Exact Price of Anarchy for Polynomial
Congestion Games∗†
Sebastian Aland1
[email protected]
Dominic Dumrauf1
[email protected]
Martin Gairing2‡
[email protected]
Burkhard Monien1
[email protected]
Florian Schoppmann1§
[email protected]
Sunday 29th May, 2011
1
Department of Computer Science, Electrical Engineering, and
Mathematics,
University of Paderborn, Fürstenallee 11, 33102 Paderborn, Germany
2
Computer Science Department, University of Liverpool,
Ashton Building, Ashton Street, Liverpool L69 3BX, U.K.
Abstract.
We show exact values for the worst-case price of anarchy in weighted
and unweighted (atomic unsplittable) congestion games, provided that all cost functions are bounded-degree polynomials with nonnegative coefficients. The given values
also hold for weighted and unweighted network congestion games.
Keywords.
chy
atomic unsplittable congestion games, selfish routing, price of anar-
AMS subject classification.
∗
†
‡
§
68Q99, 90B18, 91A10, 91A43
A preliminary version [1] of this paper appeared in the Proceedings of the 23rd International Symposium on
Theoretical Aspects of Computer Science, February 2006.
This work has been partially supported by the DFG-SFB 376 and by the European Union within the 6th
Framework Programme under contract 001907 (DELIS).
Partially supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service
(DAAD).
Partially supported by a fellowship of the International Graduate School Dynamic Intelligent Systems, Paderborn.
1
1 Introduction
1.1 Motivation and Framework
Large-scale communication networks like, e.g., the Internet often lack a central regulation for
several reasons: The size of the network may be too large, or the players may be free to act
according to their private interests. Even cooperation among the players may be impossible due
to the fact that players may not even know each other. Such an environment—where players
neither obey some central control instance nor cooperate with each other—can be modeled as a
non-cooperative game [28].
Given certain rationality assumptions on players, game-theoretic equilibria provide predictions for the outcome of non-cooperative games. Arguably the best-known equilibrium concept
is the Nash equilibrium, which is a “stable” outcome where no player can improve his cost by
unilaterally changing his strategy. In a seminal paper, Koutsoupias and Papadimitriou [24] initiated the study of worst-case approximation guarantees for Nash equilibria, i.e., they suggested
to systematically quantify the degradation of some global objective due to the players’ selfinterested behavior and the lack of cooperation. Specifically, Koutsoupias and Papadimitriou
defined the price of anarchy (PoA) as the worst-case ratio between the objective function in a
Nash equilibrium and its value in any other outcome.
Congestion games are a special class of non-cooperative games that provide a simple, yet
powerful model of self-interested resource sharing: There is a measurable set of players and
finitely many resources, and the strategies available to the players consist of subsets of the
given resources. The archetype of congestion games is selfish routing [36] where strategies are
paths in a network. Since players’ strategies may overlap, congestion effects occur and strategic
interaction arises. Specifically, using a resource comes at a cost that depends on the total mass
of the players using this resource. Each player strives to choose a strategy that minimizes the
sum of his used resources’ costs.
In this work, we study the atomic unsplittable variant of congestion games, as introduced by
Rosenthal [34]: There are finitely many players, who each have a non-negligible influence in the
game (in terms of measure theory, each player is an atom in the set of players). Each player
must choose a single strategy (i.e., he cannot perform load balancing over several strategies). We
also study a generalization by Milchtaich [29]: He defined weighted congestion games in which
the players have weights and thus different influences on the congestion on the resources.
In the context of selfish routing in road traffic systems, the non-atomic variant of congestion
games was already studied in the 1950’s (see, e.g., [41, 4]). Here, there are infinitely many players
who each have infinitesimal weight (in terms of measure theory, the set of players contains no
atoms). In the computer science literature, non-atomic congestion games were the first variant
of congestion games for which a complete characterization of the PoA was shown [39, 40].
The PoA directly depends on the definition of the global objective—often called social cost:
In their seminal paper, Koutsoupias and Papadimitriou [24] considered a very simple weighted
(atomic unsplittable) congestion game with singleton strategy sets, now known as the KPmodel. For this model, they defined the social cost as the expected maximum resource cost. For
non-atomic congestion games, in contrast, Roughgarden and Tardos [40] considered social cost
defined as the total cost—which in the context of selfish routing is proportional to the average
travel time.
For atomic unsplittable congestion games, Awerbuch et al. [2] and Christodoulou and Koutsoupias [7] also considered the total cost. In this setting, they showed asymptotic bounds on the
worst-case PoA for weighted and unweighted games, provided that all resource cost functions are
2
bounded-degree polynomials with nonnegative coefficients. For the case of affine cost functions
they gave exact values for the worst-case PoA.
1.2 Contribution and Comparison
In this work we prove exact bounds on the worst-case PoA for unweighted and weighted (atomic
unsplittable) congestion games1 , provided that all cost functions are bounded-degree polynomials
with nonnegative coefficients. We use the total cost as social-cost measure. This improves on
results by Awerbuch et al. [2] and Christodoulou and Koutsoupias [7], who gave non-matching
upper and lower bounds. We now describe our findings in more detail.
• For weighted congestion games we show that the worst-case PoA is exactly
Φd+1
,
d
where Φd is the unique nonnegative real solution to√ (x + 1)d = xd+1 . Observe that Φd is a
natural generalization of the golden ratio Φ1 = 1+2 5 to higher degrees. We show that Φd
is irrational for all d ∈ N and asymptotically grows as Θ( lndd ). This result closes the gap
between the previous best upper and lower bounds on the worst-case PoA of O(2d dd+1 )
and Ω(dd/2 ) from [2].
• For unweighted congestion games we show that the worst-case PoA is exactly
(k + 1)2d+1 − k d+1 (k + 2)d
,
(k + 1)d+1 − (k + 2)d + (k + 1)d − k d+1
where k = bΦd c. The worst-case PoA is bounded from below by bΦd cd+1 and from above
by Φd+1
d . Prior to this paper, the best known upper and lower bounds were of the form
d(1−o(1))
d
[7]. However, the term o(1) still hides a gap between the upper and the lower
bound.
We show that our bounds hold whenever no player can improve his expected cost by making a
unilateral deviation. Consequently, the worst-case prices of anarchy with respect to pure Nash,
mixed Nash, correlated and coarse-correlated equilibria always coincide.2 Moreover, we show
that the above values also hold for the subclasses of unweighted and weighted network congestion
games.
For our upper bounds, we use a similar technique as Christodoulou and Koutsoupias [7], yet
with a more elaborate analysis. The core of our analysis for unweighted games is to minimize
λ
1−µ under the following constraint: For all polynomials ` with nonnegative coefficients and
maximum degree d, and for all nonnegative integers x, y it must hold that
y · `(x + 1) ≤ λ · y · `(y) + µ · x · `(x) .
(1.1)
For the case of weighted players, a slight variant of (1.1) has to hold for all reals x, y ≥ 0.
In order to prove their upper bound for unweighted games, Christodoulou and Koutsoupias [7]
1
2
In the presence of other qualifiers (weighted, unweighted, network), we will omit “atomic unsplittable” when
there is no confusion.
In the conference version of this paper [1], we only considered pure Nash and mixed Nash equilibria. The fact
that our bounds on the PoA also hold, without any quantitative degradation, with respect to correlated and
coarse-correlated equilibria was first observed by Roughgarden [37] (see Section 1.4).
3
looked at (1.1) with µ = 12 and gave an asymptotic estimate for λ. In our analysis, we optimize
both parameters λ, µ. This optimization process requires new ideas and is non-trivial.
Table 1.1 shows a numerical comparison between our exact values for the worst-case PoA
and the previous results of Awerbuch et al. [2] and Christodoulou and Koutsoupias [7]. The
lower bounds in the table stem from construction schemes given in the cited works, before any
estimates are applied. Missing values denote cases in which the lower bound for affine functions
is larger than the PoA in the respective instance that
corresponds to d. The construction scheme
1 P∞ kd
in [2, Theorem 4.3] yields a PoA approximating e k=1 k! , which is the value of the d-th Bell
d+2
number.3 In [7, Theorem 10], a network with PoA (N −1)
is given, where N is the largest
N
integer for which (N −1)d+2 ≤ N d holds. The column with the upper bound from [7] is computed
by using (1.1) with µ = 12 and optimizing λ with the help of our analysis. Thus, the column
shows the best possible bounds that can be shown with µ = 12 .
Table 1.1: The price of anarchy—comparison of our results to Christodoulou and Koutsoupias
[7], Awerbuch et al. [2], and Roughgarden and Tardos [40]
Atomic Unsplittable Congestion Games
d
Φd
1
2
3
4
5
6
7
8
1.618
2.148
2.630
3.080
3.506
3.915
4.309
4.692
Unweighted Players
Exact
Upper
Lower
Bound [7]
Bound [7]
2.500
2.500
2.500
9.583
10.00
41.54
47.00
267.6
269.0
21.33
1,514
2,154
42.67
12,345
15,187
85.33
98,734
169,247
170.7
802,603
1,451,906
14,762
Weighted Players
Exact
Lower
Bound [2]
2.618
2.618
9.909
47.82
5.000
277.0
15.00
1,858
52.00
14,099
203.0
118,926
877.0
1,101,126
4,140
Nonatomic
Congestion
Games
Exact
[40]
1.333
1.626
1.896
2.151
2.394
2.630
2.858
3.081
1.3 Previous Work
The papers most closely related to our work are those of Awerbuch et al. [2] and Christodoulou
and Koutsoupias [7, 8]. For unweighted congestion games and social cost defined as total cost, it
was shown that the worst-case PoA with respect to pure Nash equilibria is 52 for affine resource
cost functions and dΘ(d) for polynomial cost functions with maximum degree
√ d and nonnegative
3+ 5
coefficients [2, 7]. For weighted congestion games, the worst-case PoA is 2 for affine resource
cost functions and dΘ(d) for polynomial cost functions with maximum degree d and nonnegative
coefficients [2].
The price of anarchy [32] was first introduced and studied by Koutsoupias and Papadimitriou
[24] (the original paper used the term coordination ratio). As a starting point of their investigation, they considered a simple weighted congestion game with singleton strategy sets, now
known as KP-model. In the KP-model, cost functions are linear and social cost is defined as
3
The d-th Bell number is defined as the number of partitions of a set of d elements into nonempty subsets.
4
the maximum expected resource cost. In this setting, there exist tight bounds on the worst-case
m
PoA of Θ( logloglogmm ) for identical resource cost functions [10, 25] and Θ( log log
log log m ) for linear
cost functions [10]. The PoA has also been studied for variations of the KP-model, namely for
non-linear resource cost functions [11, 19], for the case of restricted strategy sets [3, 21], for the
case of incomplete information [20], and for different social-cost measures [18, 27, 16]. In particular, Lücking et al. [27] studied the total cost (they called it quadratic social cost) for atomic
unsplittable congestion games with singleton strategies and linear resource cost functions. For
this model they showed that the worst-case PoA with respect to pure Nash equilibria is exactly
4
9
3 for the case of identical player weights, and 8 for the case of identical resource cost functions
and arbitrary player weights.
Atomic unsplittable congestion games were first defined by Rosenthal [34] and extensively
studied afterwards (see, e.g., [15, 29, 30]). In Rosenthal’s (unweighted) model, the strategy of
each player is a subset of resources. Resource cost functions can be arbitrary but they only
depend on the number of players sharing the same resource. Rosenthal showed that such games
always admit a pure Nash equilibrium using a potential function. Monderer and Shapley [30]
characterized games that possess a potential function as potential games and showed their relation to (atomic unsplittable) congestion games. Milchtaich [29] considered weighted congestion
games with player-specific cost functions and showed that these games do not admit a pure
Nash equilibrium in general. Fotakis et al. [15, 14] considered the PoA for symmetric weighted
network congestion games in layered networks [15] and for symmetric (unweighted) network
congestion games in general networks [14]. In both cases they defined social cost as expected
maximum resource cost. For a survey on weighted congestion games we refer to [17].
For non-atomic congestion games, tight bounds on the worst-case PoA predate most of the
above works on atomic unsplittable games. They were proven by Roughgarden and Tardos
[39, 40] already shortly after the PoA was introduced in [24]. With the total cost as socialcost measure, the worst-case PoA was shown to be 43 for affine resource cost functions [39] and
(1 − d · (d + 1)−(d+1)/d )−1 for polynomial cost functions with maximum degree d and nonnegative
coefficients [40]. The asymptotic growth of the latter expression is Θ( logd d ) as d → ∞. An
overview on results for this model can be found in [35]. Subsequent to [39, 40], Perakis [33]
considered a generalization of non-atomic congestion games where costs are non-separable in
terms of the resource loads. Similar to (but independent of) our work, she also used a twoparameter optimization to determine tight bounds on the price of anarchy.
1.4 Subsequent Work
After the conference version of this paper [1], several subsequent works have shaped a more
complete picture of the worst-case PoA in the various congestion-game variants:
• Blum et al. [6] were the first to show that for unweighted congestion games with affine
resource cost functions, the worst-case PoA holds with no quantitative degradation also
with respect to coarse-correlated equilibria.
• Roughgarden [37] distilled and generalized our two-parameter optimization into what he
dubbed smoothness framework. This framework unifies essentially all previous upperbound proofs for atomic unsplittable congestion games. Roughgarden also generalized the
above result by Blum et al. [6]: He proved for arbitrary classes of resource cost functions that the worst-case PoA in unweighted congestion games holds without quantitative
degradation also with respect to coarse-correlated equilibria. On the lower-bound side,
5
Roughgarden [37] showed that our unweighted lower-bound construction can be generalized to arbitrary cost functions as well.
• Gairing and Schoppmann [16], and later Bhawalkar et al. [5] in more general form, showed
that the worst-case PoA for both weighted and unweighted congestion games is already
attained on parallel links. Bhawalkar et al. [5] moreover showed that Roughgarden’s
smoothness framework is equally applicable to weighted congestion games, and they gave
a matching lower-bound construction for all sets of resource cost functions (under mild
restrictions).
• Using our two-parameter approach, Harks [22] obtained the first tight upper bounds on the
PoA for atomic splittable congestion games, where players can split their weight arbitrarily
among their strategies. (Harks’ result improved previous bounds by Cominetti et al. [9].)
Later, Roughgarden and Schoppmann [38] generalized the smoothness framework to “local
smoothness” and showed that the two-parameter optimization yields exact values for the
worst-case PoA in all games with convex strategy sets and convex cost functions. Using
that, they established exact values for the worst-case PoA in atomic splittable congestion
games with arbitrary cost functions.
1.5 Roadmap
The rest of this paper is organized as follows. In Section 2, we formally define the model. We
establish exact bounds on the worst-case PoA for weighted congestion games in Section 3 and
for unweighted congestion games in Section 4. All purely analytical tools that we believe to be
of independent interest are given in Section 5.
2 The Model
Notation For all d ∈ N, let [d] := {1, . . . , d} and [d]0 := [d]∪{0}. For a vector x = (x1 , . . . , xn ),
let (y i , x−i ) := (x1 , . . . , xi−1 , y i , xi+1 , . . . , xn ). We define Φd as a natural generalization of the
golden ratio such that Φd is the (only) positive real solution to (x + 1)d = xd+1 .
For d ∈ N, we denote by Pd the set of all polynomials with maximum degree d and
P nonnegative
coefficients. That is, for all ` ∈ Pd there are a0 , . . . , ad ∈ R≥0 such that `(x) = dj=0 aj · xj .
Congestion Games In an (atomic unsplittable) congestion game, a finite ground set of resources E has to be shared among n ∈ N players: Specifically, each player i ∈ [n] commits to
using one particular subset, out of a collection Si of non-empty subsets of E. The contribution of
player i to the load (“congestion”) on each resource he uses is determined by his weight wi ∈ R>0 .
In the special case where all weights are 1, we say the congestion game is unweighted, otherwise
we say the congestion game is weighted. Using a resource e ∈ E incurs a load-dependent cost
`e : R≥0 → R≥0 . In this work, we only consider games with cost functions from Pd ; all these
cost functions are non-decreasing.
Network congestion games are specified by a graph G = (V, E), together with origin-destination
pairs oi , di ∈ V for each player i ∈ [n]. Each edge in the graph is a resource, and each player i’s
collection Si consists of all paths from oi to di .
Strategy Profiles, Load, and Cost The collection Si is called strategy set of player i, and
any particular choice xi ∈ Si is called a strategy. A strategy profile is a vector of strategies
6
x = (x1 , . . . , xn ) ∈ S := S1 × · · · × Sn that captures a particular outcome of a game. Given
a strategy profile x and a resource e, we denote by xie the load player i places on resource e in
i
i
i
strategy
Pprofilei x, i.e., xe := wi if e ∈ x and xe := 0 otherwise. For convenience, we define
xe :=
i∈[n] xe as the total load on resource e. The cost of player i is defined as ci (x) :=
P
P
i
e∈E xe · `e (xe ) = wi ·
e∈xi `e (xe ). We use a utilitarian measure for the cost for society, and
define the social cost as the total cost,
X
X
ci (x) =
xe · `e (xe ) .
SC(x) :=
e∈E
i∈[n]
Equilibria We are interested in the equilibria of congestion games, i.e., strategy profiles (or
probability distributions over strategy profiles) that leave no player an incentive to unilaterally
change his behavior. Formally, a pure Nash equilibrium [31] is a strategy profile x where for all
players i ∈ [n] and all strategies y i ∈ Si the Nash conditions ci (x) ≤ ci (y i , x−i ) hold. More
generally, we look at probability distributions P over S where the Nash conditions hold in
expectation, i.e.,
∀i ∈ [n] : ∀y i ∈ Si : Ex∼P [ci (x)] ≤ Ex∼P [ci (y i , x−i )] .
(2.1)
While not the focus of this work, we remark that the set of probability distributions satisfying
(2.1) is commonly divided into the following hierarchy (see [42] for an introduction and also
[37] and its references). In a mixed Nash equilibrium P , all players’ strategies are stochastically
independent. Note here that we can consider a pure Nash equilibrium x as a mixed Nash
equilibrium P that places all probability on x. A correlated equilibrium P satisfies for all players
i ∈ [n] and all functions δi : Si → Si that Ex∼P [ci (x)] ≤ Ex∼P [ci (δi (xi ), x−i )]. As the most
general notion in this hierarchy, a coarse-correlated equilibrium P is just characterized by (2.1).
To summarize, we have the following set of inclusions for the sets of equilibria in a game: pure
Nash ⊆ mixed Nash ⊆ correlated ⊆ coarse-correlated. While the set of pure Nash equilibria may
be empty in weighted congestion games [26, 23] it is always non-empty in unweighted congestion
games [34]. The sets of mixed Nash, correlated, and coarse-correlated equilibria are never empty
[31].
Price of Anarchy The price of anarchy (PoA) in a congestion game, with respect to any
of the above equilibrium concepts, is the largest possible ratio between the social cost of an
equilibrium and that of a minimum-cost strategy profile. Formally,
PoA :=
Ex∼P [SC(x)]
.
SC(y)
is equilibrium
sup
P
y∈S
The interpretation of the worst-case PoA is here that it bounds the cost of pure Nash equilibria in
instances where pure Nash equilibria exist. We remark that going to a more general equilibrium
concept corresponds to dropping constraints and thus can only increase the PoA.
3 Weighted Congestion Games
In this section, we establish exact values for the worst-case PoA in weighted congestion games,
provided that all cost functions are bounded-degree polynomials with nonnegative coefficients.
7
No quantitative differences exist with respect to pure Nash, mixed Nash, correlated, and coarsecorrelated equilibria.4
3.1 Upper Bound
To upper-bound the PoA, we give an upper bound on the social cost of all probability distributions over strategy profiles that satisfy the Nash conditions in expectation (2.1). The upper
bound crucially relies on a purely analytical result that we state first. We defer its proof to
Section 5.3.
Theorem 3.1. Let d ∈ N. Then
λ min
∀x, y ∈ R≥0 , ` ∈ Pd : y · `(x + y) ≤ λ · y · `(y) + µ · x · `(x) = Φd+1
.
d
1−µ
(λ,µ)∈R×(0,1)
Theorem 3.2. Let d ∈ N. In an arbitrary weighted congestion game with cost functions in Pd ,
let P be a probability distribution over strategy profiles such that, for every player, the Nash
conditions are satisfied in expectation (2.1). Moreover, let y be a minimum-cost strategy profile.
Then
Ex∼P [SC(x)]
.
≤ Φd+1
d
SC(y)
Proof. We first bound the expected cost of each player i ∈ [n] in the equilibrium P . By the
Nash conditions, player i cannot improve by switching to the pure strategy y i . That is,
Ex∼P [ci (x)] ≤ Ex∼P [ci (y i , x−i )]
#
"
X
i
i
i
ye · `e (xe − xe + ye )
= Ex∼P
e∈E
"
≤ Ex∼P
#
X
e∈E
yei · `e (xe + ye ) ,
where the last inequality is due to the fact that the cost functions considered in this paper are
non-decreasing. By exploiting the linearity of expectation twice, we can now bound the expected
social cost by
" n
#
"
#
n
X
X
X
Ex∼P [SC(x)] = Ex∼P
ci (x) ≤
Ex∼P
yei · `e (xe + ye )
i=1
i=1
"
= Ex∼P
e∈E
#
X
e∈E
ye · `e (xe + ye ) .
Choosing (λ, µ) ∈ R × (0, 1) as the minimizer in Theorem 3.1 gives the bound
"
#
"
#
X
X
Ex∼P [SC(x)] ≤ Ex∼P
ye · `e (xe + ye ) ≤ Ex∼P
λ · ye · `e (ye ) + µ · xe · `e (xe )
e∈E
e∈E
= λ · SC(y) + µ · Ex∼P [SC(x)] ,
where the last equality is again due to the linearity of expectation. Rearranging yields the
theorem.
t
u
4
See also Section 1.4.
8
3.2 Lower Bound
We proceed by providing a matching lower bound, which holds even for network congestion
games. We first give a general construction of a (non-network) instance (Theorem 3.3) and then
present a network congestion game that is based on this construction (Theorem 3.4).
Theorem 3.3. Let d ∈ N. There is a weighted congestion game that has a pure Nash equilibrium
x and some other strategy profile y such that
SC(x)
= Φd+1
.
d
SC(y)
Proof. Let k ∈ N such that k ≥ maxj∈[d]0 dj and k ≥ 2. We construct a weighted congestion
game with n = (d + 1) · k players and |E| = n resources. The goal of our construction is to
create a pure Nash equilibrium x in which each resource has Φd times as much load as at the
other strategy profile y.
We divide the set E into d + 1 disjoint subsets: For i ∈ [d]0 , let Ei := {gi,1 , . . . , gi,k }, where
each gi,j has the cost function `(z) := ai · z d . The values of the coefficients ai will be determined
later. It will be convenient in the following to have a second name for each resource: For
2k ≥ j > k, we therefore denote gi,j := gi,j−k .
Like the resources, we also partition the set of players [n]: For i ∈ [d]0 , let Ni := {ui,1 , . . . , ui,k }.
(To keep the notation human-parsable, we will only write i, j instead of ui,j in sub- and superscripts.) Each player ui,j ∈ Ni has weight wi,j = Φid and a 2-element strategy set Si = {xi,j , y i,j },
where
(
{gd,j+1 , . . . , gd,j+(d) , gi−1,j } if i ∈ [d]
i,j
i
x :=
and
y i,j := {gi,j } .
{gd,j+1 }
if i = 0
Consider now the pure strategy profile x P
:= (x1 , .. . , xn ). Due to the symmetry of our definitions,
each resource e ∈ Ed is used by exactly dl=0 dl players (for each l ∈ [d]0 , it is used by exactly
d
l players from Nl ). For i ∈ [d − 1]0 , each resource e ∈ Ei is used by exactly one player from
Ni+1 . Consequently, we can compute the load on each resource as shown in the following table.
It follows that all resources have Φd times as much load at profile x as they have at the strategy
profile y := (y 1 , . . . , y n ).
i
d
0 to d − 1
load on every resource e ∈ Ei
xe
ye
Pd
d
d+1
l
d
Φdd
l=0 l Φd = (Φd + 1) = Φd
Φi+1
Φid
d
For x to be a pure Nash equilibrium, the following Nash conditions for each set Ni of players
need to be fulfilled:
i
1 to d
0
Nash condition to fulfill
d
d
i d
ci,j (x) = i · ad · (Φd+1
d ) + ai−1 · (Φd )
≤ ai · (Φdi+1 + Φid )d = ci,j (y i,j , x−(i,j) )
d
d
0,j
−(0,j) )
c0,j (x) = ad · (Φd+1
d ) ≤ a0 · (Φd + 1) = c0,j (y , x
9
Replacing “≤” by “=” yields a homogeneous system of linear equations, that is, the system
Bd · a = 0 where Bd is the following (d + 1) × (d + 1) matrix:










Bd = 









−Φdd
2 +d+1
d
d−1
+ Φdd
2
Φdd +d
..
.
..
.
d2 +d
d
i Φd
..
.
..
.
d2 +d
Φd
2 +d
2
Φdd
2 +1
−Φdd
0
..
.
0
..
.
..
.
0
0 ···
..
.
..
···
0
..
.
.
..
···
..
.
.
id+d+1
0 −Φd
Φid
d
..
.
0
..
.
..
.
···
0
0 ···
..
.
..
.
..
.
···
0
..
.
0
..
.
0
Φdd
−Φd+1
d




















(3.1)
and a := (ad . . . a0 )t . Obviously, a solution to this system fulfills the initial Nash conditions.
Note that
+ Φid )d = (Φid )d · (Φd + 1)d = Φdid+d+1 .
(Φi+1
d
We proceed by showing a property of Bd .
Claim. The (d + 1) × (d + 1) matrix Bd from (3.1) has rank d.
Proof (of the claim). 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 first column and on the principal diagonal.
For the top left element of Cd we get
!
d d X
X
d j
d d2 +j
d+1
d2 +d+1
d2
Φd = 0.
= Φd · −Φd +
+
Φd
− Φd
j
j
j=0
j=0
| {z }
(Φd +1)d
Since all elements on the principal diagonal of Cd —with the just shown exception of the first
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 dependent and thus
there are—with degree of freedom 1—infinitely 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 ∈ [d]0 ) that all coefficients ai must have the same sign and
thus we can always find 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
i+1
d
i d
as ai−1 and ad , for (Φd+1
+ Φid )d are all positive.
d ) , (Φd ) , and (Φd
Choosing a 6= 0 with all components being positive, all coefficients of the resource cost functions are positive. We get
P
d+1
k · di=0 ai (Φi+1
SC(x)
d )
=
= Φd+1
t
u
Pd
d .
SC(y)
k · i=0 ai (Φid )d+1
10
Theorem 3.4. The previous Theorem 3.3 also holds for network congestion games.
Proof. Each instance of the congestion game in Theorem 3.3 can be characterized by two parameters: The degree
d of the polynomial resource cost functions and the number of resources
d
k ≥ max{ bd/2c , 2} in each class Ei , where i ∈ [d]0 . Recall that both the total number of
resources as well as the number of players are given by (d + 1) · k. Given such an instance we
construct an equivalent network congestion game.
While following our construction description it will be helpful to keep half an eye on Figure 3.1,
which shows an example for d = 2 (quadratic resource cost functions) and k = 2. For each
player ui,j , 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 `e (x) = 0 as their cost function. We call these edges free edges. All other
edges have the associated cost function as in Theorem 3.3.
u 2,2
u 1,2
u 0,2
u 0,1
1
g 2,
u 1,2
u 0,2
g 0,2
g 0,1
g
2,
2
u 2,1
g 1,2
u 1,1
u 2,2
g 1,1
u 2,1
u 1,1
u 0,1
Figure 3.1: Weighted network congestion game for d = 2 and k = 2
The network corresponding to an instance (d, k) can be constructed as follows: There is a
circle of 2 · k edges where every other edge represents 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) connects to a free edge, which then
i
in turn connects to edge gi−1,j . (In case i = 0, the latter is simply another free edge.) From
there, another free edge to the destination node of player ui,j exists. 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 the respective player’s “Nash” strategy which has been denoted as xi,j in
Theorem 3.3.
We can now add two more free edges for each player ui,j that allow him to also use his “optimal”
strategy y i,j : From ui,j ’s origin node add a free link to gi,j (we call this an A-link ), and from
gi,j add a free link to ui,j ’s destination node (called a B-link ). Note that in Figure 3.1, A- and
B-links are only shown for player u1,1 , while the figure is complete otherwise. The thick gray
path denotes player u1,1 ’s strategy in the system optimum, whereas the dotted path indicates
his strategy in the worst-case Nash equilibrium.
Player ui,j ’s A-link obviously cannot create shortcuts for other players because player ui,j ’s
origin node only has outgoing edges. Similarly, his destination node only has incoming edges
11
and therefore his B-link cannot create shortcuts for other players, either. Finally, neither player
ui,j ’s A- nor B-link can create a shortcut for his strategy xi,j because the A-link, B-link, and
xi,j do not share any nodes except for the origin and destination nodes.
Note, however, that B-links do create additional paths: In Figure 3.1, for instance, player u1,1
now has the further option of using a path consisting of five 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.
t
u
4 Unweighted Congestion Games
In this section, we establish exact values for the worst-case PoA of unweighted congestion games,
provided that all cost functions are bounded-degree polynomials with nonnegative coefficients.
No quantitative differences exist with respect to pure Nash, mixed Nash, correlated, and coarsecorrelated equilibria.5 The structure of this section is very similar to the previous Section 3, yet
there are some subtle differences in the proofs. In particular, the worst-case PoA turns out to
be slightly smaller than for the weighted case.
4.1 Upper Bound
To upper-bound the PoA, we give an upper bound on the social cost of all probability distributions over strategy profiles that satisfy the Nash conditions in expectation (2.1). Again, the
upper bound crucially relies on a purely analytical result that we state first. We defer its proof
to Section 5.4.
Theorem 4.1. Let d ∈ N and define k := bΦd c. Then
λ min
∀x, y ∈ N0 , ` ∈ Pd : y · `(x + 1) ≤ λ · y · `(y) + µ · x · `(x)
1−µ
(λ,µ)∈R×(0,1)
=
(k + 1)2d+1 − k d+1 (k + 2)d
.
(k + 1)d+1 − (k + 2)d + (k + 1)d − k d+1
Theorem 4.2. Let d ∈ N. In an arbitrary unweighted congestion game with cost functions in
Pd , let P be a probability distribution over strategy profiles such that, for every player, the Nash
conditions are satisfied in expectation (2.1). Moreover, let y be a minimum-cost strategy profile.
Then
Ex∼P [SC(x)]
(k + 1)2d+1 − k d+1 (k + 2)d
≤
.
SC(y)
(k + 1)d+1 − (k + 2)d + (k + 1)d − k d+1
Proof. We first bound the expected cost of each player i ∈ [n] in the equilibrium P . By the
Nash conditions, player i cannot improve by switching to the pure strategy y i . That is,
Ex∼P [ci (x)] ≤ Ex∼P [ci (y i , x−i )]
"
#
X
i
i
i
= Ex∼P
ye · `e (xe − xe + ye )
e∈E
"
≤ Ex∼P
5
#
X
e∈E
See also Section 1.4.
12
yei · `e (xe + 1) ,
where the last inequality is due to the fact that the cost functions considered in this paper are
non-decreasing. By exploiting the linearity of expectation twice, we can now bound the expected
social cost by
" n
#
"
#
n
X
X
X
Ex∼P [SC(x)] = Ex∼P
ci (x) ≤
Ex∼P
yei · `e (xe + 1)
i=1
i=1
"
= Ex∼P
e∈E
#
X
e∈E
ye · `e (xe + 1) .
Choosing (λ, µ) ∈ R × (0, 1) as the minimizer in Theorem 4.1 gives the bound
"
#
"
#
X
X
Ex∼P [SC(x)] ≤ Ex∼P
ye · `e (xe + 1) ≤ Ex∼P
λ · ye · `e (ye ) + µ · xe · `e (xe )
e∈E
e∈E
= λ · SC(y) + µ · Ex∼P [SC(x)] ,
where the last equality is again due to the linearity of expectation. Rearranging yields the
theorem.
t
u
4.2 Lower Bound
We proceed by providing a matching lower bound, which holds even for network congestion
games. We first give a general construction of a (non-network) instance (Theorem 4.3) and then
present a network congestion game that is based on this construction (Theorem 4.4).
Theorem 4.3. Let d ∈ N. There is an unweighted congestion game that has a pure Nash
equilibrium x and some other strategy profile y such that
(k + 1)2d+1 − k d+1 (k + 2)d
SC(x)
=
.
SC(y)
(k + 1)d+1 − (k + 2)d + (k + 1)d − k d+1
Proof. Let k := bΦd c. We construct an unweighted congestion game with n ≥ k + 2 players and
|E| = 2 · n resources.
We partition the set E into two subsets E1 := {g1 , . . . , gn } and E2 := {h1 , . . . , hn }. Each
player i ∈ [n] has two strategies in his strategy set Si := {xi , y i }, where
xi := {gi+1 , . . . , gi+k , hi+1 , . . . , hi+k+1 }
and
y i := {gi , hi } .
Here we denoted gj := gj−n and hj := hj−n for 2n ≥ j > n.
All resources e ∈ E1 have the cost function `e (z) := a · z d for an a ∈ R>0 yet to be determined,
whereas all resources e ∈ E2 incur cost `e (z) := z d . In the following, we choose a such that the
strategy profile x := (x1 , . . . , xn ) becomes a pure Nash equilibrium. The Nash conditions, i.e.,
ci (x) ≤ ci (y i , x−i ) for each player i ∈ [n], are equivalent to
k · a · k d + (k + 1) · (k + 1)d ≤ a · (k + 1)d + (k + 2)d .
Solving for a gives
a≥
(k + 1)d+1 − (k + 2)d
> 0,
(k + 1)d − k d+1
13
(4.1)
where the second inequality is due to the irrationality of Φd (Theorem 5.3), which implies k < Φd .
We now compare to the strategy profile y := (y 1 , . . . , y n ). The costs of any player i ∈ [n] are
ci (x) = a · k d+1 + (k + 1)d+1 and ci (y) = a + 1. Hence,
Pn
ci (x)
SC(x)
a · k d+1 + (k + 1)d+1
= Pi=1
=
.
(4.2)
n
SC(y)
a+1
i=1 ci (y)
Choosing a to fulfill the left-hand side of (4.1) with equality gives the desired result.
t
u
Theorem 4.4. The previous Theorem 4.3 also holds for network congestion games.
Proof. Instances of the congestion games in Theorem 4.3 can be characterized by two parameters:
The maximum degree d of the cost functions and the number of players n ≥ k+2 where k := bΦd c.
The number of resources is then 2n.
3
2
h3
h2
g2
1
2
v1
4
g3
4
v4
v1
h4
g1
v2
3
g4
h1
�e (x)
�e (x)
v2
v3
1
Figure 4.1: Network congestion game for d = 2 and 4 players
Figure 4.1 (left) shows an example of the network congestion game for quadratic resource cost
functions (i.e., d = 2) and for n = 4 players. Unlabeled edges e ∈ E have `e (x) = 0 as their cost
function. As in the proof of Theorem 3.4, we say that these edges are free. All other edges have
the associated cost function as in Theorem 4.2.
The network corresponding to an instance characterized by (d, n) can be constructed as follows:
There is a circle of 2n undirected edges g1 , h1 , g2 , h2 , . . . , gn , hn . Each undirected edge (v1 , v2 )
has to be replaced by the construction shown in Figure 4.1 (right). This ensures that no matter
in which direction a player uses edge (v1 , v2 ) it produces 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 origin node is connected with a free edge to the node incident
to gi and hi−1 . The destination node of each player i is the node incident to gi+1 and hi ,
represented by a thick outline in the figure. To also allow each player i to choose strategy xi
(which essentially goes the other way round inside the circle), we finally add one more free edge
from i’s origin node to the connecting node of gi+k+1 and hi+k+1 .
Note that our construction creates additional strategies for each player (relative to the abstract
congestion game). Only two of these additional strategies for each player are a simple path: They
correspond to {hi−1 } ∪ xi and {hi−1 } ∪ y i (roughly speaking, taking the “wrong” free edge from
the origin node and then making a “sharp turn”). Clearly, these are dominated strategies, and
they can be ignored.
t
u
14
4.3 Comparison to Weighted Games
A simple calculation show that the worst-case PoA for unweighted games is not “too far” away
from the one for weighted games. Precisely, the worst-case PoA for unweighted games is bounded
from below by bΦd cd+1 and by Φd+1
from above. The upper bound holds trivially as unweighted
d
games are a subset of weighted games. To see the lower bound, define A := (k + 1)d − k d+1 and
B := (k + 1)d+1 − (k + 2)d . Note that A, B > 0 by choice of k and that
(k + 1)2d+1 − k d+1 (k + 2)d
(A + k d+1 )(k + 1)d+1 − k d+1 (k + 2)d
=
A+B
(k + 1)d+1 − (k + 2)d + (k + 1)d − k d+1
d+1
+B
A · k+1
k
= k d+1 ·
≥ bΦd cd+1 .
A+B
5 Analytical Tools
In this section, we prove the irrationality of Φd , establish the asymptotic behavior of Φd , and
give the proofs of Theorems 3.1 and 4.1. These results are based on several technical lemmas
that we state and prove in advance.
5.1 Preliminaries
Lemma 5.1. Let µ ∈ (0, 1] and x ∈ R≥0 . Define g : R≥0 → R, g(r) := (x + 1)r − µ · xr+1 . Then
it holds for all d, r ∈ R≥0 with d > r and g(r) ≥ 0 that g(d) ≥ g(r).
Proof. Let d, r ∈ R≥0 with d > r and g(r) ≥ 0. Since for x = 0 we have g(d) = g(r) = 1, we
only consider the case x > 0 in the following. We get
g 0 (r) = (x + 1)r · ln(x + 1) − µ · xr+1 · ln(x)
> ln(x + 1) (x + 1)r − µ · xr+1 ≥ 0 .
By way of contradiction, assume g(d) ≤ g(r) and let ξ ∈ (r, d) be minimal with g 0 (ξ) = 0. Such
a ξ must exist due to the intermediate and mean value theorems. By definition of ξ, however,
we have g(ξ) > g(r) = 0. Hence, g 0 (ξ) > 0. A contradiction.
t
u
Lemma 5.2. Let µ ∈ (0, 1] and d ∈ N. Define g : R≥0 → R, g(x) := (x+1)d −µ·xd+1 . Then, it
holds that g has exactly one local maximum, at some ξ ∈ R>0 . Moreover, g is strictly increasing
in [0, ξ), and strictly decreasing in (ξ, ∞).
Proof. We have as necessary first-order condition for a local extremum that
µ · (d + 1)
(x + 1)d−1
0
d−1
d
. (5.1)
g (x) = d(x + 1)
− µ(d + 1)x = 0 , i.e.,
x > 0 and
=
d
xd
d−1
Now, limx↓0 (x+1)
= ∞, limx→∞
xd
and for all x ∈ R>0 it holds that
(x+1)d−1
xd
= 0, x 7→
(x+1)d−1
xd
is continuously differentiable,
∂ (x + 1)d−1
(x + 1)d−2 · (x + d)
=
−
< 0.
∂x
xd
xd+1
d−1
As a result, there is a unique ξ ∈ R>0 with (ξ+1)
= µ·(d+1)
. Since limx→∞ g(x) = −∞, it
d
ξd
follows that g has both a local and a global maximum at ξ and that g is strictly increasing in
[0, ξ) and strictly decreasing in (ξ, ∞).
t
u
15
5.2 Properties of Φd
Theorem 5.3. For all d ∈ N it holds that Φd is irrational.
Proof. Let d ∈ N. From Lemmata 5.1 and 5.2 it follows that Φd is well-defined and Φd > 1. By
way of contradiction, suppose that Φd is rational, i.e., there are coprime a, b ∈ N with
Φd =
a
.
b
Dividing (Φd + 1)d = Φd+1
by Φdd gives
d
1 d
Φd = 1 +
.
Φd
Hence,
a
(a + b)d
.
=
b
ad
Since a and b are coprime, so are ad and b. Moreover, ab is in reduced form, and thus b is a
divisor of ad . Consequently, it must hold that a = b = 1. This is a contradiction to Φd > 1. t
u
Theorem 5.4. Φd = Θ( lndd ).
d
Proof. We show that there is some d0 so that for all d ≥ d0 it holds that ln2dd > Φd and 2 ln
d < Φd .
c·d
We first show the upper bound on Φd . Let c > 0. Due to Lemma 5.2, it holds that ln d > Φd
if and only if
d c·d
c · d d+1
+1 <
.
ln d
ln d
d
Taking the d-th root, multiplying with ln
c·d , and raising to the d-th power again yields the
equivalent statement
ln d d c · d
1+
<
.
c·d
ln d
Now recall that the tailor series expansion of f (x) := (1 + x)n about 0 is
f (x) =
n
X
f (i) (0)
i=0
i!
· xi =
n
X
i=0
n! · xi
,
(n − i)! · i!
where f (i) denotes the i-th derivative of f . Hence,
d
d!
ln d d X
1+
=
·
i
c·d
d · (d − i)!
ln d i
c
i!
i=0
.
Consequently, when c = 2,
d
ln d d X
1+
<
2·d
i=0
ln d i
2
i!
<
∞
X
ln d i
2
i!
i=0
16
= exp
ln d
2
=
√
2
d<
2·d
ln d
for all d ≥ 2. This proves the upper bound ln2dd > Φd .
We now proceed to show the lower bound on Φd . Define n(d) := b d2 c. Then,
ln d
1+
2·d
d
=
d
X
i=0
n(d)
n(d)
i=0
i=0
lni d X 2i · d!
lni d X lni d
2i · d!
·
>
·
>
= d − Rn(d)+1 (ln d) ,
di · (d − i)!
i!
di · (d − i)!
i!
i!
where
Rn(d)+1 (ln d) ≤ 2 ·
lnn(d)+1 d
→ 0 for d → ∞ .
(n(d) + 1)!
(See [12], for instance.) Hence, when d is sufficiently large6 then
This proves the lower bound
d
2 ln d
ln d
1+
2·d
d
>
d
.
2 ln d
t
u
< Φd .
5.3 Weighted Games: Proof of Theorem 3.1
As a first step of determining the exact value of Theorem 3.1, we simplify the expression to get
rid of quantification over all polynomials.
Lemma 5.5. Let d ∈ N. Then,
λ ∀x,
y
∈
R
,
`
∈
P
:
y
·
`(x
+
y)
≤
λ
·
y
·
`(y)
+
µ
·
x
·
`(x)
I :=
inf
≥0
d
1−µ
(λ,µ)∈R×(0,1)
(x + 1)d − µ · xd+1
.
= inf
max
1−µ
µ∈(0,1) x∈R≥0
Proof. We first note that for any µ ∈ (0, 1) the maximum on the right-hand side exists (Lemma 5.2).
Assume there is a pair (λ, µ) ∈ R × (0, 1) with
∀x, y ∈ R≥0 , ` ∈ Pd : y · `(x + y) ≤ λ · y · `(y) + µ · x · `(x) .
(5.2)
Since each ` ∈ Pd is a linear combination of monomials, (5.2) may be simplified such that `
ranges only over all monomials of maximum degree d. Moreover, when y = 0 the condition is
fulfilled trivially. Therefore, (5.2) is equivalent to
∀x ∈ R≥0 , y ∈ R>0 , r ∈ [d]0 : y · (x + y)r ≤ λ · y r+1 + µ · xr+1 .
Dividing by y r+1 and rearranging yields the equivalent condition
∀x ∈ R≥0 , y ∈ R>0 , r ∈ [d]0 : λ ≥
x
+1
y
r
r+1
x
−µ·
.
y
(5.3)
By substituting for xy , this can be further simplified to the equivalent condition
∀x ∈ R≥0 , r ∈ [d]0 : λ ≥ (x + 1)r − µ · xr+1 .
6
Numerically, it can be verified that this holds for all d ≥ 2.
17
(5.4)
When x = 0, we have for all r ∈ [d]0 that (x+1)r −µ·xr+1 = 1 > 0. Therefore, using Lemma 5.1,
we can again simplify (5.4) to
∀x ∈ R≥0 : λ ≥ (x + 1)d − µ · xd+1 .
(5.5)
Hence, we have that
λ d
d+1
∀x ∈ R≥0 : λ ≥ (x + 1) − µ · x
I=
inf
1−µ
(λ,µ)∈R×(0,1)
(x + 1)d − µ · xd+1
= inf
max
.
1−µ
µ∈(0,1) x∈R≥0
t
u
Now, a straightforward way of evaluating the right-hand side of Lemma 5.5 would be to determine
the value of the max-term with respect to any arbitrary µ ∈ (0, 1). According to Lemma 5.2,
the maximum is even unique. Hence, there is temptation to treat the max-term as a function
of µ and then use ordinary calculus for computing the infimum I.
Unfortunately, this naive approach turns out to be infeasible as evaluating the max-term
using first-order conditions requires solving for the roots of x 7→ (x + 1)d − µ · xd+1 , i.e., a
polynomial of degree d − 1. Our proof is therefore based on the following observation, which
exploits the first-order condition (5.1) of Lemma 5.2: For each x ∈ R≥0 , the condition yields a
unique µ ∈ (0, 1) so that x becomes a maximizer (on R≥0 ) of the aforementioned polynomial.
Consequently, inserting into the right-hand side of Lemma 5.5 immediately gives an upper bound
on the infimum I. In the following, we show that choosing x = Φd (and µ accordingly) makes
the upper bound and I coincide.
Lemma 5.6. Let d ∈ N. Then,
I = inf
µ∈(0,1)
max
x∈R≥0
(x + 1)d − µ · xd+1
1−µ
= Φd+1
.
d
Proof. Define g : (0, 1) × R → R and h : R>0 → R,
g(µ, x) :=
(x + 1)d − µ · xd+1
1−µ
and h(x) :=
d · (x + 1)d−1
.
(d + 1) · xd
Note that g is differentiable. The upper bound I ≤ Φd+1
follows from the following two claims.
d
i) h(Φd ) ∈ (0, 1)
Proof of claim (i): Since (Φd + 1)d = Φd+1
d , it holds that
h(Φd ) =
d · (Φd + 1)d
d · Φd
=
∈ (0, 1) .
d
(d + 1) · (Φd + 1)
(d + 1) · Φd · (Φd + 1)
(5.6)
ii) Suppose there is a pair (b
µ, x
b) ∈ (0, 1)×R>0 with µ
b = h(b
x). Then g(b
µ, x
b) = maxx∈R≥0 {g(b
µ, x)}.
Proof of claim (ii): Clearly, x
b fulfills the necessary first order condition (5.1) for local
extrema of g(b
µ, ·). By Lemma 5.2, we get that x
b is in fact the unique positive local
maximum and that g(b
µ, ·) is strictly increasing in (0, x
b) and strictly decreasing in (b
x, ∞).
Clearly, this implies g(b
µ, x
b) = maxx∈R≥0 {g(b
µ, x)}.
18
Now fix µ
b := h(Φd ). Up to this point, we have shown that
(i)
(ii)
I ≤ max {g(b
µ, x)} = g(b
µ, Φd ) =
x∈R≥0
(Φd + 1)d − µ
b · Φd+1
d
= Φd+1
.
d
1−µ
b
To see that also I ≥ Φd+1
holds, note that by definition of g, we have for all µ ∈ (0, 1) that
d
d+1
g(µ, Φd ) = Φd .
t
u
Finally, we remark that the infima in the previous lemmas are also minima: From the proof of
Lemma 5.5, we know that for any fixed µ there is an “optimal” λ = maxx∈R≥0 {(x+1)d −µ·xd+1 },
see (5.5). From the proof of Lemma 5.6 we get that there is an “optimal” µ ∈ (0, 1), see (5.6).
5.4 Unweighted Games: Proof of Theorem 4.1
The proof of Theorem 4.1 is similar to the proof of Theorem 3.1. However, special care is
necessary to account for the discrete variables. We start with a technical lemma that will allow
us to make a simplification corresponding to the one from (5.3) to (5.4) in Lemma 5.5.
Lemma 5.7. Let d ∈ N0 . Then it holds for all µ ∈ (0, 1) that
(
d+1 )
n
o
x
x+1 d
max
−µ·
= max (x + 1)d − µ · xd+1 .
x∈N0 ,y∈N
x∈N0
y
y
Proof. We first note that the maximum on the right-hand side exists (Lemma 5.2).
Define g : N0 × N × [0, 1] → R,
g(x, y, µ) :=
x+1
y
d
d+1
x
−µ·
.
y
We will show that for all x ∈ N0 , y ∈ N, there exists some x
b ∈ N0 such that for all µ ∈ [0, 1]:
g(b
x, 1, µ) ≥ g(x, y, µ) .
Fix now x ∈ N0 and y ∈ N. If y ≥ x + 1 then we can choose x
b := 0 to see that for all µ ∈ [0, 1]
we have g(0, 1, µ) = 1 ≥ g(x, y, µ). In the following we therefore assume w.l.o.g. that y ≤ x.
Define x
b as the smallest integer such that g(b
x, 1, 0) ≥ g(x, y, 0), that is,
x+1−y
x
b :=
.
y
Since x ≥ y, we can write x = b1 · y + b2 for some b1 ∈ N and b2 ∈ [y − 1]0 . This shows that
b2 + 1
x
x
b = b1 − 1 +
= b1 =
.
y
y
Hence,
x+1
g(b
x, 1, 1) =
y
d
d+1
−x
b
x+1
≥
y
d
d+1 d+1
x
x+1 d
x
−
≥
−
= g(x, y, 1) .
y
y
y
Since g(b
x, 1, ·) is an affine function, this completes the proof.
19
t
u
Lemma 5.8. Let d ∈ N. Then,
λ ∀x, y ∈ N0 , ` ∈ Pd : y · `(x + 1) ≤ λ · y · `(y) + µ · x · `(x)
I :=
inf
1−µ
(λ,µ)∈R×(0,1)
(x + 1)d − µ · xd+1
= inf
max
1−µ
µ∈(0,1) x∈N0
Proof (Sketch). The proof is essentially a carbon copy of the proof for Lemma 5.5. Corresponding
λ
to equation (5.3), we have that I is the infimum of 1−µ
subject to
∀x ∈ N0 , y ∈ N, r ∈ [d]0 : λ ≥
x+1
y
r
r+1
x
−µ·
.
y
(5.7)
We can now apply Lemma 5.7 by which (5.7) is equivalent to
∀x ∈ N0 , r ∈ [d]0 : λ ≥ (x + 1)r − µ · xr+1 .
(5.8)
Applying the same reasoning as for Lemma 5.5 completes the proof.
t
u
Recall from the previous remarks for the proof of Theorem 3.1 that for each x ∈ N0 , there is
some µ ∈ (0, 1) so that x becomes a maximizer (on N0 ) of the polynomial x 7→ (x + 1)d − µ · xd+1 .
Consequently, inserting any such pair into the right-hand side of Lemma 5.8 immediately gives
an upper bound on the infimum I. In the following, we show how to choose such a pair optimally.
Lemma 5.9. Let d ∈ N and define k := bΦd c. Then,
(x + 1)d − µ · xd+1
(k + 1)2d+1 − k d+1 (k + 2)d
I := inf
max
=
.
1−µ
(k + 1)d+1 − (k + 2)d + (k + 1)d − k d+1
µ∈(0,1) x∈N0
Proof. Define g : (0, 1) × R → R,
g(µ, x) :=
(x + 1)d − µ · xd+1
.
1−µ
Note that g is differentiable on (0, 1) × R. The proof proceeds similar to Lemma 5.6. The upper
bound on I follows from the following two claims.
i) There is a µ
b ∈ (0, 1) with g(b
µ, k) = g(b
µ, k + 1)
Proof of claim (i): Solving the equality g(b
µ, k) = g(b
µ, k + 1) for µ
b gives
(k + 1)d − µ
b · k d+1 = (k + 2)d − µ
b · (k + 1)d+1
⇐⇒ µ
b=
(k + 2)d − (k + 1)d
.
(k + 1)d+1 − k d+1
(5.9)
Since both numerator and denominator of this fraction are positive, and since (k + 2)d <
(k + 1)d+1 and (k + 1)d > k d+1 , we get that µ
b ∈ (0, 1).
ii) Suppose there is a pair (b
µ, x
b) ∈ (0, 1) × N0 with g(b
µ, x
b) = g(b
µ, x
b + 1). Then g(b
µ, x
b) =
maxx∈N0 {g(b
µ, x)}.
Proof of claim (ii): From the mean value theorem, we know that there is a ξ ∈ (b
x, x
b + 1)
∂
with ∂x
g(b
µ, x)|x=ξ = 0. By Lemma 5.2, we get that g(b
µ, ·) has its only positive local
maximum at ξ and that g(b
µ, ·) is strictly increasing in [0, ξ) and strictly decreasing in
(ξ, ∞). Clearly, this implies g(b
µ, x
b) = g(b
µ, x
b + 1) = maxx∈N0 {g(b
µ, x)}.
20
Now fix µ
b as in claim (i). Up to this point, we have shown that
(i)
(ii)
µ, x)} = g(b
µ, k) =
I ≤ max{g(b
x∈N0
(k + 1)2d+1 − k d+1 (k + 2)d
.
(k + 1)d+1 − (k + 2)d + (k + 1)d − k d+1
To show that I is also lower-bounded by the right-hand-side expression, we show as last claim:
iii) It holds for all µ ∈ (0, µ
b) that g(µ, k + 1) > g(b
µ, k) and for all µ ∈ (b
µ, 1) that g(µ, k) >
g(b
µ, k).
Proof of claim (iii): For all x ∈ N0 it holds that g(·, x) is monotonic as can be seen by
looking at the partial derivatives:
∂
(x + 1)d − xd+1
g(µ, x) =
> 0 ⇐⇒ x < Φd ⇐⇒ x ≤ k
∂µ
(1 − µ)2
∂
g(µ, x) < 0 ⇐⇒ x > k .
∂µ
and
This immediately implies I ≥ g(b
µ, k).
20
g(·, 3)
10
g(µ, 2) = g(µ, 3)
g(·, Φ2 )
g(·, 2)
g(·, 1)
g(·, 0)
0
µ
0.5
1
Figure 5.1: Plot of functions g(·, x) for selected values of x, when d = 2. Note that g(·, Φ2 ) is
a constant and lies slightly above the intersection of g(·, 2) and g(·, 3). This is the
difference between the worst-case PoA for weighted and unweighted games.
Figure 5.1 illustrates the selection of µ
b: Here, the bold curve shows the function µ 7→
maxx∈N0 {g(µ, x)}. Note that Φ2 ≈ 2.148 and (Φ2 + 1)2 = Φ32 ≈ 9.909.
t
u
Finally, we remark that the infima in the previous lemmas are also minima: From the proof of
Lemma 5.8, we know that for any fixed µ there is an “optimal” λ = maxx∈N0 {(x + 1)d − µ · xd+1 }.
From the proof of Lemma 5.9 we get that there is an “optimal” µ ∈ (0, 1), see (5.9).
6 Conclusion
In this work, we gave exact values for the worst-case PoA in atomic unsplittable congestion
games, provided that all resource cost functions are bounded-degree polynomials with nonnegative coefficients. It had been known before that the asymptotic growth of the PoA is exponential
21
in the degree of the polynomials. Yet, it came as a surprise that this manifests already in rather
simple networks with only quadratic or cubic resource cost functions, where the PoA can be as
large as 9 or 40, respectively.
Qualitatively, we learned several things that could not be deduced from previous works: The
difference between the unweighted and weighted cases is small (though non-zero); there is no
difference at all between the worst-case PoA of network and abstract congestion games; and
we gave a structure for worst-case examples that—as we now know from hindsight—could be
modified and generalized to a worst-case example for arbitrary cost functions [37].
As an interesting direction for further research, it remains an open problem to rigorously
characterize what structures provide immunity against a high PoA and what structures cause
it. While, for instance, it is known that symmetric unweighted network congestion games on
so-called extension-parallel graphs always enjoy a PoA as low as in non-atomic games [13],
the worst-case PoA is unknown even for series-parallel graphs. Quite remarkably, for weighted
games, the worst-case PoA is known and no difference exists between parallel links and arbitrary
complex structures [5].
Acknowledgment We thank the anonymous reviewers for very valuable remarks.
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