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)+ge(w(ˆ
t))
.
Furthermore, due to the arrival of type agent
(r, ˆ
t)
on the resources
e∈σ0
r(ˆ
t)\
σr(ˆ
t)
,
110 6 Bayesian Routing Games
∆2(Φ)
=X
e∈σ0
r(ˆ
t)\σr(ˆ
t)X
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
Ψ(i, t)·w(t)
·ge(δ−i
e(σ0,(Ψ|ti=t)) + w(t))+ge(w(t))
−ge(δ−i
e(σ,(Ψ|ti=t))+w(t)) −ge(w(t))
=X
e∈σ0
r(ˆ
t)\σr(ˆ
t)X
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
Ψ(i, t)·w(t)
·ge(δ−i
e(σ0,(Ψ|ti=t))) −ge(δ−i
e(σ,(Ψ|ti=t)))
=X
e∈σ0
r(ˆ
t)\σr(ˆ
t)X
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
Ψ(i, t)·w(t)
·aeδ−i
e(σ0,(Ψ|ti=t)) −δ−i
e(σ,(Ψ|ti=t))
=X
e∈σ0
r(ˆ
t)\σr(ˆ
t)X
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
Ψ(i, t)·w(t)·ae
·X
(t1,...,tn)∈T:
ti=t
Ψ(t1, . . . , ti−1, ti+1, . . . , tn|ti=t)·
X
s∈[n]\{i}:
e∈σ0
s(ts)
w(ts)−X
s∈[n]\{i}:
e∈σs(ts)
w(ts)
.
Note that
X
s∈[n]\{i}:
e∈σ0
s(ts)
w(ts)−X
s∈[n]\{i}:
e∈σs(ts)
w(ts) = w(ˆ
t),
for all
(t1, . . . , tn)∈T
where
tr=ˆ
t,
0,
else.
Hence,
∆2(Φ)
=X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
aeX
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
Ψ(i, t)·w(t)
·X
(t1,...,tn)∈T:
ti=t,tr=ˆ
t
Ψ(t1, . . . , ti−1, ti+1, . . . , tn|ti=t)·w(ˆ
t)
=w(ˆ
t)X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
aeX
i∈[n]\{r}X
t∈Ti:
e∈σi(t)
w(t)X
(t1,...,tn)∈T:
ti=t,tr=ˆ
t
Ψ(t1, . . . , tn)
=w(ˆ
t)X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
aeX
i∈[n]\{r}X
t∈Ti:
e∈σi(t)X
(t1,...,tn)∈T:
ti=t,tr=ˆ
t
Ψ(t1, . . . , tn)·w(t)
6.2 Pure Bayesian Nash Equilibria 111
=Ψ(r, ˆ
t)·w(ˆ
t)X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
ae
·X
i∈[n]\{r}X
t∈Ti:
e∈σi(t)X
(t1,...,tn)∈T:
ti=t,tr=ˆ
t
Ψ(t1, . . . , tr−1, tr+1, . . . , tn|tr=ˆ
t)·w(t)
=Ψ(r, ˆ
t)·w(ˆ
t)X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
ae
·X
i∈[n]\{r}X
(t1,...,tn)∈T:
e∈σi(ti),tr=ˆ
t
Ψ(t1, . . . , tr−1, tr+1, . . . , tn|tr=ˆ
t)·w(ti)
=Ψ(r, ˆ
t)·w(ˆ
t)X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
ae
·X
(t1,...,tn)∈T:
tr=ˆ
t
Ψ(t1, . . . , tr−1, tr+1, . . . , tn|tr=ˆ
t)X
i∈[n]\{r}:
e∈σi(ti)
w(ti)
=Ψ(r, ˆ
t)·w(ˆ
t)·X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
ae·δ−r
e(σ,(Ψ|tr=ˆ
t)).
Similarly, since type agent
(r, ˆ
t)
left the resources
e∈σr(ˆ
t)\σ0
r(ˆ
t)
, we obtain that
∆3(Φ) = −Ψ(r, ˆ
t)·w(ˆ
t)·X
e∈σr(ˆ
t)\σ0
r(ˆ
t)
ae·δ−r
e(σ,(Ψ|tr=ˆ
t)).
Hence,
∆(Φ)
=∆1(Φ) + ∆2(Φ) + ∆3(Φ)
=Ψ(r, ˆ
t)·w(ˆ
t)
·
X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t)) + w(ˆ
t)+ge(w(ˆ
t)) + ae·δ−r
e(σ,(Ψ|tr=ˆ
t))
−X
e∈σr(ˆ
t)\σ0
r(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)+ge(w(ˆ
t))+ae·δ−r
e(σ,(Ψ|tr=ˆ
t))
= 2 ·Ψ(r, ˆ
t)·w(ˆ
t)·
X
e∈σ0
r(ˆ
t)\σr(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)
−X
e∈σr(ˆ
t)\σ0
r(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t)) + w(ˆ
t)
= 2 ·Ψ(r, ˆ
t)·w(ˆ
t)·
X
e∈σ0
r(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)
112 6 Bayesian Routing Games
−X
e∈σr(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)
= 2 ·Ψ(r, ˆ
t)·w(ˆ
t)·
X
e∈σ0
r(ˆ
t)
geδ−r
e(σ0,(Ψ|tr=ˆ
t))+w(ˆ
t)
−X
e∈σr(ˆ
t)
geδ−r
e(σ,(Ψ|tr=ˆ
t))+w(ˆ
t)
= 2 ·Ψ(r, ˆ
t)·w(ˆ
t)·v(r,ˆ
t)(σ0,Ψ)−v(r,ˆ
t)(σ,Ψ)
<0.
Thus, any unilateral strategy change of a type agent that decreases its private
cost also decreases the value of the function
Φ
. Since the number of possible
strategy proles in
Γ
is nite, it follows that there is a pure strategy prole
that minimizes
Φ
. In this strategy prole, no type agent can decrease its private
cost by unilaterally changing its strategy. Hence,
Γ
has a pure Bayesian Nash
equilibrium, as needed.
This generalizes a result by Fotakis et al. [38, Theorem 1] to the Bayesian setting.
In particular our function
Φ
reduces to their potential function if each player has
only a single type.
6.2.2 Computation
We now turn to the model of identical links and show how a pure Bayesian
Nash equilibrium can be computed in polynomial time if the type distribution
is independent. An algorithm, called
PureBayesian
, that performs this task is
depicted in Figure 6.1. The algorithm computes a
normal
pure Bayesian Nash
PureBayesian
(
Γ
)
Input:
Bayesian routing game
Γ= (n, m, 1, T, Ψ)
with identical links and independent
type distribution
Output:
pure Bayesian Nash Equilibrium
σ
1: Calculate for each player
i∈[n]
its expected trac
W(i)
;
2: Construct a complete information routing game
ΓCI = (n, m, 1,{(t0
1, ..., t0
n)},1)
where
w(t0
i) = W(i)
for all
i∈[n]
.
3: Compute a pure Nash equilibrium
L= (`1,...,`n)
for
ΓCI
in polynomial time with the LPT
scheduling algorithm which assigns the players in order of non-increasing player trac
to minimum load links (see [37,56]).
4: Set
σi(t) = `i
for all players
i∈[n]
and types
t∈Ti
.
5:
return
σ
;
Fig. 6.1.
PureBayesian
equilibrium, that is, for a xed player, it assigns all types to the same link.
6.2 Pure Bayesian Nash Equilibria 113
PureBayesian
rst computes for each player its expected trac. Then, it uses
Graham's LPT scheduling algorithm [56] to assign (all types of) the players
in order non-increasing expected trac to minimum load links. Gairing et al.
[49] showed that the resulting strategy prole is a (normal) pure Bayesian Nash
equilibrium.
Theorem 6.2 (Gairing et al. [49]).
Let
Γ= (n, m, 1, T, Ψ)
be a Bayesian
routing game on identical links with independent type distribution. Then, a (nor-
mal) pure Bayesian Nash equilibrium for
Γ
can be computed in time polynomial
in the size of
Γ
even if
Ψ
is represented in a compact form by a set of probabilities
Ψ(i, t)
for
i∈[n]
and
t∈Ti
.
Algorithm
PureBayesian
cannot be used to compute pure Bayesian Nash equi-
libria for the more general classes of Bayesian routing games either on related
links or with correlated type distribution. The reason is that it always computes
a
normal
pure Bayesian Nash equilibrium, while the following counter-examples
show that a normal pure Bayesian Nash equilibrium does not exist in general.
Proposition 6.3.
There is a Bayesian routing game
Γ
on related links with in-
dependent type distribution that does not have a normal pure Bayesian Nash
equilibrium.
Proof.
Consider the Bayesian routing game
Γ= (2,2,c, T1×T2,Ψ)
with two
links of capacity
c1= 1
and
c2= 5
. The two players have type sets
T1={t1, t0
1}
and
T2={t2}
, where
w(t1) = 1
,
w(t0
1) = 5
,
w(t2) = 10
, and
Ψ(1, t1) = Ψ(1, t0
1) =
1
2
. We will now study the structure of pure Bayesian Nash equilibria for
Γ
and
nally recognize that it has no
normal
pure Bayesian Nash equilibrium.
Let
σ
be an arbitrary pure Bayesian Nash equilibrium. Then,
λ1
(2,t2)(σ,Ψ) = δ−2
1(σ,Ψ) + w(t2)
c1≥w(t2)
c1
= 10
while
λ2
(2,t2)(σ,Ψ) = δ−2
2(σ,Ψ) + w(t2)
c2≤
1
2·w(t1) + 1
2·w(t0
1) + w(t2)
c2
=13
5<10.
Thus,
σ
assigns
t2
to link 2, so
σ2(t2)=2
. Consider now the types of player 1.
We have
λ1
(1,t1)(σ,Ψ) = w(t1)
c1
= 1
and
λ2
(1,t1)(σ,Ψ) = w(t2) + w(t1)
c2
=11
5,
λ1
(1,t0
1)(σ,Ψ) = w(t0
1)
c1
= 5
and
λ2
(1,t0
1)(σ,Ψ) = w(t2) + w(t0
1)
c2
= 3.
So
σ
assigns
t1
to link
1
and
t0
1
to link
2
. It follows that
σ
is the unique pure
Bayesian Nash equilibrium. However,
σ
is not a
normal
pure Bayesian Nash
equilibrium. The claim follows.
114 6 Bayesian Routing Games
Proposition 6.4.
There is a Bayesian routing game
Γ
on identical links with
correlated type distribution that does not have a normal pure Bayesian Nash equi-
librium.
Proof.
Consider the Bayesian routing game
Γ= (3,2,1, T1×T2×T3,Ψ)
with 2
identical links and 3 players where the type sets are
T1={t1, t0
1}
,
T2={t2, t0
2}
and
T3={t3, t0
3}
. The types are of trac
w(t1) = w(t0
2) = w(t3) = w(t0
3) = 1
and
w(t0
1) = w(t2) = 2
. The correlated distribution
Ψ
is given by
Ψ(t1, t2, t3) =
Ψ(t0
1, t0
2, t0
3) = 1
2
.
Assume, by way of contradiction, that a
normal
pure Bayesian Nash equilib-
rium
σ
exists; so,
σ1(t1) = σ1(t0
1)
,
σ2(t2) = σ2(t0
2)
, and
σ3(t3) = σ3(t0
3)
. Let
j
and
k
be the two links. Without loss of generality, set
σ1(t1) = σ1(t0
1) = j
. Then,
clearly
λj
(2,t0
2)(σ,Ψ)≥w(t0
1) + w(t0
2) = 3
while
λk
(2,t0
2)(σ,Ψ)≤w(t0
3) + w(t0
2) = 2.
Thus,
σ2(t0
2) = k
; hence,
σ2(t2) = σ2(t0
2) = k
for all normal pure Bayesian Nash
equilibria
σ
. For the types of player 3, note that
λj
(3,t3)(σ,Ψ) = w(t1) + w(t3) = 2
while
λk
(3,t3)(σ,Ψ) = w(t2) + w(t3) = 3,
and
λj
(3,t0
3)(σ,Ψ) = w(t0
1) + w(t0
3) = 3
while
λk
(3,t0
3)(σ,Ψ) = w(t0
2) + w(t0
3) = 2.
Since
σ
is a Bayesian Nash equilibrium,
σ3(t3) = j
and
σ3(t0
3) = k
. Hence,
σ
is
not normal. A contradiction.
6.3 Properties of Fully Mixed Bayesian Nash Equilibria
In this section, we study fully mixed Bayesian Nash equilibria for the case of
identical links. We start by proving a technical lemma that will be handy later
on (Lemma 6.5). With the help of this lemma, we prove a simple expression
on the private cost of the players in a fully mixed Bayesian Nash equilibrium
(Theorem 6.6). Then, we show that fully mixed Bayesian Nash equilibria max-
imize private costs (Theorem 6.7). This result will be of particular interest in
Section 6.4. We proceed with an exact characterization of fully mixed Bayesian
Nash equilibria (Theorem 6.9). Finally, we determine the dimension of space of
fully mixed Bayesian Nash equilibria (Theorem 6.10).
We start with the following technical lemma.
Lemma 6.5.
Consider a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on identical
links and an associated mixed strategy prole
P
. Then, for each player
i∈[n]
,
X
j∈[m]
δ−i
j(P,(Ψ|ti=t)) = X
s∈[n]{i}
W(s|ti=t).
6.3 Properties of Fully Mixed Bayesian Nash Equilibria 115
Proof.
Clearly,
X
j∈[m]
δ−i
j(P,(Ψ|ti=t))
=X
j∈[m]X
σ∈ΣY
s∈[n]
p(s, σs)·δ−i
j(σ,(Ψ|ti=t))
=X
j∈[m]X
σ∈ΣY
s∈[n]
p(s, σs)·X
(t1,...,tn)∈T:
ti=t
Ψ(t1, . . . , ti−1, ti+1, . . . , tn|ti=t)X
s∈[n]\{i}:
σs(ts)=j
w(ts)
=X
σ∈ΣY
s∈[n]
p(s, σs)·X
(t1,...,tn)∈T:
ti=t
Ψ(t1, . . . , ti−1, ti+1, . . . , tn|ti=t)X
s∈[n]\{i}
w(ts)
=X
(t1,...,tn)∈T:
ti=t
Ψ(t1, . . . , ti−1, ti+1, . . . , tn|ti=t)X
s∈[n]\{i}
w(ts)
=X
s∈[n]\{i}X
(t1,...,tn)∈T:
ti=t
Ψ(t1, . . . , ti−1, ti+1, . . . , tn|ti=t)w(ts)
=X
s∈[n]\{i}
W(s|ti=t),
as needed.
We continue to prove a simple expression for the private cost of each player in a
fully mixed Bayesian Nash equilibrium.
Theorem 6.6.
Consider a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on iden-
tical links and an associated fully mixed Bayesian Nash equilibrium
F
. Then for
each player
i∈[n]
,
PCi(F,Ψ) = W
m+m−1
mW(i).
Proof.
Fix any player
i∈[n]
. Clearly, for any link
k∈supporti(F)=[m]
, and
by Lemma 6.5,
PCi(F,Ψ) = X
t∈Ti
Ψ(i, t)·v(i,t)(F,Ψ)
=X
t∈Ti
Ψ(i, t)·w(t) + δ−i
k(F,(Ψ|ti=t))
=X
t∈Ti
Ψ(i, t)·w(t) + X
t∈Ti
Ψ(i, t)·δ−i
k(F,(Ψ|ti=t))
=W(i) + X
t∈Ti
Ψ(i, t)·1
m·X
j∈[m]
δ−i
j(F,(Ψ|ti=t))
=W(i) + 1
mX
t∈Ti
Ψ(i, t)X
s∈[n]\{i}
W(s|ti=t)
(6.1)
116 6 Bayesian Routing Games
=W(i) + 1
mX
s∈[n]\{i}X
t∈Ti
Ψ(i, t)·W(s|ti=t)
=W(i) + 1
mX
s∈[n]\{i}
W(s)
=W
m+m−1
mW(i),
as needed.
We now prove that the private cost of each player is maximized in a fully mixed
Bayesian Nash equilibrium. For the special case of complete information routing
games this result was shown by Gairing et al. [47].
Theorem 6.7.
Consider a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on identi-
cal links and an associated fully mixed Bayesian Nash equilibrium
F
and Bayesian
Nash equilibrium
P
. Then for each player
i∈[n]
,
PCi(P,Ψ)≤PCi(F,Ψ).
Proof.
Fix any player
i∈[n]
. Then, for any link
j∈[m]
,
PCi(P,Ψ) = X
t∈Ti
Ψ(i, t)·v(i,t)(P,Ψ)
≤X
t∈Ti
Ψ(i, t)·w(t) + δ−i
j(P,(Ψ|ti=t)),
since
P
is a Bayesian Nash equilibrium. In particular,
PCi(P,Ψ)≤X
t∈Ti
Ψ(i, t)w(t) + min
j∈[m]nδ−i
j(P,(Ψ|ti=t))o
≤X
t∈Ti
Ψ(i, t)
w(t) + 1
mX
j∈[m]
δ−i
j(P,(Ψ|ti=t))
=X
t∈Ti
Ψ(i, t)
w(t) + 1
mX
s∈[n]\{i}
W(s|ti=t)
=W(i) + 1
mX
t∈Ti
Ψ(i, t)X
s∈[n]\{i}
W(s|ti=t)
=PCi(F,Ψ),
by Equation (6.1), as needed.
We proceed to dene a particular fully mixed strategy prole
F∗
.
Denition 6.8.
The
standard
fully mixed strategy prole
F∗
is the fully mixed
strategy prole that assigns every type agent to every link with probability
1
m
.
6.3 Properties of Fully Mixed Bayesian Nash Equilibria 117
It is easy to see that for any Bayesian routing game
Γ
on identical links, the
standard fully mixed strategy prole is a Bayesian Nash equilibrium. For the
special case of complete information routing games, this fact was rst stated in
[74].
In general, there exists more than one fully mixed Bayesian Nash equilibrium.
In the remainder of this section, we study the structure of fully mixed Bayesian
Nash equilibria for Bayesian routing games on identical links with independent
type distribution. We start with an exact characterization of fully mixed Bayesian
Nash equilibria.
Theorem 6.9.
Consider a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on identi-
cal links with independent type distribution and an associated fully mixed strategy
prole
F
. Then,
F
is a fully mixed Bayesian Nash equilibrium if and only if
1
m·W(i) = X
σi∈Σi
f(i, σi)X
t∈Ti:
σi(t)=j
Ψ(i, t)·w(t)
for all players
i∈[n]
and links
j∈[m]
.
Proof.
For any player
i∈[n]
and link
j∈[m]
, set
µ(F, i, j) = X
σi∈Σi
f(i, σi)X
t∈Ti:
σi(t)=j
Ψ(i, t)·w(t).
Observe that for any player
i∈[n]
and link
j∈[m]
,
δ−i
j(F,Ψ) = X
σ∈ΣY
s∈[n]
f(s, σs)X
(t1,...,tn)∈T
Ψ(t1, . . . , tn)X
k∈[n]\{i}:
σk(tk)=j
w(tk)
=X
σ∈ΣY
s∈[n]
f(s, σs)X
k∈[n]\{i}X
tk∈Tk:
σk(tk)=j
Ψ(k, tk)·w(tk)
=X
k∈[n]\{i}X
σ∈ΣY
s∈[n]
f(s, σs)X
tk∈Tk:
σk(tk)=j
Ψ(k, tk)·w(tk)
=X
k∈[n]\{i}X
σ0
k∈Σk
f(k, σ0
k)X
σ∈Σ:
σk=σ0
kY
s∈[n]\{k}
f(s, σs)X
tk∈Tk:
σk(tk)=j
Ψ(k, tk)·w(tk)
=X
k∈[n]\{i}X
σ0
k∈Σk
f(k, σ0
k)X
tk∈Tk:
σk(tk)=j
Ψ(k, tk)·w(tk)
=X
k∈[n]\{i}
µ(F, k, j).
Consider rst an arbitrary fully mixed strategy prole
F
that satises
µ(F, i, j) =
1
m·W(i)
for all players
i∈[n]
and links
j∈[m]
. Then, for all players
i∈[n]
,
types
t∈Ti
, and links
j∈[m]
,
118 6 Bayesian Routing Games
λj
(i,t)(F,Ψ) = δ−i
j(F,(Ψ|ti=t))+w(t)
=δ−i
j(F,Ψ) + w(t)
=X
k∈[n]\{i}
µ(F, k, j) + w(t)
=X
k∈[n]\{i}
1
m·W(k) + w(t).
Hence, we get for the private cost of type agent
(i, t)
,
v(i,t)(F,Ψ) = X
σi∈Σi
f(i, σi)·λσi(t)
(i,t)(F,Ψ)
=X
k∈[n]\{i}
1
m·W(k) + w(t).
So,
v(i,t)(F,Ψ) = λj
(i,t)(F,Ψ)
for all players
i∈[n]
, types
t∈Ti
, and links
j∈[m]
. Thus,
F
is a fully mixed Bayesian Nash equilibrium.
We will now show the opposite direction. Assume that
F
is a fully mixed
Bayesian Nash Equilibrium. It follows that
supportt(P) = [m]
for all players
i∈[n]
and types
t∈Ti
. Since
F
is a fully mixed Bayesian Nash Equilibrium and
Ψ
is independent, it follows that for all links
j∈supportt(P) = [m]
,
v(i,t)(F,Ψ) = λj
(i,t)(F,Ψ)
=δ−i
j(F,(Ψ|ti=t))+w(t)
=δ−i
j(F,Ψ) + w(t).
So, for all players
i∈[n]
and pair of links
j, l ∈[m]
,
δ−i
j(F,Ψ) = δ−i
l(F,Ψ).
Since
δ−i
j(F,Ψ) = Pk∈[n]\{i}µ(F, k, j)
for any player
i
and link
j
, it follows that
for an arbitrary pair of players
i1, i2∈[n]
with
i16=i2
and an arbitrary pair of
links
j1, j2∈[m]
with
j16=j2
,
X
k∈[n]\{i1}
µ(F, k, j1) = X
k∈[n]\{i1}
µ(F, k, j2)
(6.2)
and
X
k∈[n]\{i2}
µ(F, k, j1) = X
k∈[n]\{i2}
µ(F, k, j2).
(6.3)
Subtracting (6.3) from (6.2) yields that
µ(F, i2, j1)−µ(F, i1, j1) = µ(F, i2, j2)−µ(F, i1, j2),
or equivalently
6.3 Properties of Fully Mixed Bayesian Nash Equilibria 119
0 = µ(F, i2, j1)−µ(F, i2, j2) + µ(F, i1, j2)−µ(F, i1, j1).
Summing up over all players
i2∈[n]\{i1}
yields that
0 = X
i2∈[n]\{i1}
µ(F, i2, j1)−X
i2∈[n]\{i1}
µ(F, i2, j2)
+X
i2∈[n]\{i1}
µ(F, i1, j2)−X
i2∈[n]\{i1}
µ(F, i1, j1)
=δ−i1
j1(F,Ψ)−δ−i1
j2(F,Ψ)+(n−1) ·µ(F, i1, j2)−(n−1) ·µ(F, i1, j1)
= (n−1) ·(µ(F, i1, j2)−µ(F, i1, j1)) .
It follows that for all players
i1∈[n]
and pair of links
j1, j2∈[m]
,
µ(F, i1, j1) = µ(F, i1, j2).
Clearly, for any player
i∈[n]
,
W(i) = X
t∈Ti
Ψ(i, t)·w(t)
=X
σi∈Σi
f(i, σi)X
t∈Ti
Ψ(i, t)·w(t)
=X
σi∈Σi
f(i, σi)X
j∈[m]X
t∈Ti:
σi(t)=j
Ψ(i, t)·w(t)
=X
j∈[m]X
σi∈Σi
f(i, σi)X
t∈Ti:
σi(t)=j
Ψ(i, t)·w(t)
=X
j∈[m]
µ(F, i, j)
=m·µ(F, i, j),
for any link
j∈[m]
. This implies that for all players
i∈[n]
and links
j∈[m]
,
µ(F, i, j) = 1
m·W(i)
or
1
m·W(i) = X
σi∈Σi
f(i, σi)X
t∈Ti:
σi(t)=j
Ψ(i, t)·w(t),
as needed.
We nally determine a lower bound on the dimension of the space of fully mixed
Bayesian Nash equilibria.
Theorem 6.10.
Consider a Bayesian routing game
Γ
on identical links with
independent type distribution. Then, the dimension of the space of fully mixed
Bayesian Nash equilibria for
Γ
is at least
Pi∈[n]mτi−nm
.
120 6 Bayesian Routing Games
Proof.
Let
F
be a fully mixed Bayesian Nash equilibrium. By Theorem 6.9, this
is equivalent to
F
being a fully mixed strategy prole and
X
σi∈Σi
f(i, σi)X
t∈Ti:σi(t)=j
Ψ(i, t)·w(t) = 1
m·W(i)
for all players
i∈[n]
and links
j∈[m]
. So,
F
is a solution to the system of linear
equations and inequalities:
(1) f(i, σi)>0∀i∈[n],∀σi∈Σi
(2) X
σi∈Σi
f(i, σi) = 1 ∀i∈[n]
(3) X
σi∈Σi
f(i, σi)X
t∈Ti:σi(t)=j
Ψ(i, t)·w(t) = 1
m·W(i)∀i∈[n],∀j∈[m].
The dimension of the solution space of this system is the number of variables
minus the number of independent equations. For each player
i∈[n]
we have
mτi
variables. Thus, the total number of variables is
Pi∈[n]mτi
. We now show an
upper bound on the number of independent equations. Fix any player
i∈[n]
.
Summing up the equations (3) for all links
j∈[m]
yields
X
j∈[m]X
σi∈Σi
f(i, σi)X
t∈Ti:σi(t)=j
Ψ(i, t)·w(t) = X
j∈[m]
1
m·W(i)
⇔X
σi∈Σi
f(i, σi)X
t∈Ti
Ψ(i, t)·w(t) = W(i)
⇔X
σi∈Σi
f(i, σi)·W(i) = W(i)
⇔X
σi∈Σi
f(i, σi) = 1
It follows that all equations (2) are implied by a linear combination of equations
in (3). Therefore,
nm
is an upper bound on the number of independent equations.
The claim follows.
6.4 Social Cost and Price of Anarchy
In this section, we present bounds on the price of anarchy for three dierent social
cost measures. All these results are for the case of
identical
links. In Section 6.4.1,
we summarize our results for makespan social cost. In Section 6.4.2, we consider
social cost as sum of private cost and in Section 6.4.3, we present our ndings for
social cost as maximum of private costs.
6.4 Social Cost and Price of Anarchy 121
6.4.1 Makespan Social Cost
In this section, we study the price of anarchy for
makespan social cost
. For the
special case of complete information routing games this social cost measure was
introduced in [67] and asymptotic tight bounds on the price of anarchy were
given by Czumaj and Vöcking [23] and Koutsoupias et al. [66]. Their techniques
use Cherno bounds to show that for identical links the quotient between the
expected maximum load and the maximum expected load on a link is at most
O(log m
log log m)
. We prove that previous techniques cannot be applied to prove an
upper bound on the price of anarchy which is better than
O(m)
.
Proposition 6.11.
For any
> 0
, there is a Bayesian routing game
Γ=
(n, m, 1, T, Ψ)
on identical links with independent type distribution and an as-
sociated pure Bayesian Nash equilibrium
σ
with
SCMSP(Γ, σ) = OPTMSP(Γ)
,
such that for each link
j∈[m],
SCMSP(Γ, σ)
δj(σ,Ψ)≥m
1 + .
Proof.
Set
n=m
. For each player
i∈[n]
, set
Ti={ti, t0
i}
with
w(ti) = 0
and
w(t0
i) = a
; set also for each player
i∈[n]
,
Ψ(i, ti)=1−1
a
and
Ψ(i, t0
i) = 1
a
. Let
σ
be the pure Bayesian Nash equilibrium that maps both types of player
i
to
link
i
, where
i∈[n]
. Since each player is assigned to a dierent link, we have
OPTMSP(Γ) = SCMSP(Γ, σ)
. Clearly, on the one hand,
δj(σ,Ψ)=1
for all links
j∈[m]
. On the other hand,
SCMSP(Γ, σ) = 1−1−1
ama.
Note that
lim
a→∞1−1−1
ama= lim
a→∞ 1−
m
X
i=0 m
i−1
ai!a!
= lim
a→∞ 1−1−
m
X
i=1 m
i(−1)i1
ai!a!
= lim
a→∞ m
X
i=1 m
i(−1)i−11
ai−1!
= lim
a→∞ m+
m
X
i=2 m
i(−1)i−11
ai−1!
=m.
The claim follows.
We now turn our attention to the standard fully mixed Bayesian Nash equi-
librium on identical links. We prove:
122 6 Bayesian Routing Games
Theorem 6.12.
Consider a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on iden-
tical links and an associated standard fully mixed Bayesian Nash equilibrium
F∗
.
Then,
SCMSP(Γ, F∗)
OPTMSP(Γ)=Olog m
log log m.
Proof.
Consider an arbitrary type prole
t= (t1, ..., tn)∈T
. Given
t
, we
dene the game
ΓCI(t) = (n, m, 1,{(t1, ..., tn)},1)
. Recall that for this com-
plete information routing game
ΓCI(t)
, the unique fully mixed Nash equilibrium
P(t)
assigns each player to each link with probability
1
m
(see [74, Lemma 15]).
By [66, Theorem 4.4] or [23, Theorem 1.1], it holds that
SCMSP(ΓCI(t),P(t))
OPTMSP(ΓCI(t)) =Olog m
log log m.
Recall that
F∗
assigns every type agent to every link with probability
1
m
. Thus,
SCMSP(Γ, F∗) = X
t∈T
Ψ(t)·X
(σ1(t1),...,σn(tn))∈[m]n1
mn
·max
j∈[m]
X
i∈[n]:
σi(ti)=j
w(ti)
=X
t∈T
Ψ(t)·SCMSP(ΓCI(t),P(t))
=X
t∈T
Ψ(t)·OPTMSP(ΓCI(t)) ·Olog m
log log m
=OPTMSP(Γ)·Olog m
log log m,
as needed.
Theorem 6.12 implies that for the standard fully mixed Nash equilibrium, incom-
plete information has no impact on the price of anarchy if social cost is taken as
makespan social cost.
Since, in general, there is more than one fully mixed Bayesian Nash equilib-
rium, the natural question arises whether they have all the same makespan social
cost. As we see now, this is not the case.
Proposition 6.13.
There exists a Bayesian routing game
Γ
on identical links
and an associated fully mixed Bayesian Nash equilibrium
F
such that
SCMSP(Γ, F)>SCMSP(Γ, F∗).
Proof.
Consider the Bayesian routing game
Γ= (n, m, 1, T, Ψ)
with
n= 2, m =
3
and
Ti={ti, t0
i}
with
w(ti) = 2, w(t0
i) = 1
for all players
i∈ {1,2}
; set
Ψ(i, ti) = Ψ(i, t0
i) = 1
2
for all players
i∈ {1,2}
. Consider the standard fully
mixed Bayesian Nash equilibrium
F∗
and some other fully mixed Bayesian Nash
equilibrium
F
which we dene below:
6.4 Social Cost and Price of Anarchy 123
•F∗
assigns each type to each link with a probability of
1
3
. Thus, the two
players are assigned to the same link with a probability of
1
3
. In this case, the
maximum latency can be 2, 3, or 4. With a probability of
2
3
, the players are
assigned to dierent links. In this case the maximum latency can be 1 or 2.
Hence, the social cost of the standard fully mixed Bayesian Nash equilibrium
F∗
is
SCMSP(Γ, F∗) = 1
3·1
4·2 + 1
2·3 + 1
4·4+2
3·1
4·1 + 3
4·2=13
6
.
•
The fully mixed strategy prole
F
assigns each type of trac 1 to link 1 with
a probability of
1
2
, to link 2 with a probability of
1
4
and to link 3 with a
probability of
1
4
. Each type of trac 2 is assigned to link 1 with a probability
of
1
4
, to link 2 with a probability of
3
8
and to link 3 with a probability of
3
8
. Observe that for all
i∈ {1,2}
we get
δ−i
1(F) = 1
2·1
2·1 + 1
2·1
4·2 = 1
2
and
δ−i
2(F) = δ−i
3(F) = 1
2·1
4·1 + 1
2·3
8·2 = 1
2
. Thus,
F
is a Bayesian Nash
equilibrium.
With probability
1
4
both players are of trac 1. In this case they use the
same link with probability
(1
2)2+ 2 ·(1
4)2=3
8
. With probability
1
2
, exactly
one of the two players is of trac 1. In this case, the players use the same
link with probability
1
2·1
4+ 2 ·1
4·3
8=5
16
. With the remaining probability
1
4
,
both players are of trac 2. In this case, the players use the same link with
probability
(1
4)2+ 2 ·(3
8)2=11
32
. Hence we get that the social cost of
F
is
SCMSP(Γ, F) = 1
4·3
8·2 + 5
8·1+1
2·5
16 ·3 + 11
16 ·2+1
4·11
32 ·4 + 21
32 ·2
=139
64
.
Observe that
SCMSP(Γ, F) = 139
64 =417
192 >416
192 =13
6=SCMSP(Γ, F∗).
It is known (see [73, Section 8.E]) that mixed Nash equilibria in games with
complete information are related to pure Bayesian Nash equilibria in a Bayesian
game, where for each player all its types are identical. The following denition
and theorem applies this to Bayesian routing games.
Denition 6.14.
A
CI-like game
is a Bayesian routing game with an independent
type distribution such that
w(t) = w(t0)
for all types
t, t0∈Ti
, where
i∈[n]
.
We call these games CI-like games (where CI stands for complete information)
since they are similar to complete information routing games in the sense that the
trac of a player does not depend on its type. For complete information routing
games, there exist asymptotically tight upper bounds on the price of anarchy for
the cases of identical links [23,66] and related links [23]. We use these bounds to
prove:
Theorem 6.15.
Let
Γ= (n, m, c, T, Ψ)
be a CI-like game with an associated
pure Bayesian Nash equilibrium
σ
. Then
(a)
SCMSP(Γ,σ)
OPTMSP(Γ)=Olog m
log log m
, for the case of identical links,
(b)
SCMSP(Γ,σ)
OPTMSP(Γ)=Olog m
log log log m
, for the case of related links,
and there are CI-like games for which both bounds are asymptotically tight.
124 6 Bayesian Routing Games
Proof.
The proof is structured as follows: We rst dene a construction that
maps any CI-like game
Γ
with an associated pure strategy prole
σ
to a com-
plete information routing
ΓCI
with associated (mixed) strategy prole
P
. For
this construction, we show that
SCMSP(Γ, σ) = SCMSP(ΓCI,P)
,
OPTMSP(Γ) =
OPTMSP(ΓCI)
, and that
P
is a Nash equilibrium if
σ
is a Bayesian Nash equilib-
rium. From these properties of our construction, we derive that the corresponding
upper bounds on the price of anarchy [23,66] for complete information routing
games also hold for CI-like games. To prove tightness, we show that for every com-
plete information routing game
ΓCI
with associated (mixed) Nash equilibrium
P
,
we can dene a CI-like game
Γ
with associated pure Bayesian Nash equilibrium
σ
, such that our construction maps
Γ
and
σ
to
ΓCI
and
P
, respectively. This
implies that also the lower bounds on the price of anarchy can be carried over to
the CI-like games.
We start by dening our construction.
Construction
Γ7→ ΓCI
:
Let
Γ= (n, m, c, T, Ψ)
be a CI-like game. For each
i∈[n]
,
denote by
wi=w(t)
the trac of all types
t∈Ti
. Dene a complete information
routing game
ΓCI = (n, m, c, T0,1)
where
T0={(t0
1, . . . , t0
n)}
and
w(t0
i) = wi
for
all
i∈[n]
.
Let
σ= (σ1, . . . , σn)
be a pure strategy prole for the CI-like game
Γ
. Denote
by
Σ0
the set of all pure strategy proles for
ΓCI
; thus,
Σ0=Σ0
1×. . . ×Σ0
n
,
where for each player
i∈[n]
, the set
Σ0
i
consists of all possible pure strategies
σ0
i:{t0
i} → [m]
for player
i
.
Dene a mixed strategy prole
P
for
ΓCI
, where for each player
i∈[n]
and all pure strategies
σ0
i∈Σ0
i
the probability
p(i, σ0
i)
is given by
p(i, σ0
i) =
Pt∈Ti:σi(t)=σ0
i(t0
i)Ψ(i, t)
.
We proceed by showing properties of our construction.
•SCMSP(Γ, σ) = SCMSP(ΓCI,P)
:
To show that the strategy proles
σ
for
Γ
and
P
for
ΓCI
are of the same social cost observe that
SCMSP(Γ, σ)
=X
(t1,...,tn)∈T
Ψ(t1, . . . , tn)·max
j∈[m]
1
cjX
i∈[n],
σi(ti)=j
w(ti)
=X
(t1,...,tn)∈TY
i∈[n]
Ψ(i, ti)·max
j∈[m]
1
cjX
i∈[n],
σi(ti)=j
wi
=X
(σ0
1,...,σ0
n)∈Σ0
X
(t1,...,tn)∈T:
σi(ti)=σ0
i(t0
i)∀i∈[n]Y
i∈[n]
Ψ(i, ti)
·max
j∈[m]
1
cjX
i∈[n],
σ0
i(t0
i)=j
wi
6.4 Social Cost and Price of Anarchy 125
=X
(σ0
1,...,σ0
n)∈Σ0
Y
i∈[n]X
t∈Ti:
σi(t)=σ0
i(t0
i)
Ψ(i, t)
·max
j∈[m]
1
cjX
i∈[n],
σ0
i(t0
i)=j
wi
=X
(σ0
1,...,σ0
n)∈Σ0Y
i∈[n]
p(i, σ0
i)·max
j∈[m]
1
cjX
i∈[n],
σ0
i(t0
i)=j
wi
=SCMSP(ΓCI,P).
•OPTMSP(Γ) = OPTMSP(ΓCI)
:
To show
OPTMSP(Γ)≥OPTMSP(ΓCI)
observe
that our construction maps a pure strategy prole for
Γ
of optimum social
cost to a strategy prole for
ΓCI
that has the same social cost.
For the other direction
OPTMSP(Γ)≤OPTMSP(ΓCI)
, observe that there
always exists a
pure
strategy prole
ˆ
σ0
for
ΓCI
of optimum social cost,
i.e.
SCMSP(ΓCI,ˆ
σ0) = OPTMSP(ΓCI)
. Consider the normal pure strategy prole
ˆ
σ
for
Γ
that assigns for each
i∈[n]
all types of player
i
to the link to that
ˆ
σ0
assigns player
i
, so
ˆσi(t) = ˆσi0(t0
i)
for all players
i∈[n]
and all types
t∈Ti
.
Notice that our construction transforms
Γ
and
ˆ
σ
back to
ΓCI
and
ˆ
σ0
. Thus,
SCMSP(Γ, ˆ
σ) = SCMSP(ΓCI,ˆ
σ0)
. We get that
OPTMSP(Γ)≤SCMSP(Γ, ˆ
σ)
=SCMSP(ΓCI,ˆ
σ0)
=OPTMSP(ΓCI).
•
Mapping of Equilibria:
Clearly, for all players
i∈[n]
, types
t∈Ti
, and links
j∈[m]
,
λj
(i,t)(σ,Ψ)
=1
cj·w(t) + δ−i
j(σ,Ψ)
=1
cj·
w(t) + X
(t1,...,tn)∈T
Ψ(t1, . . . , tn)·X
s∈[n]\{i}:
σs(ts)=j
w(ts)
=1
cj·
wi+X
(t1,...,tn)∈TY
s∈[n]
Ψ(s, ts)·X
s∈[n]\{i}:
σs(ts)=j
ws
=1
cj·
wi+X
(σ0
1,...,σ0
n)∈Σ0
X
(t1,...,tn)∈T:
σs(ts)=σ0
s(t0
s)∀s∈[n]Y
s∈[n]
Ψ(s, ts)
·X
s∈[n]\{i}:
σ0
s(t0
s)=j
ws
126 6 Bayesian Routing Games
=1
cj·
wi+X
(σ0
1,...,σ0
n)∈Σ0
Y
s∈[n]X
ts∈Ts:
σs(ts)=σ0
s(t0
s)
Ψ(s, ts)
·X
s∈[n]\{i}:
σ0
s(t0
s)=j
ws
=1
cj·
w(t0
i) + X
(σ0
1,...,σ0
n)∈Σ0Y
s∈[n]
p(s, σ0
s)·X
s∈[n]\{i}:
σ0
s(t0
s)=j
w(t0
s)
=1
cj·w(t0
i) + δ−i
j(P,1)
=λj
(i,t0
i)(P,1).
We now use this property to show that
P
is a Nash equilibrium for
ΓCI
if
σ
is
a pure Bayesian Nash equilibrium for
Γ
. So, let
σ
be a pure Bayesian Nash
equilibrium for
Γ
. Fix an arbitrary player
i∈[n]
. Remember that in
Γ
all
types of player
i
have the same trac. Thus,
v(i,t)(σ,Ψ) = v(i,ˆ
t)(σ,Ψ)
for all pairs of types
t, ˆ
t∈Ti
. Since
σ
is a pure Bayesian Nash equilibrium for
Γ
, this implies that for all types
t∈Ti
,
v(i,t)(σ,Ψ) = λj
(i,t)(σ,Ψ)
for all
j∈supporti(σ)
and
v(i,t)(σ,Ψ)≤λj
(i,t)(σ,Ψ)
for all
j6∈ supporti(σ).
By denition of
P
,
supporti(σ) = supportt0
i(P).
It follows that
v(i,t0
i)(P,1) = λj
(i,t0
i)(P,1)
for all
j∈supportt0
i(P)
and
v(i,t0
i)(P,1) ≤λj
(i,t0
i)(P,1)
for all
j6∈ supportt0
i(P),
so that
P
is a Nash equilibrium.
Upper bounds on price of anarchy:
Recall that by our construction, we have that
SCMSP(Γ, σ) = SCMSP(ΓCI,P)
and
OPTMSP(Γ) = OPTMSP(ΓCI)
. Thus, resorting
to the corresponding upper bounds on the price of anarchy from [66] and [23],
we get
SCMSP(Γ, σ)
OPTMSP(Γ)=SCMSP(ΓCI,P)
OPTMSP(ΓCI)
=
Olog m
log log m,
for the case of identical links,
Olog m
log log log m,
for the case of related links.
6.4 Social Cost and Price of Anarchy 127
This completes the proof of the upper bounds.
Tightness of the upper bounds:
From [66] and [23], there exist complete informa-
tion routing games
ΓCI
with an associated mixed Nash equilibrium
P
such that
SCMSP(ΓCI,P)
OPTMSP(ΓCI)=
Ωlog m
log log m
, for the case of identical links,
Ωlog m
log log log m
, for the case of related links.
Let
ΓCI = (n, m, c, T0,1)
,
T0={(t0
1, . . . , t0
n)}
, be such a complete information
routing game with an associated mixed Nash equilibrium
P
. With a slight abuse
of notation, we denote
P= (p(i, j))i∈[n],j∈[m]
where
p(i, j)
is the probability that
type
t0
i∈T0
i
is assigned to link
j∈[m]
.
We dene a CI-like game
Γ= (n, m, c, T, Ψ)
and an associated pure strategy
prole
σ
as follows:
For each player
i∈[n]
,
Ti
consists of
|supporti(P)|
types, where we have
a type
tj
i
for every link
j∈supporti(P)
. For all players
i∈[n]
and links
j∈supporti(P)
, dene
Ψ(i, tj
i) = p(i, j)
and
σi(tj
i) = j
.
Notice that our construction
Γ7→ ΓCI
transforms the CI-like game
Γ
with
associated pure strategy prole
σ
back to the complete information routing game
ΓCI
with associated (mixed) Nash equilibrium
P
. It follows that
SCMSP(Γ, σ) =
SCMSP(ΓCI,P)
,
OPTMSP(Γ) = OPTMSP(ΓCI)
and
λl
(i,tj
i)(σ,Ψ) = λl
(i,t0
i)(P,1)
for
all players
i∈[n]
, for all links
l∈[m]
, and for all
j∈supporti(P)
. Since
P
is a
Nash equilibrium we have
v(i,t0
i)(P,1) = λj
(i,t0
i)(P,1)
for all
j∈supportt0
i(P)
and
v(i,t0
i)(P,1) ≤λj
(i,t0
i)(P,1)
for all
j6∈ supportt0
i(P).
Furthermore,
supporti(σ) = supporti(P)
for all
i∈[n]
, and
λl
(i,tj
i)(σ,Ψ) =
λl
(i,t0
i)(P,1)
for all players
i∈[n]
, for all links
l∈[m]
, and for all
j∈supporti(P)
.
It follows that
σ
is a pure Bayesian Nash equilibrium with
SCMSP(Γ, σ)
OPTMSP(Γ)=SCMSP(ΓCI,P)
OPTMSP(ΓCI)
=
Ωlog m
log log m
, for the case of identical links,
Ωlog m
log log log m
, for the case of related links.
This completes the proof.
We conclude with a lower bound on the price of anarchy for social cost as expected
maximum latency if we restrict to normal pure Bayesian Nash equilibria.
Theorem 6.16.
There exists a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on
identical links and an associated normal pure Bayesian Nash equilibrium
σ
such
that
SCMSP(Γ, σ)
OPTMSP(Γ)=Ωlog m
log log m.
128 6 Bayesian Routing Games
Proof.
Let
m∈N
be a perfect square. Consider a Bayesian routing game
Γ=
(n, m, 1, T, Ψ)
on identical links with independent type distribution
Ψ
. There are
two classes of players,
U1
and
U2
:
•
The class
U1
consists of
m
players with type set
Ti={ti, t0
i}
, where
w(ti) =
1, w(t0
i) = 0, Ψ(i, ti) = 1
√m
and
Ψ(i, t0
i) = 1 −1
√m
for all players
i∈ U1
.
•
The class
U2
consists of
(√m−1)m
players with type set
Ti={ti}
, where
w(ti) = 1
√m
and
Ψ(i, ti) = 1
all players
i∈ U2
.
Consider the pure strategy prole
σ0
that assigns to each link one player from
U1
and
√m−1
players from
U2
. By analyzing the social cost of
σ0
, we get
SCMSP(Γ, σ0)≤1+(√m−1) ·1
√m<2.
Now consider the normal pure strategy prole
σ
where
√m
players from
U1
are assigned to each link
j∈[√m]
and
√m
players from
U2
to each of the
remaining
m−√m
links. Clearly,
σ
is a normal pure Bayesian Nash equilibrium.
To show a lower bound on
SCMSP(Γ, σ)
we consider any link
j∈[√m]
. The
actual load, say
Xj
, on link
j∈[√m]
is a random variable which is a sum of
√m
independent random variables with
E(Xj) = 1
. Let
1≤k≤√m, k ∈N
; the
precise choice of
k
will be made later. Clearly,
Pr(Xj≥k)≥Pr(Xj=k)
=√m
k·1
√mk
·1−1
√m√m−k
≥√m
k·1
√mk
·1
e
(since
k≥1
)
=√m·. . . ·(√m−k+ 1)
√mk·1
k!·1
e.
Now, observe that
√m·...·(√m−k+1)
√mk
is monotonically increasing in
√m
and
√m≥
k
. Thus,
√m·. . . ·(√m−k+ 1)
√mk≥k!
kk.
It follows that
Pr(Xj≥k)≥k!
kk·1
k!·1
e
=1
e·kk,
so that
Pr(Xj< k)≤1−1
e·kk.
Now, since the actual loads
X1, . . . , X√m
are independent of each other, we have
6.4 Social Cost and Price of Anarchy 129
Pr((X1< k)∧... ∧(X√m< k)) = Y
j∈[√m]
Pr(Xj< k)
≤1−1
e·kk√m
≤e−1
e·kk·√m.
Dene now
α > 0
so that
α
eα=m
. Then, clearly,
α=Θlog m
log log m
. Choose
k=α
e
. Then
kk=m1
e
which implies
Pr((X1< k)∧... ∧(X√m< k)) ≤e−1
e·kk·√m
=e−1
e·m1
2−1
e
≤1
m,
for suitably large
m
. This implies that
SCMSP(Γ, σ)≥Pr((X1≥k)∨... ∨(X√m≥k)) ·k
=1−Pr((X1< k)∧... ∧(X√m< k))·k
≥1−1
m·α
e
=Θlog m
log log m.
Thus,
SCMSP(Γ, σ)
OPTMSP(Γ)≥SCMSP(Γ, σ)
SCMSP(Γ, σ0)=Ωlog m
log log m,
as needed.
6.4.2 Social Cost as Sum of Private Costs
In this section, we study the price of anarchy for social cost as the sum of private
costs.
Theorem 6.7 implies that fully mixed Bayesian Nash equilibria maximize social
cost as sum of private costs. Hence, we obtain:
Theorem 6.17.
Consider a Bayesian routing game
Γ
on identical links and an
associated fully mixed Bayesian Nash equilibrium
F
and a Bayesian Nash equi-
librium
P
. Then,
SCSUM(Γ, P)≤SCSUM(Γ, F).
We now use Theorem 6.17 to prove an asymptotically tight bound on the price
of anarchy for the case of identical links (and social cost as sum of private costs).
130 6 Bayesian Routing Games
Theorem 6.18.
Consider a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on iden-
tical links and an associated Bayesian Nash equilibrium
P
. Then,
SCSUM(Γ, P)
OPTSUM(Γ)≤m+n−1
m,
and this bound is tight up to a factor of
(1 + )
for any
> 0
, even if
Γ
is a
complete information routing game.
Proof.
By Theorem 6.17, it suces to prove the upper bound for a fully mixed
Bayesian Nash equilibrium
F
. Clearly, on the one hand,
SCSUM(Γ, F) = X
i∈[n]
PCi(F,Ψ)
=X
i∈[n]W
m+m−1
mW(i)
(by Proposition 6.6)
=nW
m+m−1
mW
=m+n−1
mW.
On the other hand,
PCi(P,Ψ)≥W(i)
for any player
i∈[s]
and any strategy
prole
P
; hence,
OPTSUM(Γ)≥X
i∈[n]
W(i) = W.
The upper bound follows.
We now prove that this upper bound is tight even for complete information
routing games. To do so, we will prove that for any
ε > 0
, there is a complete
information routing game
ΓCI = (n, m, 1, T, 1)
such that
OPTSUM(ΓCI)≤(1 + ε)·W.
We proceed by case analysis on the relation between
n
and
m
.
•
Assume rst that
n≤m
. Let
ΓCI
be an arbitrary complete information routing
game with
n≤m
. Then we can assign each player to a separate link which
yields
OPTSUM(ΓCI) = W
.
•
Assume now that
n > m
. Dene the complete information routing game
ΓCI
as follows:
There are two sets of players
U1
, and
U2
. The set
U1
consists of
n−m+1
players with
w(ti)=1
for all
i∈ U1
, and
U2
consists of
m−1
players
with
w(ti) = k
for all
i∈ U2
where
k∈N
is a constant to be determined
later.
6.4 Social Cost and Price of Anarchy 131
For the (expected) total trac, we get
W=n−m+1+(m−1)k.
Let
σ
be the pure strategy prole that assigns all players from
U1
to link
m
and each of the
m−1
players from
U2
separately to a link from
[m−1]
. Thus,
OPTSUM(ΓCI)≤SCSUM(ΓCI,σ)
= (n−m+ 1)2+ (m−1)k
=(n−m+ 1)2+ (m−1)k
n−m+1+(m−1)k·W
=(n−m+ 1) ·(n−m)+(n−m+1)+(m−1) ·k
n+ (m−1)(k−1) ·W
=1 + (n−m)(n−m+ 1)
n+ (m−1)(k−1) ·W.
Clearly, for any
ε > 0
, there is a
k∈N
such that
(n−m)(n−m+1)
n+(m−1)(k−1) ≤ε
. Hence,
for any
ε > 0
, there is a complete information routing game
ΓCI
such that
OPTSUM(ΓCI)≤(1 + ε)·W
. This completes the proof for the case
n>m
.
In all cases, there is a complete information routing game
ΓCI
such that
OPTSUM(ΓCI)≤(1 + ε)·W
. Since
SCSUM(ΓCI,F) = m+n−1
mW
, it follows that
SCSUM(ΓCI,F)
OPTSUM(ΓCI)≥1
1 + ε·m+n−1
m,
as needed.
Berenbrink et al. [11] showed that the price of anarchy for complete information
routing games and social cost as sum of private costs grows at least linearly with
the number of players. In particular they proved that
n
5
is a lower bound on the
price of anarchy. Theorem 6.18 implies that the price of anarchy increases at most
linear with
n
and also shows the impact of the number of links.
Another interesting insight of Theorem 6.18 is that the price of anarchy does
not
increase if we allow incomplete information. This is not the case if social cost
is dened as the maximum of private costs, as we will see next.
6.4.3 Social Cost as Maximum of Private Costs
In this section, we study the price of anarchy for social cost as the maximum of
private costs.
Theorem 6.7 implies that fully mixed Bayesian Nash equilibria maximize social
cost as maximum of private costs. Hence, we obtain:
Theorem 6.19.
Consider a Bayesian routing game
Γ
on identical links and an
associated fully mixed Bayesian Nash equilibrium
F
and a Bayesian Nash equi-
librium
P
. Then,
SCMAX(Γ, P)≤SCMAX(Γ, F).
132 6 Bayesian Routing Games
We now use Theorem 6.19 to prove asymptotically tight bounds on the price
of anarchy for the case of identical links.
Theorem 6.20.
Consider a Bayesian routing game
Γ= (n, m, 1, T, Ψ)
on iden-
tical links and an associated Bayesian Nash equilibrium
P
. Then,
(a)
SCMAX(Γ, P)
OPTMAX(Γ)≤m+n−1
m,
and
(b)
SCMAX(Γ, P)
OPTMAX(Γ)≤2−1
m,
if
Γ
is a complete information routing game.
The bound from (a) is tight up to a factor of
(1 + )
for any
> 0
and the bound
from
(b)
is tight.
Proof.
Let
F
be a fully mixed Bayesian Nash equilibrium for
Γ
. Theorem 6.6
and Theorem 6.7 imply together that
PCi(P,Ψ)≤PCi(F,Ψ) = W
m+m−1
mW(i),
(6.4)
for each player
i∈[n]
. We consider the two cases from the theorem.
Case (a):
Upper bound:
Clearly, for any strategy prole
P0
and for any player
i∈[n]
,
PCi(P0,Ψ)≥W(i)
; hence,
Pi∈[n]PCi(P0,Ψ)≥W
. This implies that
OPTMAX(Γ)≥W
n.
(6.5)
Clearly,
OPTMAX(Γ)≥W(i)
for all
i∈[n]
. Fix any player
i∈[n]
. By (6.4) and
(6.5),
PCi(P,Ψ)≤W
m+m−1
m·OPTMAX(Γ)
≤n
m·OPTMAX(Γ) + m−1
m·OPTMAX(Γ)
=m+n−1
m·OPTMAX(Γ).
and the upper bound follows.
Lower bound:
Fix any arbitrary
k, a, r ∈N
, which will be determined later. Con-
sider the Bayesian routing game
Γk,a,r = (n, m, 1, T, Ψ)
with independent type
distribution and
n=k·(m−1)
players. Each player
i∈[n]
has type set
Ti={ti, t0
i}
with trac
w(ti) = 1
,
w(t0
i) = a·r
and probabilities
Ψ(i, ti) = 1 −1
a
,
Ψ(i, t0
i) = 1
a
. Clearly, for player
i∈[n]
,
W(i) = r+ 1 −1
a
.
Dene a pure strategy prole
σ
that assigns all types
t0
i
,
i∈[n]
, of trac 1 to
link
m
. The types
ti
,
i∈[n]
, are evenly distributed among the links in
[m−1]
;
so,
σ
assigns exactly
k
of these types to each link in
[m−1]
. Now for each player
i∈[n]
,
6.4 Social Cost and Price of Anarchy 133
PCi(σ,Ψ) = 1−1
a·1+(k−1) ·1−1
a+1
a·((n−1)r+r·a)
=1−1
a·1
a+k·1−1
a+r·(n−1)
a+ 1;
so, for any
0>0
, there is a suciently large
a
such that for each player
i∈[n]
,
PCi(σ,Ψ)≤(k+r)(1 + 0).
Hence,
OPTMAX(Γk,a,r)≤(k+r)(1+0).
Fix now any fully mixed Bayesian Nash
equilibrium
F
. Theorem 6.6 implies that for each player
i∈[n]
,
PCi(F,Ψ) = 1 + n−1
m·W(i)
=m+n−1
m·r+ 1 −1
a.
Thus,
SCMAX(Γk,a,r,F) = m+n−1
m·r+ 1 −1
a
and we can conclude that
SCMAX(Γk,a,r,F)
OPTMAX(Γk,a,r)≥(r+ 1 −1
a)
(k+r)(1 + 0)·m+n−1
m.
So, for any
> 0
, there is a suciently large
r
such that
SCMAX(Γk,a,r,F)
OPTMAX(Γk,a,r)≥m+n−1
m·1
1 + .
This proves that the upper bound shown before is tight up to a factor of
(1 + )
.
Case (b):
Upper bound:
Consider the complete information routing game
ΓCI = (n, m, 1,{(t1, . . . , tn)},1)
.
Here,
W(i) = w(ti)
for all
i∈[n]
. Clearly,
OPTMAX(ΓCI)≥W(i)
for all
i∈[n]
and
OPTMAX(ΓCI)≥W
m
. By Equation (6.4),
PCi(P,Ψ)≤W
m+m−1
mW(i)
≤OPTMAX(ΓCI) + m−1
mOPTMAX(ΓCI)
=2−1
mOPTMAX(ΓCI),
so that
SCMAX(Γ, P)
OPTMAX(Γ)≤2−1
m
as needed. The upper bound follows.
Lower bound:
Consider the complete information routing game
ΓCI = (n, m, 1,{(t1, . . . , tn)},1)
134 6 Bayesian Routing Games
with
n=m
and
w(t1) = . . . =w(tn)=1
. Clearly,
W(i) = w(ti)=1
for
all
i∈[n]
,
W=m
and
OPTMAX(ΓCI)=1
. Now, for the fully mixed Nash
equilibrium
F
and any player
i∈[n]
, by Equation (6.4),
PCi(F,Ψ) = W
m+m−1
mW(i)
= (2 −1
m)·OPTMAX(ΓCI),
so that
SCMAX(Γ, F)
OPTMAX(Γ)= 2 −1
m,
as needed.
6.5 Conclusion and Discussion
In this chapter, we have introduced the dimension of incomplete information into
the class of routing games on parallel links. For this setting, we have studied
the existence and computational complexity of pure Bayesian Nash equilibria,
structural properties of fully mixed Bayesian Nash equilibria and the price of
anarchy for dierent social cost measures.
Our work leaves open several interesting problems. On the most concrete level,
we would like to ask:
•
Can pure Bayesian Nash equilibria be computed in polynomial time?
•
What is the exact value of the price of anarchy for identical links if social cost
is dened as expected maximum latency?
•
What is the price of anarchy for all three considered social cost measures in
the case of related links?
References
[1] H. Ackermann, H. Röglin, and B. Vöcking. On the Impact of Combina-
torial Structure on Congestion Games. In
Proceedings of the 47th Annual
Symposium on Foundations of Computer Science (FOCS'06)
, 2006.
[2] H. Ackermann, H. Röglin, and B. Vöcking. Pure Nash Equilibria in Player-
Specic and Weighted Congestion Games. In
Proceedings of the 2nd Inter-
national Workshop on Internet and Network Economics (WINE'06)
, 2006.
[3] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin.
Network Flows: Theory,
Algorithms, and Applications
. Prentice Hall, 1993.
[4] S. Aland, D. Dumrauf, M. Gairing, B. Monien, and F. Schoppmann. Ex-
act Price of Anarchy for Polynomial Congestion Games. In B. Durand
and W. Thomas, editors,
Proceedings of the 23rd International Symposium
on Theoretical Aspects of Computer Science (STACS'06)
, Lecture Notes in
Computer Science, Vol. 3884, Springer Verlag, pages 218229, 2006.
[5] B. Awerbuch, Y. Azar, and A. Epstein. The Price of Routing Unsplittable
Flow. In
Proceedings of the 37th Annual ACM Symposium on Theory of
Computing (STOC'05)
, pages 5766, 2005.
[6] B. Awerbuch, Y. Azar, Y. Richter, and D. Tsur. Tradeos in Worst-
Case Equilibria. In K. Jansen and R. Solis-Oba, editors,
Proceedings of
the 1st International Workshop on Approximation and Online Algorithms
(WAOA'03)
, Lecture Notes in Computer Science, Vol. 2909, Springer Ver-
lag, pages 4152, 2003.
[7] M. Beckmann, C. B. McGuire, and C. B. Winsten.
Studies in the Economics
of Transportation
. Yale University Press, 1956.
[8] M. J. Beckmann. On the Theory of Trac Flow in Networks.
Trac Quar-
terly
, 21:109116, 1967.
[9] R. Beier, A. Czumaj, P. Krysta, and B. Vöcking. Computing Equilibria for
Congestion Games with (Im)perfect Information. In
Proceedings of the 15th
Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'04)
, pages
746755, 2004.
[10] E. T. Bell. Exponential Polynomials.
Annals of Mathematics
, 35:258277,
1934.
136 References
[11] P. Berenbrink, L. A. Goldberg, P. W. Goldberg, and R. Martin.
Utilitarian Resource Assignment.
Journal of Discrete Algorithms
,
(http://dx.doi.org/10.1016/j.jda.2005.06.009), to appear.
[12] J. O. Berger.
Statistical Decision Theory and Bayesian Analysis
. Second
Edition. Springer Verlag, 1980.
[13] D. Braess. Über ein Paradoxon der Verkehrsplanung.
Un-
ternehmensforschung
, 12:258268, 1968.
[14] P. Brucker, J. Hurink, and F. Werner. Improving Local Search Heuristics
for Some Scheduling Problems. Part II.
Discrete Applied Mathematics
, 72
(12):4769, 1997.
[15] X. Chen and X. Deng. Settling the Complexity of 2-Player Nash-Equilibrium.
In
Electronic Colloquium in Computational Complexity
, TR05-140, 2005.
[16] G. Christodoulou and E. Koutsoupias. On The Price of Anarchy and Sta-
bility of Correlated Equilibria of Linear Congestion Games. In S. Leonardi
G. S. Brodal, editor,
Proceedings of the 13th Annual European Symposium on
Algorithms (ESA'05)
, Lecture Notes in Computer Science, Vol. 3669 Springer
Verlag, pages 5970, 2005.
[17] G. Christodoulou and E. Koutsoupias. The Price of Anarchy of Finite Con-
gestion Games. In
Proceedings of the 37th Annual ACM Symposium on
Theory of Computing (STOC'05)
, pages 6773, 2005.
[18] R. Cole, Y. Dodis, and T. Roughgarden. Bottleneck Links, Variable Demand,
and the Tragedy of the Commons. In
Proceedings of the 17th Annual ACM-
SIAM Symposium on Discrete Algorithms (SODA'06)
, pages 668677, 2006.
[19] R. Comminetti, J. R. Correa, and N. E. Stier-Moses. Network Games with
Atomic Players. In M. Bugliesi, B. Preneel, V. Sassone, and I. Wegener, ed-
itors,
Proceedings of the 33rd International Colloquium on Automata, Lan-
guages, and Programming (ICALP'06), Part I
, Lecture Notes in Computer
Science, Vol. 4051, Springer Verlag, pages 525536, 2006.
[20] J. R. Correa, A. S. Schulz, and N. E. Stier Moses. Selsh Routing in Ca-
pacitated Networks.
Mathematics of Operations Research
, 29(4):961976,
2004.
[21] A. Czumaj. Selsh Routing on the Internet. In J. Leung, editor,
Handbook
of Scheduling: Algorithms, Models, and Performance Analysis
. CRC Press,
2004.
[22] A. Czumaj, P. Krysta, and B. Vöcking. Selsh Trac Allocation for Server
Farms. In
Proceedings of the 34th Annual ACM Symposium on Theory of
Computing (STOC'02)
, pages 287296, 2002.
[23] A. Czumaj and B. Vöcking. Tight Bounds for Worst-Case Equilibria. In
Pro-
ceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA'02)
, pages 413420, 2002. Also accepted to
Journal of Algorithms
as
Special Issue of SODA'02.
[24] D. Dumrauf and M. Gairing. Price of Anarchy for Polynomial Wardrop
Games. In
Proceedings of the 2nd International Workshop on Internet and
Network Economics (WINE'06)
, 2006.
References 137
[25] R. Elsässer, M. Gairing, T. Lücking, M. Mavronicolas, and B. Monien. A
Simple Graph-Theoretic Model for Selsh Restricted Scheduling. In X. Deng
and Y. Ye, editors,
Proceedings of the 1st International Workshop on Internet
and Network Economics (WINE'05)
, Lecture Notes in Computer Science,
Vol. 3828, Springer Verlag, pages 195209, 2005.
[26] E. Even-Dar, A. Kesselmann, and Y. Mansour. Convergence Time to Nash
Equilibria. In J. C. M. Baeten, J. K. Lenstra, J. Parrow, and G. J. Woegin-
ger, editors,
Proceedings of the 30th International Colloquium on Automata,
Languages, and Programming (ICALP'03)
, Lecture Notes in Computer Sci-
ence, Vol. 2719, Springer Verlag, pages 502513, 2003.
[27] A. Fabrikant, C. H. Papadimitriou, and K. Talwar. The Complexity of Pure
Nash Equilibria. In
Proceedings of the 36th Annual ACM Symposium on
Theory of Computing (STOC'04)
, pages 604612, 2004.
[28] G. Facchini, F. van Megan, P. Borm, and S. Tijs. Congestion Models and
Weighted Bayesian Potential Games.
Theory and Decision
, 42:193206,
1997.
[29] J. Feigenbaum, C. H. Papadimitriou, and S. Shenker. Sharing the Cost of
Multicast Transmissions.
Journal of Computer and System Sciences
, 63(1):
2141, 2001.
[30] R. Feldmann, M. Gairing, T. Lücking, B. Monien, and M. Rode. Nashica-
tion and the Coordination Ratio for a Selsh Routing Game. In J. C. M.
Baeten, J. K. Lenstra, J. Parrow, and G. J. Woeginger, editors,
Proceedings
of the 30th International Colloquium on Automata, Languages, and Program-
ming (ICALP'03)
, Lecture Notes in Computer Science, Vol. 2719, Springer
Verlag, pages 514526, 2003.
[31] R. Feldmann, M. Gairing, T. Lücking, B. Monien, and M. Rode. Selsh
Routing in Non-Cooperative Networks: A Survey. In B. Rovan and P. Vojtás,
editors,
Proceedings of the 28th International Symposium on Mathematical
Foundations of Computer Science (MFCS'03)
, Lecture Notes in Computer
Science, Vol. 2747, Springer Verlag, pages 2145, 2003.
[32] G. Finn and E. Horowitz. A Linear Time Approximation Algorithm for
Multiprocessor Scheduling.
BIT
, 19(3):312320, 1979.
[33] S. Fischer, H. Räcke, and B. Vöcking. Fast Convergence to Wardrop Equilib-
ria by Adaptive Sampling Methods. In
Proceedings of the 38th Annual ACM
Symposium on Theory of Computing (STOC'06)
, pages 653662, 2006.
[34] S. Fischer and B. Vöcking. On the Evolution of Selsh Routing. In T. Radzik
S. Albers, editor,
Proceedings of the 13th Annual European Symposium on
Algorithms (ESA'04)
, Lecture Notes in Computer Science, Vol. 3221 Springer
Verlag, pages 323334, 2004.
[35] S. Fischer and B. Vöcking. Adaptive Routing with Stale Information. In
Proceedings of the 24th Annual ACM Symposium on Principles of Distributed
Computing (PODC'05)
, pages 276283, 2005.
[36] S. Fischer and B. Vöcking. On the Structure and Complexity of Worst-Case
Equilibria. In X. Deng and Y. Ye, editors,
Proceedings of the 1st International
138 References
Workshop on Internet and Network Economics (WINE'05)
, Lecture Notes
in Computer Science, Vol. 3828, Springer Verlag, pages 151160, 2005.
[37] D. Fotakis, S. Kontogiannis, E. Koutsoupias, M. Mavronicolas, and P. Spi-
rakis. The Structure and Complexity of Nash Equilibria for a Selsh Routing
Game. In P. Widmayer, F. T. Ruiz, R. M. Bueno, M. Hennessy, S. Eidenbenz,
and R. Conejo, editors,
Proceedings of the 29th International Colloquium
on Automata, Languages, and Programming (ICALP'02)
, Lecture Notes in
Computer Science, Vol. 2380, Springer Verlag, pages 123134, 2002.
[38] D. Fotakis, S. Kontogiannis, and P. Spirakis. Selsh Unsplittable Flows. In
J. Diaz, J. Karhumäki, A. Lepistö, and D. Sannella, editors,
Proceedings of
the 31st International Colloquium on Automata, Languages, and Program-
ming (ICALP'04)
, Lecture Notes in Computer Science, Vol. 3142, Springer
Verlag, pages 593605, 2004.
[39] D. Fotakis, S. Kontogiannis, and P. Spirakis. Symmetry in Network Con-
gestion Games: Pure Equilibria and Anarchy Cost. In
Proceedings of
the 3rd International Workshop on Approximation and Online Algorithms
(WAOA'05)
, 2005.
[40] D. K. Friesen. Tighter Bounds for LPT Scheduling on Uniform Processors.
SIAM Journal on Computing
, 16(3):554560, 1987.
[41] M. Gairing, T. Lücking, M. Mavronicolas, and B. Monien. Computing Nash
Equilibria for Scheduling on Restricted Parallel Links. In
Proceedings of the
36th Annual ACM Symposium on Theory of Computing (STOC'04)
, pages
613622, 2004.
[42] M. Gairing, T. Lücking, M. Mavronicolas, and B. Monien. The Price of An-
archy for Polynomial Social Cost. In J. Fiala, V. Koubek, and J. Kratochvíl,
editors,
Proceedings of the 29th International Symposium on Mathematical
Foundations of Computer Science (MFCS'04)
, Lecture Notes in Computer
Science, Vol. 3153, Springer Verlag, pages 574585, 2004.
[43] M. Gairing, T. Lücking, M. Mavronicolas, and B. Monien. The Price of
Anarchy for Polynomial Social Cost.
Theoretical Computer Science
, 369:
116135, 2006.
[44] M. Gairing, T. Lücking, M. Mavronicolas, and B. Monien. The Price of
Anarchy for Restricted Parallel Links.
Parallel Processing Letters
, 16(1):
117131, 2006.
[45] M. Gairing, T. Lücking, M. Mavronicolas, B. Monien, and M. Rode. Nash
Equilibria in Discrete Routing Games with Convex Latency Functions. In
J. Diaz, J. Karhumäki, A. Lepistö, and D. Sannella, editors,
Proceedings of
the 31st International Colloquium on Automata, Languages, and Program-
ming (ICALP'04)
, Lecture Notes in Computer Science, Vol. 3142, Springer
Verlag, pages 645657, 2004.
[46] M. Gairing, T. Lücking, M. Mavronicolas, B. Monien, and P. Spirakis. Ex-
treme Nash Equilibria. In C. Blundo and C. Laneve, editors,
Proceedings
of the 8th Italian Conference on Theoretical Computer Science (ICTCS'03)
,
References 139
Lecture Notes in Computer Science, Vol. 2841, Springer Verlag, pages 120,
2003.
[47] M. Gairing, T. Lücking, M. Mavronicolas, B. Monien, and P. Spirakis. Struc-
ture and Complexity of Extreme Nash Equilibria.
Theoretical Computer
Science
, 343:133157, 2005.
[48] M. Gairing, T. Lücking, B. Monien, and K. Tiemann. Nash Equilibria, the
Price of Anarchy and the Fully Mixed Nash Equilibrium Conjecture. In
L. Caires, G. F. Italiano, L. Monteiro, C. Palamidessi, and M. Yung, editors,
Proceedings of the 32nd International Colloquium on Automata, Languages,
and Programming (ICALP'05)
, Lecture Notes in Computer Science, Vol.
3580, Springer Verlag, pages 5165, 2005.
[49] M. Gairing, B. Monien, and K. Tiemann. Selsh Routing with Incomplete
Information. In
Proceedings of the 17th Annual ACM Symposium on Par-
allel Algorithms and Architectures (SPAA'05)
, pages 203212, 2005. Also
accepted to
Theory of Computing Systems
.
[50] M. Gairing, B. Monien, and K. Tiemann. Routing (Un-) Splittable Flow
in Games with Player-Specic Linear Latency Functions. In M. Bugliesi,
B. Preneel, V. Sassone, and I. Wegener, editors,
Proceedings of the
33rd International Colloquium on Automata, Languages, and Programming
(ICALP'06), Part I
, Lecture Notes in Computer Science, Vol. 4051, Springer
Verlag, pages 501512, 2006.
[51] M. Gairing, B. Monien, and A. Woclaw. A Faster Combinatorial Approxi-
mation Algorithm for Scheduling Unrelated Parallel Machines. In L. Caires,
G. F. Italiano, L. Monteiro, C. Palamidessi, and M. Yung, editors,
Proceed-
ings of the 32nd International Colloquium on Automata, Languages, and
Programming (ICALP'05)
, Lecture Notes in Computer Science, Vol. 3580,
Springer Verlag, pages 828839, 2005. Also accepted to
Theoretical Computer
Science
.
[52] M. R. Garey and D. S. Johnson.
Computers and Intractability: A Guide to
the Theory of NP-Completeness
. W.H. Freeman and Company, 1979.
[53] C. Georgiou, T. Pavlides, and A. Philippou. Network Uncertainty in Self-
ish Routing. In
Proceedings of the 20th IEEE International Parallel &
Distributed Processing Symposium (IPDPS'06), CD-ROM Proceedings
, page
105, 2006.
[54] P. W. Goldberg. Bounds on the Convergnece Rate of Randomized Local
Search in a Multiplayer Load-balancing Game. In
Proceedings of the 23th An-
nual ACM Symposium on Principles of Distributed Computing (PODC'04)
,
pages 131140, 2004.
[55] G. H. Gonnet. Expected Length of the Longest Probe Sequence in Hash
Code Searching.
Journal of the ACM
, 28(2):289304, 1981.
[56] R. L. Graham. Bounds on Multiprocessing Timing Anomalies.
SIAM Journal
of Applied Mathematics
, 17(2):416429, 1969.
[57] R. L. Graham, D. E. Knuth, and O. Patashnik.
Concrete Mathematics: A
Foundation for Computer Science
. Addison-Wesley, 1994.
140 References
[58] J. C. Harsanyi. Games with Incomplete Information Played by Bayesian
Players, I, II, III.
Management Science
, 14:159182, 320332, 468502, 1967.
[59] J. C. Harsanyi. Games with Randomly Disturbed Payos.
Internation Jour-
nal on Game Theory
, 21:123, 1973.
[60] J. Havil.
Gamma: Exploring Euler's Constant
. Princeton University Press,
2003.
[61] D. S. Hochbaum and D. B Shmoys. Using Dual Approximation Algorithms
for Scheduling Problems: Theoretical and Practical Results.
Journal of the
ACM
, 34(1):144162, 1987.
[62] D. S. Hochbaum and D. B. Shmoys. A Polynomial Approximation Scheme
for Scheduling on Uniform Processors: Using the Dual Approximation Ap-
proach.
SIAM Journal on Computing
, 17(3):539551, 1988.
[63] K. Jain and V. Vazirani. Applications of Approximation Algorithms to Co-
operative Games. In
Proceedings of the 33rd Annual ACM Symposium on
Theory of Computing (STOC'01)
, pages 364372, 2001.
[64] R. M. Karp. Reducibility Among Combinatorial Problems. In R. E. Miller
and J. W. Thatcher, editors,
Complexity of Computer Computations
, Plenum
Press, pages 85103, 1972.
[65] E. Koutsoupias. Selsh Task Allocation.
Bulletin of the EATCS
, 81:7988,
2003.
[66] E. Koutsoupias, M. Mavronicolas, and P. Spirakis. Approximate Equilibria
and Ball Fusion.
Theory of Computing Systems
, 36(6):683693, 2003.
[67] E. Koutsoupias and C. H. Papadimitriou. Worst-Case Equilibria. In
C. Meinel and S. Tison, editors,
Proceedings of the 16th International Sym-
posium on Theoretical Aspects of Computer Science (STACS'99)
, Lecture
Notes in Computer Science, Vol. 1563, Springer Verlag, pages 404413, 1999.
[68] L. Libman and A. Orda. The Designer's Perspective to Atomic Noncoop-
erative Networks.
IEEE/ACM Transactions on Networking
, 7(6):875884,
1999.
[69] L. Libman and A. Orda. Atomic Resource Sharing in Noncooperative Net-
works.
Telecommunication Systems
, 17(4):385409, 2001.
[70] T. Lücking. Analysing Models for Scheduling and Routing, PhD-thesis, Uni-
versity of Paderborn, 2005.
[71] T. Lücking, M. Mavronicolas, B. Monien, and M. Rode. A New Model for
Selsh Routing. In V. Diekert and M. Habib, editors,
Proceedings of the
21st International Symposium on Theoretical Aspects of Computer Science
(STACS'04)
, Lecture Notes in Computer Science, Vol. 2996, Springer Verlag,
pages 547558, 2004.
[72] T. Lücking, M. Mavronicolas, B. Monien, M. Rode, P. Spirakis, and I. Vrto.
Which is the Worst-Case Nash Equilibrium? In B. Rovan and P. Vojtás,
editors,
Proceedings of the 28th International Symposium on Mathematical
Foundations of Computer Science (MFCS'03)
, Lecture Notes in Computer
Science, Vol. 2747, Springer Verlag, pages 551561, 2003.
References 141
[73] A. Mas-Colell, M. D. Whinston, and J. R. Green.
Microeconomic Theory
.
Oxford University Press, 1995.
[74] M. Mavronicolas and P. Spirakis. The Price of Selsh Routing. In
Proceedings
of the 33rd Annual ACM Symposium on Theory of Computing (STOC'01)
,
pages 510519, 2001.
[75] V. Mazalov, B. Monien, F. Schoppmann, and K. Tiemann. Wardrop Equi-
libria and Prize of Stability for Bottleneck Games with Splittable Trac.
In
Proceedings of the 2nd International Workshop on Internet and Network
Economics (WINE'06)
, 2006.
[76] R. D. McKelvey and A. McLennan. Computation of Equilibria in Finite
Games. In H. Amman, D. Kendrick, and J. Rust, editors,
Handbook of
Computational Economics
, 1996.
[77] I. Milchtaich. Congestion Games with Player-Specic Payo Functions.
Games and Economic Behavior
, 13(1):111124, 1996.
[78] D. Monderer and L. S. Shapley. Potential Games.
Games and Economic
Behavior
, 14(1):124143, 1996.
[79] R. B. Myerson.
Game Theory: Analysis of Conict
. Harvard University
Press, 1997.
[80] J. F. Nash. Equilibrium Points in
n
-Person Games.
Proceedings of the
National Academy of Sciences of the United States of America
, 36:4849,
1950.
[81] J. F. Nash. Non-Cooperative Games.
Annals of Mathematics
, 54(2):286295,
1951.
[82] N. Nisan. Algorithms for Selsh Agents. In C. Meinel and S. Tison, editors,
Proceedings of the 16th International Symposium on Theoretical Aspects of
Computer Science (STACS'99)
, Lecture Notes in Computer Science, Vol.
1563, Springer Verlag, pages 115, 1999.
[83] N. Nisan and A. Ronen. Algorithmic Mechanism Design. In
Proceedings
of the 31st Annual ACM Symposium on Theory of Computing (STOC'99)
,
pages 129140, 1999.
[84] A. Orda, R. Rom, and N. Shimkin. Competitive Routing in Multiuser Com-
munication Networks.
IEEE/ACM Transactions on Networking
, 1(5):510
521, 1993.
[85] M.J. Osborne and A. Rubinstein.
A Course in Game Theory
. MIT Press,
1994.
[86] C. H. Papadimitriou. Algorithms, Games, and the Internet. In
Proceedings
of the 33rd Annual ACM Symposium on Theory of Computing (STOC'01)
,
pages 749753, 2001.
[87] A. C. Pigou.
The Economics of Welfare
. Macmillan and Company, 1920.
[88] R. W. Rosenthal. A Class of Games Possessing Pure-Strategy Nash Equi-
libria.
International Journal of Game Theory
, 2:6567, 1973.
[89] T. Roughgarden. The Price of Anarchy is Independent of the Network Topol-
ogy.
Journal of Computer and System Sciences
, 67(2):341364, 2003.
142 References
[90] T. Roughgarden.
Selsh Routing and the Price of Anarchy
. MIT Press,
2005.
[91] T. Roughgarden. Selsh Routing with Atomic Players. In
Proceedings of the
16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'05)
,
pages 11841185, 2005.
[92] T. Roughgarden and É. Tardos. How Bad Is Selsh Routing?
Journal of the
ACM
, 49(2):236259, 2002.
[93] P. Schuurman and T. Vredeveld. Performance Guarantees of Local Search
for Multiprocessor Scheduling. In K. Aardal and B. Gerards, editors,
Pro-
ceedings of the 8th Conference on Integer Programming and Combinatorial
Optimization (IPCO'01)
, Lecture Notes in Computer Science, Vol. 2081,
Springer Verlag, pages 370382, 2001.
[94] J. Stirling. Methodus Dierentialis Sive Tractatus de Summatione et Inter-
polatione Serierum Innitarum.
London
, 1730.
[95] S. Suri, C. D. Tóth, and Y. Zhou. Selsh Load Balancing and Atomic
Congestion Games. In
Proceedings of the 16th Annual ACM Symposium on
Parallel Algorithms and Architectures (SPAA'04)
, pages 188195, 2004.
[96] M. Voorneveld, P Borm, F. van Megan, S. Tijs, and G. Facchini. Congestion
Games and Potentials Reconsidered.
International Game Theory Review
, 1:
283299, 1999.
[97] J. G. Wardrop. Some Theoretical Aspects of Road Trac Research. In
Proceedings of the Institute of Civil Engineers, Pt. II, Vol. 1
, pages 325378,
1952.
