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Chapter 4

Congestion Games

In this section, we introduce several classes of games that model the competitionof players over a finite set of resources whose cost change with the demand. Tra-ditionally, these games are phrased as cost minimization games. Minimizationgames are triplets G = (N,S,⇡) where ⇡ = (⇡i)i2N and ⇡i(s) is the private costof player i in strategy profile s.In a congestion game, each player i strives to solve a combinatorial optimizationproblem of the form

minX

e2A

ce(s) s.t. A 2 A,

where A ✓ 2E is the set of feasible allocations for player i. The players’ optimiza-tion problems are coupled via the cost functions of the resources as the cost ceof resource e is a function of the strategy profile. In the most basic model, oneassumes that the cost of a resource is a function of its users, i.e., ce(s) = ce(t)

whenever |{j 2 N : e 2 sj}| = |{j 2 N : e 2 tj}|. Such coupled optimization problemsare captured by the class of unweighted congestion games.

4.1 Unweighted Congestion Games

Definition 4.1 (Unweighted Congestion Game). For a finite set of resources E

with cost functions (ce)e2E, and an allocation vector A = (Ai)i2N, the corre-sponding unweighted congestion game is the strategic game G(A) = (N,S,⇡),where- N is a finite and non-empty set of n players,- Si = Ai for all players i,- ⇡i(s) =

Pe2si ce(xe(s)) for all players i, where xe(s) = |{j 2 N : e 2 sj}|.

Example 4.2. Consider the game in Figure 4.1. There are four resources e1,e2, e3and e4 with cost functions defined as ce1(x) = ce4(x) = 2x3 and ce2(x) = ce3(x) =

(x + 1)3 for all x � 0. The feasible allocations of player 1 are A = {e1, e2} andA 0 = {e3, e4}; for player 2 the feasible allocations are B = {e1, e3} and B 0 = {e2, e4}.

61

62 | Chapter 4. Congestion Games

A 0

A

B B 0

e3

e1

e4

e2

(a) Resource representation

u1

u2

v1

v2

e1 e2

e3 e4

(b) Graph representation

B B 0

A (24, 24) (29, 29)

A 0 (29, 29) (24, 24)

(c) Strategic representation

Figure 4.1: Congestion Game

Setting A = (Ai)i2{1,2} with A1 = {A,A 0} and A2 = {B,B 0} this defines a congestiongame. Inspecting the strategic representation of the game in Figure 4.4c, weobserve that the game has two pure Nash equilibria, i.e., (A,B) and (A 0, B 0).

Congestion games in which the feasible allocations of each player i can be repre-sented as the set of paths between two designated vertices ui and vi of a givengraph D = (V, E) are of particular interest.

Definition 4.3 (Network Congestion Game). A congestion game G = (N,S,⇡)

is called a network congestion game if there is a graph D = (V, E) and, for eachplayer i, two designated vertices ui, vi 2 V such that

Si = {P ✓ E : P is a ui, vi-path in D.}

for all players i.

The game from Example 4.2 obviously is a network congestion game as the graphrepresentation in Figure 4.4b shows.

Example 4.4 (Braess’ Paradox). A famous network congestion game is Braessparadox. There are 100 players striving to travel along a simple (u, v)-path inthe network in Figure 4.2a. In the unique equilibrium, there are 50 players oneach path resulting in a cost of 151 per player. After adding an additional edgewith travel time 0, in the unique equilibrium the cost of all players is 200, seeFigure 4.2b.

Theorem 4.5 (Rosenthal, 1973). Every unweighted congestion game has a pureNash equilibrium.

Preliminary version of February 16, 2016.

4.1. Unweighted Congestion Games | 63

u v

x 101

101 x

(a) Original graph

u1 v1

x 101

101 x

0

(b) Augmented graph

Figure 4.2: Breass’ paradox

0 1 2 3 4 5

P(x)

x

Figure 4.3: Potential function

Proof. Consider the function P : S ! R defined as

P(s) =X

e2E

xe(s)X

k=1

ce(k).

Intuitively, the function P corresponds to an approximation of the integral overthe cost functions of the resources, see Figure 4.3.Let s 2 S be arbitrary and let i 2 N and ti 2 Si. We then obtain

P(ti, s-i) =X

e2E

xe(ti,s-i)X

k=1

ce(k)

=X

e2E

xe(s)X

k=1

ce(k) +X

e2ti\si

ce(xe(s) + 1)-X

e2si\ti

ce(xe(s))

= P(s) + ui(ti, s-i)- ui(s)

Thus, P(ti, s-i)- P(s) = ⇡i(ti, s-i)- ⇡i(s) for all s 2 S, i 2 N and ti 2 Si.In particular, P(ti, s-i) < P(s) when ⇡i(ti, s-i) < ⇡i(s). Consider a sequence

s

0, s1, s2, . . . (4.1)


where for all l = 0, 1, . . . there is a player il such that s

l+1 = (til, sl-il

) for sometil 2 Si and ⇡il(til, s-il) < ⇡(s). Then, P is decreasing along the sequence andusing that S is finite, every such sequence must be finite. The endpoint of suchsequence is a pure Nash equilibrium. ⇤

4.2 Exact Potential Games

Definition 4.6 (Exact Potential Game). A strategic game G = (N,S,⇡) is calledan exact potential game if there is a function P : S ! R such that

P(ti, s-i)- P(s) = ⇡i(ti, s-i)- ⇡i(s)

for all s 2 S, i 2 N and ti 2 Si.

Definition 4.7 (Coordination Game). A strategic game G = (N,S,⇡) is a coor-dination game if ⇡i(s) = ⇡j(s) for all s 2 S.

Definition 4.8 (Dummy Game). A strategic game G = (N,S,⇡) is a dummygame if pii(si, s-i) = ⇡i(ti, s-i) for all i 2 N, s 2 S and ti 2 Si.

Theorem 4.9. A strategic game G = (N,S,⇡) is an exact potential game ifand only if there are functions ⇡

c : S ! R and ⇡

d : S ! R such that (N,S,⇡c)

is a coordination game, (N,S,⇡d) is a dummy game and ⇡(s) = ⇡c(s) + ⇡d(s)

for all s 2 S.

Proof. “(”: We claim that P = ⇡c

i = ⇡Cj is an exact potential function. To see

this, note that

⇡i(ti, s-i)- ⇡i(s) = ⇡c

i(ti, s-i) + ⇡d

i (ti, s-i)- ⇡c

i(s)- ⇡d

i (s)

= ⇡c

i(ti, s-i)- ⇡c

i(s)

for all s 2 S, i 2 N and ti 2 Si.‘‘ ) 00: Let P be an exact potential function for G. For all players i and all strategyprofiles s, let ⇡c

i(s) = P(s) and let ⇡c

i(s) = ui(s)- P(s). Then, G = (N,S,⇡c

i) is acoordination game. Moreover,

⇡i(s)- ⇡i(ti, s-i) = P(s)- P(ti, s-i)

which is equivalent to

⇡i(s)- P(s) = ⇡i(ti, s-i)- P(ti, s-i).

Thus, G = (N,S,⇡D) is a dummy game. ⇤


4.2. Exact Potential Games | 65

Definition 4.10 (Isomorphism Between Games). Two games G = (N,S,⇡) andG 0 = (N,S 0,⇡ 0) are isomorphic if, for all i 2 N, there is a bijection � : Si ! S 0

i

such that

⇡i(⇥i2Nsi) = ⇡ 0i(⇥i2N�i(si))

for all players i and strategy vectors s 2 S.

Theorem 4.11. Every coordination game is isomorphic to a congestion game.

Proof. Let G = (N,S,⇡) be an n-player coordination game. We proceed todefine a congestion game G = (NS

0,⇡ 0) isomorphic to G. For all s 2 S, introducea resource e(s) with cost function

ce(s)(x) =

�⇡1(s), if x = n

0, else.

Further, for a player i and a strategy si 2 Si, let

�i(si) =[

s-i2S-i

�e(si, s-i)

and let

S 0i =

⌦�i(si) : si 2 Si

↵

For all s 2 S, let s

0 = ⇥i2N�i(si). We obtain

⇡ 0i(s

0) =X

e2�(si)

ce�xe(s

0)�=

X

e2s 0i :xe(s 0)=n

ce�xe(s

0)�= ⇡1(s) = ⇡i(s),

where the last equality used that G is a coordination game. To obtain an isomor-phic congestion game we introduce a resource e(s) for all strategy profiles s 2 S,see Figure 4.4b. ⇤

Example 4.12 (Congestion Game Isomorphic to a Coordination Game). Consider thecoordination game G in Figure 4.4a. To obtain an isomorphic congestion game,we introduce a resource for each strategy profile s of G and define the players’strategies as in Figure 4.4b.

Theorem 4.13. Every dummy game is isomorphic to a congestion game.

Proof. Let G = (N,S,⇡) be a dummy game. For all players i and all s-i 2 S-i,we introduce a resource e(s-i) with cost function

ce(s-i)(x) =

�⇡i(si, s-i) if x = 1

0, else,


s2 s 02

s1 (0, 0) (1, 1)

s 01 (2, 2) (3, 3)

(a) Coordination game

�1(s 01)

�1(s1)

�2(s2) �2(s 02)

e(s 01, s2)

e(s1, s2)

e(s 01, s02)

e(s1, s 02)

(b) Isomorphic congestion game

where si 2 Si is arbitrary. Let E be the set of resources introduced this way.Further, for a player i with strategy si 2 Si, let

�i(si) =[

t-i2S-i

⌦e(t-i)

↵[[

j2N\{i}

[t-j2S-j:ti 6=si

⌦e(t-j)

↵

and let

S 0i =

��(si) : si 2 Si

Let i 2 N, si 2 Si and s-i 2 S-i be arbitrary and consider the strategy profiles = (si, s-i). We claim that for any resource e 2 E we have

�j 2 N : e 2 �j(sj)

= {i}, if e = e(s-i) for some player i,

��j 2 N : e 2 �j(sj) �� 2, else.

First, note that e(s-i) 2 �i(si). Moreover, for any player j 6= i, we have that

e(s-i) /2[

t-i2S-i:tj 6=sje(t-i)

and, thus, also e(s-i) /2 �j(sj). On the other hand, for any other resource e 2E \

�Si2N e(s-i)

�, we have that e = e(t-i) for some j 2 N and t-j 2 S-j and,

thus, e 2 �j(sj). Using that t-j 6= s-j, we further derive the existence of aplayer k 2 N \ {j} with tk 6= sk implying that e(t-j) 2 �k(sk).For all s 2 S, let s

0 = ⇥i2N�i(si). We then obtain

⇡ 0i(s

0) =X

e2�(si)

ce�xe(s

0)�

=X

t-i2S-i

ce(t-i)

�xe(t-i)(s

0)�+X

j2N\{i}

Xt-j2S-j:ti 6=si

ce(t-i)

�xe(t-i)(s

0)�

= ⇡i(si, s-i),


4.2. Exact Potential Games | 67

s2 s 02

s1 (0, 2) (1, 2)

s 01 (0, 3) (1, 3)

(a) Coordination game

�1(s 01)

�1(s1)

�2(s2) �2(s 02)

e(s1)

e(s 02)

e(s 01)

e(s2)

�1(s 01)

�1(s1)

�2(s2) �2(s 02)

e(s1)

e(s 02)

e(s 01)

e(s2)

(b) Isomorphic congestion game

which concludes the proof. ⇤

Example 4.14 (Congestion Game Isomorphic to a Dummy Game). Consider thedummy game G = (N,S,⇡) in Figure 4.4a. For each i 2 N and each strategypartial strategy profile s-i 2 S-i we introduce a resource e(s-i), which leads tothe strategies shown in Figure 4.4b.

We are now in position to state the main isomorphism result for congestion games.

Theorem 4.15. Every exact potential game is isomorphic to a unweighted con-gestion game.

Proof. By Theorem 4.9, we can write each exact potential game G = (N,S,⇡)

as the sum of a coordination game Gc = (N,S,⇡c) and a dummy game Gd =

(N,S,⇡d). By Theorem 4.11 and Theorem 4.13, there are congestion games G 0 =

(N,S 0,⇡ 0) and G 00 = (N,S 00,⇡ 00) with resources E 0 and E 00 that are isomorphic toGc and Gd respectively, i.e., there are bijections � 0

i : Si ! S 0i and � 00

i : Si ! S 00i for

all players i such that

⇡C

i (⇥i2Nsi) = ⇡ 0i(⇥i2N�

0i(si)) and ⇡D

i (⇥i2Nsi) = ⇡ 00i (⇥i2N�

00i (si)).

By renaming resources, we may assume that E and E 0 are disjoint. Consider thecongestion game G̃ = (N, S̃, ⇡̃) with resources E = E 0 [ E 00 and cost functionsas in G 0 and G 00 and consider the bijection �i : Si ! 2E defined as �i(si) =


A 0

A

B B 0

e3

e1

e4

e2

(a) Resource representation

u1

u2

v1

v2

e1 e2

e3 e4

(b) Graph representation

B B 0

A (62, 162) (66, 160)

A 0 (66, 160) (62, 162)

(c) Strategic representation

Figure 4.4: Weighted congestion Game

� 0i(si) [ � 00

i (si). We obtain

⇡̃i

�⇥i2N�i(si)

�= ⇡i(⇥i2N�

0(si)) + ⇡i(⇥i2N�00(si))

= ⇡C

i (⇥i2Nsi) + ⇡D

i (⇥i2Nsi)

= ⇡i(s),

where the first equation uses that E 0 and E 00 are disjoint. We conclude that G̃ isisomorphic to G. ⇤

4.3 Weighted Congestion Games

Weighted congestion games are a straightforward generalization of unweightedcongestion in which the player contribute differently to the congestion on theresources.

Definition 4.16 (Weighted Congestion Game). For a finite set of resources E withcost functions (ce)e2E, and allocation vector A = (Ai)i2N and a demand vectord = (di)i2N > 0, the corresponding unweighted congestion game is the strategicgame G(A) = (N,S,⇡), where- N is a finite and non-empty set of n players,- Si = Ai for all players i,- ⇡i(s) =

Pe2si dice

�xe(s)

�for all players i, where xe(s) =

Pj2N:e2sj dj.

Example 4.17. Consider a weighted variant of the congestion game from Exam-ple 4.2 the game in Figure 4.4. Cost functions are defined as ce1(x) = ce4(x) = 2x3

and ce2(x) = ce3(x) = (x+ 1)3 for all x � 0. The players’ demands are d1 = 1 andd2 = 2. This game does not have a pure Nash equilibrium.


4.3. Weighted Congestion Games | 69

0 1 2 3 4 5

P(x)x

Figure 4.5: For an affine function the potential function is independent of the ordering of theplayers.

Theorem 4.18. Weighted congestion games with affine cost functions have apure Nash equilibrium.

Proof. Consider the function P : S ! R defined as

P(s) =X

e2E

X

i2N:e2si

dice

⇣Xj2{1,...,i}:e2sj

dj

⌘.

For players of unit weight this function is equal to the potential function givenfor unweighted congestion games. Again, the contribution of a resource to thepotential function corresponds to a discrete approximation of the integral. How-ever, this approximation is not uniform and, thus, depends on the indices of theplayers (respectively, their demands), in general. The crucial observation is thatfor affine function, the potential is independent of the ordering of the players, seeFigure 4.5.In fact, we obtain for cost functions of the form ce(x) = aex + be with ae, be 2 Rthat

P(s) =X

e2E

X

i2N:e2si

dice

⇣Xj2{1,...,i}:e2sj

dj

⌘

=X

e2E

bexe(s) +

Xi,j2N:e2si\sj,ij

aedidj

!

=X

e2E

bexe(s) +

1

2

Xi,j2N:e2si\sj

aedidj +1

2

X

i2N:e2si

aed2i

!

=1

2

X

e2E

⇣ce�xe(s)

�+ ce(0)

⌘xe(s) +

1

2

X

i2N

X

e2si

⇣ce(di)- ce(0)

⌘di.

We claim that P is an exact potential, i.e., P(ti, s-i)-P(s) = ⇡i(ti, s-i)-⇡i(s) forall i 2 N, s 2 S and ti 2 Si. Since the potential function is independent of the


ordering of the players, it is without loss of generality to assume that i = n. Wethen calculate

P(tn, s-n) = P(s) + dn

X

e2tn\sn

ce�xe(tn, s-n)

�- dn

X

e2sn\tn

ce�xe(s)

�

= P(s) + ⇡i(tn, s-i)- ⇡i(s).

We conclude that P is a potential function and, thus, the game has a pure Nashequilibrium. ⇤

For � 2 R, let

Cexp

(�) = {f : R�0 ! R | f(x) = ae�x + b with a, b 2 R}.

Theorem 4.19. For every � 2 R, every weighted congestion game with costfunctions in C

exp

(�) has a pure Nash equilibrium.

Proof. For � = 0, the theorem follows from Theorem 4.18 so we may assume that� 6= 0. Consider the function P : S ! R defined as

P(s) =X

e2E

X

i2N:e2si

sgn(�)�1- e-�di

�ce

⇣Xj2{1,...,i}:e2sj

dj

⌘.

For cost functions of the form ce(x) = aee�x + be, this give

P(s) =X

e2E

X

i2N:e2si

⇣sgn(�)

�1- e-�di

�ae exp

⇣�X

j2{1,...,i}:e2sjdj

⌘+ be

⌘

=X

e2E

X

i2N:e2si

⇣sgn(�)aee

�P

j2{1,...,i}:e2sjdj - sgn(�)aee

�P

j2{1,...,i-1}:e2sjdj

+ besgn(�)(1- e�di)⌘.

Collapsing the telescoping sum, we obtain

P(s) = sgn(�)X

e2E

⇣ae

�e�x(s) - 1

�+ be

Xi2N:e2si

(1- e-�di)⌘,

which is independent of the ordering of the players. For �i = sgn(�)�1 - e-�di

�

we then obtain

P(tn, s-n) = P(s) + �n

X

e2tn\sn

ce�xe(tn, s-n)

�- �n

X

e2sn\tn

ce�xe(s)

�

= P(s) +dn

�n

�⇡i(tn, s-i)- ⇡i(s)

�.

Using that dn/�n > 0, we conclude that every sequence of profitable unilateralimprovements is finite and, thus, the game has a pure Nash equilibrium. ⇤



Definition 4.20 (Weighted Potential Game). A strategic game G = (N,S,⇡) iscalled weighted potential game if there is vector (wi)i2N > 0 and a functionP : S ! R such that

P(ti, s-i)- P(s) = wi

⇣⇡i(ti, s-i)- ⇡i(s)

⌘

for all s 2 S, i 2 N and ti 2 Si.

In the following, we show that the set of affine cost functions and the set ofexponential cost functions are the only sets of cost functions for which generalexistence results as in Theorem 4.18 and Theorem 4.19 are possible. In order tomake the statement precise, we introduce the notion of consistency of a set of costfunctions.

Definition 4.21 (Consistent Set of Cost Functions). A set C of cost functionsc : R�0 ! R is called consistent (for weighted congestion games) if every weightedcongestion game with the property that ce 2 C for all e 2 E has a pure Nashequilibrium.

The following lemma states that when characterizing consistent sets of cost func-tions, it is without loss of generality to assume that the set is closed under positiveinteger scalar multiplication.

Lemma 4.22. A set C of cost functions is consistent if and only if the set

C̄ := {g : R�0 ! R | f(x) = �f(x) for some � 2 N, f 2 C}

is consistent.

Proof Sketch. Every resource e with cost function �f where � 2 N and f 2 C

can be replaced by � resources with cost function f. ⇤

In order to characterize the sets of consistent cost functions, the following lemmawill be useful.

Lemma 4.23. Let C be a consistent set of strictly increasing functions. Then,

f(x+ y)- f(x)

f(x+ y)- f(y)=

g(x+ y+ t)- g(x+ t)

g(x+ y+ t)- g(y+ t)(4.2)

for all f, g 2 C and x, y, t 2 R�0.

Proof. By Lemma 4.22, it is without loss of generality to assume that C is closedunder scalar multiplication. Let , � 2 N be arbitrary and consider a weighted


{e3, e4}

{e1, e2}

{e1, e3} {e2, e4}{e1}

{e4}

e3

e1

e4

e2

Figure 4.6: Congestion game constructed in the proof of Lemma 4.23.

congestion game with four resources E = {e1, e2, e3, e4} with cost functions ce1 =

ce4 = g, ce2 = ce3 = �f and four players i, j, k, k 0 with demands di = x, dj = y,dk = dk 0 = t and strategy sets Si =

�{e1, e2}, {e3, e4}

, Sj =

�{e1, e3}, {e2, e4}

,

Sk =�{e1}

, Sk 0 =

�{e4}

.

Players i and j have two strategies each and players k and k 0 have one strategy eachresulting in four distinct strategy profiles. There are only two types of strategyprofiles: in the first type players i and j are together on a resource with costfunction g, but are not together on a resource with cost function �f. For thesecond type, it is the other way around. In a strategy profile s

1 of the first typewe have

⇡i(s1) = g(x+ y+ t) + �f(x) ⇡j(s

1) = g(x+ y+ t) + �f(y) (4.3a)

while in a strategy profile s

2 of the second type we have

⇡i(s2) = g(x+ t) + �f(x+ y) ⇡j(s

2) = g(y+ t) + �f(x+ y). (4.3b)

In order to have a pure Nash equilibrium, one of these two types must be beneficialto both players, i.e., at least one of the following two cases holds:

⇡`(s1) ⇡`(s

2) for ` 2 {i, j} or ⇡`(s1) � ⇡`(s

2) for ` 2 {i, j}. (4.4)

Combining (4.3) and (4.4), we obtain that at least one of the following cases holds:

� min

�f(x+ y)- f(x)

g(x+ y+ t)- g(x+ t),

f(x+ y)- f(y)

g(x+ y+ t)- g(y+ t)

✏or

�� max

�f(x+ y)- f(x)

g(x+ y+ t)- g(x+ t),

f(x+ y)- f(y)

g(x+ y+ t)- g(y+ t)

✏.



This yields

f(x+ y)- f(x)

g(x+ y+ t)- g(x+ t)=

f(x+ y)- f(y)

g(x+ y+ t)- g(y+ t).

Multiplying this equation with g(x+y+t)-g(x+t)f(x+y)-f(y) gives the claimed result. ⇤

Lemma 4.24. Let f be a continuous and strictly increasing function such that

f(x+ y)- f(x)

f(x+ y)- f(y)=

f(x+ y+ t)- f(x+ t)

f(x+ y+ t)- f(y+ t)(4.5)

for all x, y, t 2 R�0. Then, f is one of the two functional forms:

f(x) = ax+ b, or f(x) = ae�x + b.

Proof. Let ✏ > 0 be arbitrary and let x = ✏, y = 2✏, and t = m✏ for some m 2 N.Using (4.5), we obtain

� :=f(3✏)- f(1✏)

f(3✏)- f(2✏)=

f((m+ 3)✏)- f((m+ 1)✏)

f((m+ 3)✏)- f((m+ 2)✏)

for all m 2 N. We note that � 6= 1 since f(1✏) 6= f(2✏). Setting am := f(m✏) forall m 2 N and rearranging terms, we derive that the sequence (am)m2N obeys therecursive formula

0 = am+3 -�

�- 1am+2 +

1

�- 1am+1

for all m 2 N. The characteristic equation of this recurrence relation equals

x2 -�

�- 1x+

1

�- 1= (x- 1)

⇣x-

1

�- 1

⌘.

If � 6= 2, the characteristic equation has two distinct roots and am can be calculatedexplicitly and uniquely as

am = a⇣ 1

�- 1

⌘m+ b (4.6)

for some constants a, b 2 R. If, on the other hand, � = 2, we can calculate am as

am = am+ b (4.7)

for some constants a, b 2 R. Thus, f is affine or exponential for every integermultiple of ✏. As ✏ was arbitrary we may conclude that f is affine or exponentialon a dense subset of R�0. Using that f is continuous, the claimed result follows. ⇤


Theorem 4.25. A set C of strictly increasing and continuous functions is con-sistent for weighted congestion games if and only if one of the following two casesholds:1. C contains only affine functions of the form c(x) = ax+ b.2. C contains only exponential functions of the form c(x) = ae�x + b, where �

is equal for all functions in C.

Proof. The “if”-part of the statement follows from Theorem 4.18 and Theo-rem 4.19, so it suffices to show the “only if”-part.By Lemma 4.24, all functions in C are either linear or exponential. It is left toshow that C neither contains both affine and exponential functions nor C containsexponential functions with different � in the exponent. To this end, recall that,by Lemma 4.23 it is necessary that

f(x+ y)- f(x)

f(x+ y)- f(y)=

g(x+ y+ t)- g(x+ t)

g(x+ y+ t)- g(y+ t)

for all f, g 2 C and x, y, t 2 R�0. For all x, y, t 2 R�0, the ratio f(x+y+t)-f(x)f(x+y+t)-f(y+t) is

equal to x/y if f is linear, and is equal to e�(x-y) if f is of the form ae�x+b. Thus,(4.5) is satisfied for all x, y only if C is as claimed. ⇤

4.4 Computation of Equilibria

As congestion games are potential games, every sequence of unilateral improve-ments is finite. That is, in principle a pure Nash equilibrium can be computed byfollowing such a sequence of better replies, see Algorithm 4.1.

Input: Congestion game G

Output: Nash equilibrium s of GChoose s

0 2 S arbitrarilyk = 0

while s

k is not a Nash equilibrium dofind player i and ti 2 Si with ⇡i(ti, s-i) < ⇡i(s)s

k+1 = (ti, s-i)k = k+ 1

endreturn s

k

Algorithm 4.1: Better-reply algorithm

Usually, we assume that we can check in polynomial time for a player i whetherthere is a strategy ti 2 Si with ⇡i(ti, s-i) < ⇡i(s). This is trivial, if the set Si ✓ 2E

is given explicitly. If Si is given implicitly, e.g., the set Si corresponds to the set


4.4. Computation of Equilibria | 75

of si, ti-paths in a given graph, we can still solve the problem via a shortest-path-computation.However, it is not possible, to give a polynomial bound on the number of iterationsof said algorithm, even if all cost functions are affine-linear with integer coefficients,i.e., c(x) = ax+b with a, b 2 N. To see this, note that P(s0)

Pe2E

Pnk=1 ce(k) =P

e2E aen(n+1)

2+P

e2E nbe. Since ae and be are encoded in binary, this is notpolynomially bounded in the input.

4.4.1 The Class PLS

In fact, one can show that to compute the Nash equilibrium of a congestion game isas hard as any other local search problem. The fact that a local search problem hasa solution can be established via the following simple graph theoretic statement.

Every finite acyclic digraph has a sink.

In a local search problem, the acyclicity of the graph is established via a potentialfunction.Let us denote the subclass of problems in TFNP whose solutions are the locallyoptimal solutions of an objective function by PLS. The acronym PLS stands forpolynomial local search.

Definition 4.26. A problem ⇧R is in PLS if, for all instances I and all solutionsx 2 SI, Si = ⌃p(|I|), there is a neighborhood NI(x) ✓ SI and polynomial algorithmsCI : SI ! N, NI : SI ! NI [ {;}, Si = ⌃p(|I|) such that- CI : SI ! N computes the cost of solution s for instance I

- NI : SI ! NI(x) [ {;} computes s 0 with C(s 0) < C(s), or ;, when s is locallyoptimal.

- (I, x) 2 R , NI(x) = ;

CircuitFlipInput: Boolean circuit with n input bits and m output bits.Output: locally optimal input s 2 {0, 1}.

Local Max CutInput: Graph G = (V,A) with integer weights wa on each arc a.Output: partition V1, V2 2 2V such that V1 \ V2 = ;, V1 [ V2 = V such that�(V1) =

Pa=(v,w)2A:v2V1,w/2V1

wa � �(V 01) for all V 0

1 with |V1�V01| = 1.

Between problems in PLS there is the usual notion of reducability in TFNP, i.e.,⇧R reduces to problem ⇧S if there are polynomial time computable functions f

and g such that (I, g(x)) 2 R , (f(I), x) 2 S.


Theorem 4.27. Local Max Cut is PLS-complete.

Theorem 4.28. Calculating a Nash equilbrium for a congestion game (wherethe Si ✓ 2E are given explicilty) is PLS-complete.

Proof. Calculating a Nash equilibrium is clearly in PLS so we only have to arguebout completeness. For the PLS-hardness, we reduce from the PLS-complete prob-lem local max cut. Let G = (V,A) be an instance of Local Max Cut. Let W bethe sum of of the weights of all arcs in A, i.e., W =

Pe2E w(e). Observe, that for

every cut (V1, V2) we have w(�(V1)) = W-w(E(V1))-w(E(V2)), where E(U) ✓ E

is the set of all arcs having both endpoints in U ✓ V . Instead of maximizing theweight of arcs that traverse the cut, we can also minimize the weight of the arcsthe remain inside the two sets of the partition.The reduction is now as follows: For every arc a 2 A we add two resources ea,1and ea,2. For j 2 {1, 2}, we define

cea,j(x) =

�0, if x 1,wa/2, else.

Furthermore, for every node v 2 V we have a player v 2 N. Player v has twostrategies, Av := {Av,1,Av,2}, where Av,j := {ea,1 : e 2 �(v)} for j 2 {1, 2}. Thestrategies can be interpreted as follows. If player v chooses aloocation Av,1, thenthe node v 2 V belongs to the set V1 of the partition (V1, V2); if the player choosesAv,2, then the node v belongs to the set V2. An arc a = {u, v} contributes the costw(e) to the cut if and only if player u and v both belong to the same part of thepartition. This transformation can be done in polynomial time.Using the above construction, we can translate a strategy profile of the congestiongame into a feasible cut of Local Max Cut and vice versa. Let s be a Nashequilibrium of the congestion games and let (V1, V2) be the induced cut in G.As s is a Nash equilibrium, we have for any v with sv = Av,1 that

X

a=(v,w),w2V2

wa

2�

X

a=(v,w),w2V1

wa

2,

The same holds for a player v with sv = Av,2. Thus, the cut is locally optimal. ⇤

Theorem 4.28 shows that an efficient algorithm for calculating Nash equilibria insymmetric congestion games would imply the existence of efficient algorithms formany local search problems. This is very unlikely but not impossible. However,it is known that efficient algorithms cannot use the local search paradigm: Thereexist instances of Max Cut, such that every path from a node to a sink in thetransition graph has exponential length.One can also show that computing a Nash equilbrium is PLS-complete for networkcongestion games with affine linear costs, but the proof is more involved andbeyond the scope of this lecture.


4.4. Computation of Equilibria | 77

Theorem 4.29 (Ackermann et al., 2008). Calculating a Nash equilibrium for anetwork congestion games with affine-linear costs is PLS-complete.

4.4.2 Symmetric Network Congestion Games

Theorem 4.30. In symmetric network congestion games with non-decreasingcosts, a Nash equilibrium can be computed in polynomial time.

Proof. Let G be a symmetric network congestion game with resource set E.We proceed to show that the minimum of Rosenthals potential function can becomputed in polynomial time by a min-cost flow computation.To this end, we replace each edge e in the graph by n parallel edges e1, . . . , enbetween the same nodes. Edge ei is assigned cost ce(i), for i 2 {1, . . . , n}. Sincethe edge costs functions are non-decreasing, we have ce(1) ce(2) · · · ce(n). Alledges have capacity 1.By standard minimum cost flow algorithms, we can compute an integral minimumcost flow with flow value n in polynomial time. By the monotonicity of the costfunctions, it is without loss of generality to assume that the flow sends the flowonly along the cheapest k copies of each edge. Thus, the cost for sending theflow along these edges is ce(1) + · · ·+ ce(k), which corresponds to the potential thatRosenthal’s potential function assigns to edge e if k players use this edge. Conse-quently, we can translate the optimal solution of the min-cost flow problem into astate of the congestion game whose potential corresponds to the cost of the flow.Hence, the min-cost flow solution corresponds to a Nash equilibrium that globallyminimizes Rosenthal’s potential function. ⇤

4.4.3 Matroid Congestion Games

For a ground set E, let I ✓ 2E be a set system over E.

Definition 4.31 (Matroid). A pair (E, I) with I ✓ 2E is a matroid if1. ; 2 I.2. J 2 I and I ⇢ J ) I 2 I for all I, J 2 2E.3. I, J 2 I and |I| < |J| ) there is x 2 J \ I with I+ x 2 I.

Systems that only satisfy (1) and (2) are called independence system. Every setI 2 I is called independent and sets I 2 2E \ I are called dependent. Set-wiseminimal dependent sets are called circuits. Set-wise maximal independent setsare called basis. It follows from (3), that all bases of a matroid have the samecardinality. The cardinality of a basis of a matroid is called the rank of a matroid.


Example 4.32 (Graphical matroid). Let G be a graph. Then the subset I ⇢ 2E offorests of G is a matroid. If G is connected, the bases are the maximal cycle-freegraphs, i.e., the spanning trees of the graph.

Example 4.33 (Uniform matroid). Let E be a set of n elements. For an integer k,the set {I ✓ E : |I| k} is called a uniform matroid.

Example 4.34 (Partition matroid). Let E be a set of n elements and let E1, . . . , Ek

be a partition of E, i.e.,Sk

i=1 Ei = E and Ei \ Ej = ; for all i 6= j. Further, letb1, . . . , bk be non-negative integers. Then

{I ✓ E : |I \ Ei| bi for all i = 1, . . . , k}

is called a partition matroid.

Lemma 4.35. Let E be a finite set, B be the set of bases of a matroid (E, I)

and (ce)e2E a vector of real-values weights. Then, the minimization problem

min⌦X

e2B

ce : B 2 B↵

can be solved in polynomial time by the standard greedy algorithm, see Algo-rithm 4.2.Moreover, every optimal solution of this minimization problem can be obtainedby the greedy algorithm for an appropriate tie-breaking rule for edges with thesame weight.

Input: ground set E with weights (ce)e2E, matroid (E, I), given by anindependence oracle

Output: basis B 2 I minimizingP

e2B cesort elements such that c1 c2 · · · cnI := ;for e = 1, . . . , n do

if I [ {e} 2 I then I := I [ {e}

endreturn I

Algorithm 4.2: Greedy algorithm for matroids

A congestion game is a matroid congestion game, if, for every player i, there is amatroid (Ei, Ii) such that Ai is equal to the set of bases of Ii.

Theorem 4.36. Let G be a matroid congestion game. Then, every best-replysequence converges in polynomial time to a Nash equilibrium.


4.5. Non-Atomic Players | 79

Proof. The game has m resources with cost functions ce, each of which can beused by x players with x 2 {1, . . . , n}. Thus, there are at most n ·m different costvalues that can occur on the resources. Let L be a list of these cost values sortednon-decreasingly such that each value appears only once. For each resource E,consider the alternative cost function

c 0e : N ! {1, . . . , n ·m},

where c 0e(x) is equal to the index of the entry in the list L. Moreover, let ⇡ 0 denote

the private cost function based on the alternative cost functions c 0e.

Fix an arbitrary strategy profile s. Let i be a player such that there is a s 0i 2 Si

with ⇡i(s 0i, s-i) < ⇡i(s) and let

ti 2 arg mins 0i2Si

⇡i(s0i, s-i). (4.8)

be computed by the greedy algorithm. Then,

⇡ 0i(ti, s-i) < ⇡ 0

i(s),

i.e., the private cost decreases also with respect to the alternative costs after a bestreply. By Lemma 4.35, it is without loss of generality to assume that the minimumin (4.8) is computed by the greedy algorithm. It is easy to verify that this greedyalgorithm computes the same output for the original costs and the alternative costs(assuming the same tie-breaking rule). This implies ti 2 arg mins 0i2Si ⇡

0i(s

0i, s-i),

and, thus, ⇡ 0i(ti, s-i) ⇡ 0

i(s). If ⇡ 0i(ti, s-i) = ⇡ 0

i(s), then there is a tie-breakingrule such that si is the output of the greedy algorithm for the alternative costs,which implies si 2 arg mins 0i2Si ⇡i(s 0i, s-i), a contradiction.Next, consider Rosenthal’s potential function P 0 for the alternative costs. Usingthat c 0

e(x) nm for all e 2 E and x 2 {1, . . . , n}, we obtain

P 0(s) =X

e2E

xe(s)X

k=1

n2m2.

Using that every best-response reduces the potential by at least 1, the claimedresult follows. ⇤

4.5 Non-Atomic Players

In this section, we consider a slightly different model with an infinite number ofinfinitesimally small players. This model is conveniently represented as a flow.


u v

1

x

Figure 4.7: Pigou’s example

Definition 4.37 (Flow). Let G = (V, E) be a graph and let N be a set of com-modities (ui, vi, di) 2 V⇥V⇥R>0. For each commodity i, a flow is non-negativea vector fi = (fiP)P2Pi

, where Pi is the set of all paths from ui to vi. The flow isfeasible if

PP2Pi

fP = di. A multicommodity flow is a vector of flows (fi)i2N, onefor each commodity.

An alternative formulation of flows is the edge representation. An edge flow forcommodity i is a non-negative vector fi = (fie)e2E. An edge flow is feasible if

X

e2�+(v)

fie -X

e2�-(v)

fie =

8>><

>>:

di, if v = ui

-di, if v = vi

0, else.

It is a basic fact of the theory of flows that for each feasible edge flow, there isa corresponding feasible path flow and vice versa. In the following, we use pathflows and edge flows interchangeably and we write fe =

Pi2N fie.

The equilibrium concept used in this section is that of a Wardrop flow.

Definition 4.38 (Wardrop’s First Principle). Let G be a graph with flow-dependent edge costs ce : R�0 ! R and commodities (ui, vi, di). A multi-commodity flow f satisfies Wardrop’s first principle if for every commodity i

X

e2P

ce(fe) X

e2Q

ce(fe) for all P,Q 2 Pi with fP > 0.

A flow satisfying Wardrop’s first principle is called Wardrop equilibrium.

Example 4.39 (Pigou’s example). Consider the parallel link network in Fig. 4.7.The upper edge has a constant cost of 1; the lower link has a cost equal to theflow on that edge. One unit wants to travel from u to v.In the unique Wardrop equilibrium, all flow is routed along the bottom edge.To compute the System optimum, assume that we send (1 - p) units along thetop edge and p units along the bottom edge. The minimum is attained for

minp2[0,1]

(1- p) + p2,

which is minimized for p = 1/2 and gives a total cost of 3/4.



u v

x 1

1 x

(a) Original graph

u1 v1

x 1

1 x

0

(b) Augmented graph

Figure 4.8: Breass’ paradox

Example 4.40 (Braess’ Paradox reconsidered). Reconsider Braess’ paradox in anon-atomic context. There is 1 unit of flow travelling from u to v in the networkin Figure 4.8a. In the unique equilibrium, there is 1/2 units of flow on each pathresulting in a total cost of 3/2. After adding an additional edge with travel time 0,in the unique equilibrium the cost of all players is 2, see Figure 4.8b.

Theorem 4.41. In every network with continuous edge-cost functions ce, aWardrop flow exists. Moreover, this flow appears as the limit of a sequence ofpure Nash equilibria of atomic congestion games played on the same graph.

Proof. We show the result only for rational demands di. By scaling demands andcost functions, we may assume that the demand di of each commodity (ui, vi, di)

is integral. For all n 2 N, let Gn the atomic congestion game, where, for eachcommodity i, there are di · n players with demand 1/n, whose set of strategies isequal to all simple (ui, vi)-paths. By Rosenthal’s Theorem, each of these gameshas a pure Nash equilibrium. For each game Gn, choose a Nash equilibrium s

n

arbitrarily.For each such game Gn, consider the path flow fn defined as

fnPi =|{ players of commodity i that use Pi in s

n}|

n

for all commodities i and (ui, vi)-paths Pi.The sequence (fn)n2N, viewed as path flows for each commodity, is clearly bounded.Thus, by the Theorem of Bolzano and Weierstrass, there is a convergent subse-quence (fn(`))`2N. Let f⇤ be the limit of the convergent subsequence. We will arguethat this path flow f⇤ satisfies Wardrop’s first principle.First note that since for each flow fn, we have for each commodity i that the sum ofthe flow values of the (ui, vi)-paths is di, and the set of such flows is bounded, alsof⇤ is a valid multi-commodity path flow that satisfies all commodities’ demands.


In remains to show the defining inequality of Wardrop’s first principle.Suppose the inequality is not true, i.e., there is a commodity i and (ui, vi)-pathsPi, P 0

i with fPi > 0, but there is ✏ > 0 s.t.X

e2Pi

ce(f⇤e) > ✏+

X

e2P 0i

ce(f⇤e). (4.9)

Let m denote the number of edges in G. First, note that since the cost functionsce are continuous, they are uniformly continuous on [0, d], where d =

Pi di. Thus,

there is � > 0 such that |ce(x)- ce(x- �)| < ✏2m

for all x 2 [0, d] and e 2 E.Let k be the number of commodities and let l̃ 2 N be chosen such that the followinghold:1.��f⇤Pi - f

n(`)Pi

�� 2k

for all commodities i, (ui, vi)-paths Pi and ` � ˜̀,2. 1

n(˜`) �

2k, and

3. fn(`)Pi

> 0 for all ` � ˜̀.We obtain for ` � ˜̀ that

X

e2Pi

ce(f⇤e)

X

e2Pi

ce(fn(`)e ) +

✏

2X

e2P 0i

ce

⇣fn(`)e +

1

n(`)

⌘+

✏

2X

e2P 0i

ce(f⇤e) + ✏.

The first and the last inequality are satisfied since the cost functions ce are uni-formly continuous and f⇤e and f

n(`)e differ by at most �. The second inequality uses

that the flow fn(`) is a Nash equilibrium, in the sense that no atomic user (withdemand 1/n(`)) may improve when switching to path P 0

i . This contrdicts (4.9). ⇤

Lemma 4.42. A feasible flow f is a Wardrop flow if and only if it satisfies thefollowing variational inequality:

X

e2E

ce(fe)(fe - ge) 0 (4.10)

for all feasible multi-commodity flows g.

Proof. “(”: Let f be a flow satisfying (4.10), i 2 N and P,Q 2 Pi be two pathswith fiP > 0. Let � = fiP and consider the flow g defined as gj = gi for all j 2 N \ i

and

gie =

8>><

>>:

fie - �, if e 2 P \Q

fie + �, if e 2 Q \ P

fie, else.

By construction, g is feasible and, thus, (4.10) impliesX

e2E

ce(fe)(fe - ge) =X

e2P

ce(fe)�-X

e2Q

ce(fe)� 0.



This impliesP

e2P ce(fe) P

e2Q ce(fe).“)”: Let f be a Wardrop flow. Then, there is a constant ki for every commodity i

such thatP

e2P ce(fe) = ki for all P 2 Pi with fP > 0. In addition,P

e2Q ce(fe) � ki

for all Q 2 Pi with fQ = 0. For an arbitrary feasible flow g, we obtainX

e2E

ce(fe)fe =X

i2N

X

P2Pi

kifiP =

X

i2N

ki

X

P2Pi

fiP

=X

i2N

kidi =X

i2N

ki

X

P2Pi

giP =

X

i2N

X

P2Pi

kigiP

X

i2N

X

P2Pi

giP

X

e2P

ce(fe) =X

e2E

ce(fe)ge,

which concludes the proof. ⇤

Using this variational inequality, it can be shown that a Wardrop flow f can becomputed by solving a convex optimization problem.

Theorem 4.43. For continuous and non-decreasing cost functions, a feasibleflow is a Wardrop flow if and only if solves the following convex optimizationproblem:

minimizeX

e2E

Z fe

0

ce(t)dt (4.11)

subject toX

e2�+(v)

fie -X

e2�-(v)

fie =

8>><

>>:

di, if v = ui

-di, if v = vi

0, else.

for all v 2 V, i 2 N

fie � 0 for all e 2 E, i 2 N.

Proof. We only show that a Wardrop equilibrium f is an optimal solution to theabove problem. Using that the cost functions are non-decreasing, the objectivefunctions is clearly convex. Let P(f) =

Pe2E

Rfe0ce(t)dt. Recall that the first order

Taylor polynomial of a function h : Rn ! R in a 2 Rn has the form

Th(x;a) = h(a) + (x- a)Trh(a).

Consider the linear approximation (i.e., the first two terms of the Taylor series)of P in f, i.e.,

TP(x; f) = P(f) +X

e2E

ce(fe)(xe - fe).

By the convexity of P(x), the tangent plane is below the function, i.e.,

TP(x; f) = P(f) +X

e2E

ce(fe)(xe - fe) P(x)


for all feasible flows x. This implies

P(f) P(x) +X

e2E

ce(fe)(fe - xe).

P(x)

where we used variational inequality. We conclude that f minimizes the convexprogram as claimed. ⇤

With a similar approach, also the system optimal flow can be computed efficiently,we here require that the cost functions are semi-convex, i.e., that the functionx 7! xce(x) is convex for all e 2 E.

Theorem 4.44. For continuous and semi-convex cost functions, the systemoptimal flow can be computed solving the following convex problem:

minimizeX

e2E

ce(fe)fe (4.12)

subject toX

e2�+(v)

fie -X

e2�-(v)

fie =

8>><

>>:

di, if v = ui

-di, if v = vi

0, else.

for all v 2 V, i 2 N

fie � 0 for all e 2 E, i 2 N.

Note that the system (4.12) is similar to that of (4.11) except for the fact that forthe system optimum, we minimize C(f) =

Pe ce(fe)fe while for the equilibrium

we minimizeP

e

Rfe0ce(t)dt. Next observe that

ce(fe)fe =

Z fe

0

c̃e(t)dt

if

c 0e(fe)fe + fe = c̃e,

the system optimum is a Wardrop flow with respect to the modified cost functionsc̃e defined as c̃e(x) = c 0

e(x)x+ ce(x).Thus, we obtain the following direct corollary from Lemma 4.42.

Corollary 4.45. A feasible flow f solves (4.12) if and only ifX

e2E

⇣ce(fe) + c 0

e(fe)fe⌘(fe - ge) 0

for all feasible flows g.


4.6. Efficiency of Equilibria | 85

4.6 Efficiency of Equilibria

In light of Pigou’s and Braess’ network in which a Wardrop flow has higher to-tal cost than the system optimium, it is a natural question to ask how bad anequilibrium can be compared to an optimal flow.

Definition 4.46 (Price of anarchy). For a non-atomic routing game G, let f bea Wardrop equilibrium and let f⇤ be a system optimal flow. Then, the price ofanarchy is defined as

⇢(G) =C(f)

C(f⇤).

Moreover, for a set G of instances, let ⇢(G) = supG2G ⇢(G).

4.6.1 Nonatomic Games with Affine Cost Functions

Theorem 4.47. Let G be the class of non-atomic routing games with affine costfunctions. Then, ⇢(G) = 4/3.

Proof. The inequality ⇢(G) � 4/3 follows from Pigou’s example (Example 4.39.We proceed to show ⇢(G) 4/3. Let f be a Wardrop equilibrium and let g be afeasible flow. We obtain

C(f) =X

e2E

ce(fe)fe X

e2E

ce(fe)ge

by the variational inequality. Thus,

C(f) X

e2E

ce(ge)ge +

⇣ce(fe)- ce(ge)

⌘ge

!

Let us set We(fe, ge) =�ce(fe)- ce(ge)

�ge. Then, we obtain

C(f) X

e2E

ce(ge)ge +X

e2E

We(fe, ge).

We proceed to bound We(fe, ge) in terms of ce(fe)fe. To this end let

µ = supfe,ge2R�0

�ce(fe)- ce(ge)

�ge

ce(fe)fe.

First, note that for ge > fe, we have µ 0 as ce is non-decreasing, so the supremumis clearly attained for ge fe. For affine functions, this ratio is upper bounded by


1/4, see Figure . For a more formal proof, note that for affine functions of typec(x) = ax+ b with a, b 2 R�0, we have

µ supf,g,a,b2R�0

(af+ b- ag- b)g

(af+ b)f

= supf,g,a,b2R�0

(af- ag)g

(af+ b)f

= supf,g,a2R�0

(af- ag)g

af2

= supf,g2R�0

(f- g)g

f2

= supf,g2R�0

⇣1-

g

f

⌘gf

= supx2R>0

(1- x)x

=1

4

Finally, we obtain

C(f) C(g) +1

4C(f),

which implies the result. ⇤

4.6.2 Nonatomic Games with Arbitrary Cost Functions

For a cost function c : R�0 ! R�0, let

µ(c) = supf,g2R�0

�c(f)- c(g)

�g

c(f)f.

For a class C of functions, let µ(C) = supc2C µ(c).

Theorem 4.48. Let G be a non-atomic routing game with cost functions in C

where µ(C) < 1, then ⇢(G) 11-µ(C) .

Proof. As in the proof of Theorem 4.47, we obtain

C(f) =X

e2E

ce(fe)fe X

e2E

ce(fe)ge =X

e2E

⇣ce(fe)ge + ce(ge)ge - ce(ge)ge

⌘

� C(g) + µ(C)C(f),

which implies the result. ⇤

Under a mild assumption on C, we can also give a matching lower bound.


4.6. Efficiency of Equilibria | 87

Theorem 4.49. Let C be a class of functions that include all constant functionsand let G be the set of all non-atomic routing games with cost functions in C.Then ⇢(C) � 1

1-µ(C) .

Proof. Consider a Pigou-style instance with two cost functions c1 and c2 wherea total demand of d has to be routed from the origin to the destination. Letc1 = c where c 2 C is arbitrary and let c2 be the function that is constant to c(d).Then, sending a flow of d on the first edge is a Wardrop equilibrium f with costC(f) = dc(d). The cost of the system optimum f⇤ equals

C(f⇤) = minx2[0,d]

{xc(x) + c(d)(d- x)} = dc(d)- maxx2[0,d]

��c(d)- c(x)

�x .

Thus,

C(f)

C(f⇤)=

1-

maxx2[0,d]{(c(d)- c(x))x}

dc(d)

!-1

.

Taking the supremum over d 2 R�0 and c 2 C, we obtain the claimed result. ⇤

For a parameter m > 0, let Cm be the class of continuous, non-decreasing andsemi-convex functions such that c(�x) � �mc(x) for all c 2 C and x � 0 and� 2 [0, 1]. Note that for all m 2 N, the class Cm contains all polynomials withnon-negative coefficients and maximal degree at most m since

mX

k=0

ak(�x)k � �k

mX

k=0

akxk.

In addition, C1 contains all semi-convex and concave functions.

Lemma 4.50. µ(Cm) m(m+1)1+1/m .

Proof. Recall that

µ(Cm) = supc2C

supf,g2R�0

�c(f)- c(g)

�g

c(f)f.

The supremum is clearly attained for g f. Substituting x = g/f, we obtain

µ(Cm) = supc2C

supf,x2R�0

�c(f)- c(xf)

�xf

c(f)f

supc2C

supf,x2R�0

�c(f)- xmc(f)

�xf

c(f)f

= supx2R�0

�1- xm

�x


The unique solution to this maximization problem is attained for x = m

q1

m+1

which yields the claimed bound. ⇤

Corollary 4.51. Let G be a non-atomic routing games with cost functions inCm. Then, ⇢(G)

�1- m

(m+1)1+1/m

�-1.


chapter 4 congestion games - tu berlin · chapter 4 congestion games in this section, ... example...

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