FINITE COMPOSITE GAMES: EQUILIBRIA AND DYNAMICS SYLVAIN SORIN AND CHENG WAN Abstract.

arXiv:1503.07935v1 [math.OC] 27 Mar 2015

SYLVAIN SORIN AND CHENG WAN

Abstract. We introduce finite games with the following types of players:

(I) nonatomic players, (II) atomic splittable players, (III) atomic non splittable players.

We recall and compare the basic properties, expressed through variational inequalities, concerning equilibria, potential games and dissipative games, as well as evolutionary dynamics.

Then we consider composite games where the three types are present, a typical example being congestion games, and extend the previous properties of equilibria and dynamics.

Finally we describe an instance of composite potential game.

1. Introduction

We study equilibria and dynamics in finite games: there are finitely many “participants”

i ∈ I and each of them has finitely many “choices” p ∈ Sⁱ. The basic variable describing the interaction is thus a profile x = {xⁱ, i∈I}, where each xⁱ = {xⁱ_p, p∈Sⁱ} is an element of the simplex Xⁱ = ∆(Sⁱ) on Sⁱ. LetX =Q

i∈IXⁱ. We consider three frameworks:

(I) Population games where each participant i∈I corresponds to a population: a nonatomic set of agents having all the same characteristics. In this setupxⁱ_p is the proportion of agents of “type p” in populationi.

One can distinguish two kinds of I-player games where each participant i ∈ I stands for an atomic player:

(II) Splittable case: xⁱ_p is the proportion that player i allocates top. (The set of pure moves of player iisXⁱ.)

(III) Non splittable case: xⁱ_p is the probability that player ichooses p. (The set of pure moves is Sⁱ and xⁱ is a mixed strategy.)

As an example, consider the following network where a routing game of each of the three frameworks takes place. Assume that arc 1 and arc 2 are connecting oto d.

o d

arc 1

arc 2

First, in a population game, consider two groups of players going from oto d. Suppose that a proportion x¹₁ of group 1 is taking arc 1 while the rest of the group uses arc 2, and similarly for group 2.

Next, in the splittable case, consider two players who both have a stock to send from o to d and can split their stocks. Suppose that player 1 sends a fractionx¹₁ of his stock by arc 1 and the remaining by arc 2, and similarly for player 2.

Date: March 2015.

2010 Mathematics Subject Classification. 91A10, 91A22, 91A06, 91A13.

Key words and phrases. finite game, composite game, variational inequality, potential game, dissipative game, evolutionary dynamics, Lyapunov function.

1

Finally, in the non splittable case, still consider two players having to send their stock from o to d. However, they can no longer split their stock, but have to send it entirely by one arc.

Suppose that with probability x¹₁ player 1 sends it by arc 1 and with probability x¹₂ = 1−x¹₁ he sends it by arc 2, and similarly for player 2.

In all the three examples, the basic variables are (x¹₁, x²₁)∈[0,1]² which definex∈X.

2. Description of the models 2.1. Framework I: population games.

We consider here the nonatomic framework where each participanti∈I corresponds to a popu- lation of nonatomic players.

The payoff (fitness) is defined by a family of continuous functions{F_pⁱ, i∈I, p∈Sⁱ}, all fromX to R, whereF_pⁱ(x) is the outcome of a member in population ichoosingp, when the environment is given by the basic variable x.

An equilibriumis a point x∈X satisfying:

xⁱ_p>0⇒F_pⁱ(x)≥F_qⁱ(x), ∀p, q∈Sⁱ, ∀i∈I. (1) This corresponds to a Wardrop equilibrium[31].

Proposition 2.1. (Smith [23] and Dafermos [4])

An equivalent characterization of (1) is through the variational inequality:

hFⁱ(x), xⁱ−yⁱi ≥0, ∀yⁱ ∈Xⁱ,∀i∈I, (2) or alternatively:

hF(x), x−yi=X

i∈I

hFⁱ(x), xⁱ−yⁱi ≥0, ∀y∈X. (3) A special class of population games corresponds togames with external interaction where each Fⁱ depends only on x⁻ⁱ.

2.2. Framework II: atomic splittable.

In this case each participant i ∈ I corresponds to an atomic player with action set Xⁱ. Given functions F_pⁱ as introduced above, his gain is defined by:

Hⁱ(x) =hxⁱ, Fⁱ(x)i= X

p∈Sⁱ

xⁱ_pF_pⁱ(x).

In other words, it is the weighted average gain of all fractions allocated to different choices.

An equilibriumis as usual a profilex∈X satisfying:

Hⁱ(x)≥Hⁱ(yⁱ, x⁻ⁱ), ∀yⁱ ∈Xⁱ, ∀i∈I. (4) Suppose that for allp∈Sⁱ,F_pⁱ(·) is of class C¹ on a neighborhood Ω ofX, then

∂ Hⁱ

∂ xⁱ_p(x) =F_pⁱ(x) + X

q∈Sⁱ

xⁱ_q∂ F_qⁱ

∂ xⁱ_p(x).

Let∇ⁱHⁱ(x) stand for the gradient of Hⁱ(xⁱ, x⁻ⁱ) with respect toxⁱ. Then one has [13]:

Proposition 2.2. Any solution of (4) satisfies h ∇H(x), x−yi=X

i∈I

h ∇ⁱHⁱ(x), xⁱ−yⁱi ≥ 0, ∀y∈X. (5) Moreover, if each Hⁱ is concave with respect to xⁱ, there is equivalence.

Variational inequalities characterizing Nash equilibrium in atomic splittable games (Haurie and Marcotte [9]) and those characterizing Wardrop equilibrium in nonatomic games have different origins. Inequalities (5) for a Nash equilibrium are obtained as first order conditions, while (2) for a Wardrop equilibrium are derived directly from its definition.

2.3. Framework III: atomic non splittable.

We consider here anI-player game where the payoff is defined by a family of functions{Gⁱ, i∈I}, all from S = Q

i∈ISⁱ to R. We still denote by G the multilinear extension to X where each Xⁱ= ∆(Sⁱ) is considered as the set of mixed actions.

An equilibrium is a profile x∈X satisfying:

Gⁱ(xⁱ, x⁻ⁱ)≥Gⁱ(yⁱ, x⁻ⁱ), ∀yⁱ ∈Xⁱ, ∀i∈I. (6) Let VGⁱ denote the vector payoff associated toGⁱ. Explicitly, VGⁱ_p :X⁻ⁱ → R is defined by VGⁱ_p(x⁻ⁱ) =Gⁱ(p, x⁻ⁱ), for all p∈Sⁱ. HenceGⁱ(x) =hxⁱ, VGⁱ(x⁻ⁱ)i.

An equilibrium is thus a profile x∈X satisfying:

hVG(x), x−yi=X

i∈I

hVGⁱ(x⁻ⁱ), xⁱ−yⁱi ≥0, ∀y∈X. (7) 2.4. Remarks.

Framework I and framework III have been extensively studied. III corresponds to games with external interaction and the multilinearity of VGⁱ(x⁻ⁱ) will not be used.

Note that F,∇H and VG play similar roles in the three frameworks. This can be seen from the three variational characterizations of equilibrium: (2), (5) and (7). We call F, ∇H and VG evaluation functions and denote them by Φ in each of the three frameworks.

From now on, we consider the following class of games which includes i) population games where F_pⁱ are continuous on X for all i and p, ii) atomic splittable games where Hⁱ is concave and of classC¹ on a neighborhood ofXⁱ, and iii) atomic non splittable games. A typical game in this class is denoted by Γ(Φ), where Φ is its evaluation function.

Definition 2.1. NE(Φ)is the set of x∈X satisfying:

hΦ(x), x−yi ≥0, ∀y∈X. (8) NE(Φ) is the set of equilibria ofΓ(Φ).

The next result recalls general properties of a variational inequality on a convex set.

Theorem 2.1. Let C⊂R^dbe a convex set and Ψa map from C to R^d. Consider the variational inequality:

hΨ(x), x−yi ≥0, ∀y∈C. (9) Four equivalent representations are given by:

Ψ(x)∈N_C(x), (10)

where N_C(x) is the normal cˆone to C atx;

Ψ(x)∈[TC(x)]^⊥, (11)

where TC(x) is the tangent cˆone to C at x and [TC(x)]^⊥ its polar;

Π_TC(x)Ψ(x) = 0, (12)

where Π is the projection operator on a closed convex subset; and

ΠC[x+ Ψ(x)] =x. (13)

Proof. hΨ(x), x−yi ≥0 for all y ∈ C is equivalent to Ψ(x) ∈ NC(x). Hence, Ψ(x) ∈ [TC(x)]^⊥ and Π_TC(x)Ψ(x) = 0 by Moreau’s decomposition [16].

Finally the characterization of the projection gives:

x+ Ψ(x)−ΠC[x+ Ψ(x)], y−ΠC[x+ Ψ(x)]

≤0, ∀y∈X.

Therefore, ΠC[x+ Ψ(x)] =xis the solution.

Note that this result holds in a Hilbert space.

3. Potential and dissipative games 3.1. Potential games.

Definition 3.1. A real function W, of class C¹ on a neighborhood Ωof X, is a potential for Φ if for each i∈I, there is a strictly positive function µⁱ(x) defined onX such that

∇ⁱW(x)−µⁱ(x)Φⁱ(x), yⁱ

= 0, ∀x∈X,∀yⁱ∈X₀ⁱ, ∀i∈I, (14) where X₀ⁱ ={y∈R^|Si|, P

p∈Sⁱy_p= 0} is the tangent space to Xⁱ.

The game Γ(Φ) is then called apotential game and one says that Φ derives fromW. Some alternative definitions of potential games have been used.

Proposition 3.1. Suppose Ω a neighborhood ofX. Consider the following two statements:

∃W: Ω7→R,∀i∈I,∃µ_i:X 7→R^∗₊,s.t. ∂W(x)

∂xⁱ_p −∂W(x)

∂xⁱ_q =µⁱ(x)[Φⁱ_p(x)−Φⁱ_q(x)], ∀x∈Ω,∀p, q∈Sⁱ; (15)

∃W: Ω7→R,∀i∈I,∃µi:X 7→R^∗+, s.t. ∂W(x)

∂xⁱ_p =µⁱ(x)Φⁱ_p(x), ∀x∈Ω,∀p∈Sⁱ. (16) Then: (16) ⇒ (15) ⇒ (14)

Proof. (16)⇒ (15): Clear.

(15) ⇒ (14): (15) implies that the vector {^∂W_∂x^(x)i

p −µⁱΦⁱ_p(x)}_p∈Sⁱ is proportional to (1, . . . ,1)

hence is orthogonal to X₀ⁱ.

Sandholm [20] defines a population potential game by (16) with µ_i ≡1 for all i.

Monderer and Shapley [15] defines potential games for finite games, which is equivalent to our definition (15) in the frameworkIII.

Theorem 3.1. Let Γ(Φ) be a game with potential W. 1. Every local maximum of W is an equilibrium of Γ(Φ).

2. If W is concave on X, then any equilibrium ofΓ(Φ) is a global maximum of W on X.

Proof. Since a local maximumx ofW on the convex set X satisfies:

h∇W(x), x−yi ≥0, ∀y∈X, (17)

it follows from (14) thathµⁱ(x)Φⁱ(x), xⁱ−yⁱi ≥0 for all iand for all y ∈X. This further yields (8).

On the other hand, if W is concave onX, a solutionx of (17) is a global maximum of W on

X.

3.2. Dissipative games.

Definition 3.2. The game Γ(Φ)is dissipative if Φsatisfies:

hΦ(x)−Φ(y), x−yi ≤0, ∀(x, y)∈X×X.

It is strictly dissipativeif

hΦ(x)−Φ(y), x−yi<0, ∀(x, y)∈X×X with x6=y.

In the framework of population games, Hofbauer and Sandholm [11] introduce this class of games and call them “stable games”.

Notice that if Φ is dissipative and derives from a potential W, it implies that W is concave.

The set of equilibria in dissipative games has a specific structure (see [11]) described as follows.

Definition 3.3. SNE(Φ)is the set of x∈X satisfying:

hΦ(y), x−yi ≥0, ∀y∈X. (18)

Lemma 3.1.

SNE(Φ)⊂NE(Φ).

If Γ(Φ) is dissipative, then

SNE(Φ) =NE(Φ).

Proof. Given x^∗ ∈ SNE(Φ), for all x ∈ X and ε∈]0,1[, define y =εx+ (1−ε)x^∗. Then, (18) yields

hΦ(y), x^∗−xi ≥0, ∀x∈X, which implies, by continuity,

hΦ(x^∗), x^∗−xi ≥0, ∀x∈X.

On the other hand, if x^∗ ∈NE(Φ) and Φ is dissipative, by addinghΦ(y)−Φ(x^∗), x^∗−yi ≥0

and hΦ(x^∗), x^∗−yi ≥0, one hasx^∗ ∈SNE(Φ).

Corollary 3.1. If Φ is dissipative,NE(Φ) is convex.

A strictly dissipative game Γ(Φ)has a unique equilibrium.

The description is more precise in the smooth case.

Proposition 3.2. Suppose that Φis of class C¹ on a neighborhood Ωof X. Denote byJ_Φ(x)the Jacobian matrix of Φatx, i.e. J_Φ(x) = ^∂Φip

∂x^jq

q∈S^j

p∈Sⁱ. Then, Φdissipative implies that J_Φ(x) is negative semidefinite on T_X(x), the tangent cˆone toX at x.

Proof. Givenx∈Xandz∈TX(x), there existsǫ >0 such thatx+tz∈X for allt∈]0, ǫ]. Hence hΦ(x+tz)−Φ(x), x+tz−xi ≤ 0, which implies that hΦ(x+tz)−Φ(x)

x , zi ≤0. Since Φ is of class

C¹, lettingt go to 0 yieldshJ_Φ(x)z, zi ≤0.

Definition 3.4. Suppose that Φ is of class C¹ on a neighborhood Ω of X. The game Γ(Φ) is strongly dissipative if J_Φ(x) is negative definite on T_X(x).

4. Dynamics 4.1. Definitions.

The general form of a dynamics describing the evolution of the strategic interaction in game Γ(Φ) is

˙

x=BΦ(x), x∈X, whereX is invariant so that for eachi∈I,B_Φⁱ(x)∈X₀ⁱ.

First recall the definitions of several dynamics expressed in terms of Φ.

4.1.1. Replicator dynamics (RD). (Taylor and Jonker [28])

˙

xⁱ_p =xⁱ_p[Φⁱ_p(x)−Φⁱ(x)], p∈Sⁱ, i∈I, where

Φⁱ(x) =hxⁱ,Φⁱ(x)i= X

p∈Sⁱ

xⁱ_pΦⁱ_p(x) is the average evaluation for participanti.

4.1.2. Brown-von-Neumann-Nash dynamics (BNN). (Brown and von Neumann [2], Smith [24, 26], Hofbauer [10])

˙

xⁱ_p = ˆΦⁱ_p−xⁱ_p X

q∈Sⁱ

Φˆⁱ_q, p∈Sⁱ, i∈I,

where ˆΦⁱ_q= [Φⁱ_q(x)−Φⁱ(x)]⁺is called the “excess evaluation” ofp. (Recall that [t]⁺,max{t,0}.)

4.1.3. Smith dynamics (Smith). (Smith [25])

˙

xⁱ_p = X

q∈Sⁱ

xⁱ_q[Φⁱ_p(x)−Φⁱ_q(x)]⁺−xⁱ_p X

q∈Sⁱ

[Φⁱ_q(x)−Φⁱ_p(x)]⁺, p∈Sⁱ, i∈I, where [Φⁱ_p(x)−Φⁱ_q(x)]⁺ corresponds to pairwise comparison [21].

4.1.4. Local/direct projection dynamics (LP). (Dupuis and Nagurney [5], Lahkar and Sandholm [14])

˙ xⁱ= Π_T

Xi(xⁱ)[Φⁱ(x)], i∈I, where we recall thatT_Xi(xⁱ) denotes the tangent cˆone toXⁱ at xⁱ.

4.1.5. Global/target projection dynamics (GP). (Friesz et al. [6], Tsakas and Voorneveld [29])

˙

xⁱ = Π_Xⁱ[xⁱ+ Φⁱ(x)]−xⁱ, i∈I.

Recall that the two dynamics above are linked by: Π_T

Xi(xⁱ)[Φⁱ(x)] = lim_δ→0+^ΠXi(xi+δΦ_δ ^i(x))−xi. 4.1.6. Best reply dynamics (BR). (Gilboa and Matsui [7])

˙

xⁱ ∈BRⁱ(x)−xⁱ, i∈I, where

BRⁱ(x) ={yⁱ ∈Xⁱ, hyⁱ−zⁱ,Φⁱ(x)i ≥0,∀zⁱ∈Xⁱ}.

4.2. General properties.

We define here properties expressed in terms of Φ.

Definition 4.1. Dynamics B_Φ satisfies:

i) positive correlation (PC) [20] if:

hB_Φⁱ(x),Φⁱ(x)i>0, ∀i∈I,∀x∈X s.t. B_Φⁱ(x)6= 0.

(This corresponds to MAD (myopic adjustment dynamics)[27]: assuming the configuration given, an unilateral change should increase the evaluation);

ii)Nash stationarity if:

for x∈X, B_Φ(x) = 0 if and only if x is an equilibrium of Γ(Φ).

Proposition 4.1. (RD), (BNN), (Smith), (LP), (GP) and (BR) satisfy (PC).

Proof.

(1) RD:

hB_Φⁱ(x),Φⁱ(x)i= X

p∈Sⁱ

xⁱ_p

Φⁱ_p(x)−Φⁱ(x) Φⁱ_p(x)

= X

p∈Sⁱ

xⁱ_p

Φⁱ_p(x)−Φⁱ(x)2

+ X

p∈Sⁱ

xⁱ_p

Φⁱ_p(x)−Φⁱ(x) Φⁱ(x)

= X

p∈Sⁱ

xⁱ_p

Φⁱ_p(x)−Φⁱ(x)2

+ X

p∈Sⁱ

xⁱ_pΦⁱ_p(x)−Φⁱ(x) Φⁱ(x)

= X

p∈Sⁱ

xⁱ_p

Φⁱ_p(x)−Φⁱ(x)2

≥0.

The equality holds if and only if for all i ∈ I, p ∈ Sⁱ, xⁱ_p[Φⁱ(x)−Φⁱ(x)]² = 0 or, equivalently xⁱ_p[Φⁱ(x)−Φⁱ(x)] = 0, hence B_Φ(x) = 0.

(2) BNN:

hB_Φⁱ(x),Φⁱ(x)i= X

p∈Sⁱ

Φˆⁱ_p(x)−xⁱ_p X

q∈Sⁱ

Φˆⁱ_q(x) Φⁱ_p(x)

= X

p∈Sⁱ

Φˆⁱ_p(x)Φⁱ_p(x)− X

p∈Sⁱ

xⁱ_pΦⁱ_p(x) X

q∈Sⁱ

Φˆⁱ_q(x)

= X

p∈Sⁱ

Φˆⁱ_p(x)Φⁱ_p(x)−X

q∈Sⁱ

Φˆⁱ_q(x)Φⁱ(x)

= X

p∈Sⁱ

Φˆⁱ_p(x)

Φⁱ_p(x)−Φⁱ(x)

= X

p∈Sⁱ

( ˆΦⁱ_p(x))²

≥0.

The equality holds if and only if for alli∈I and allp∈Sⁱ, ˆΦⁱ_p(x) = 0, in which caseB_Φ(x) = 0.

(3) Smith:

hBⁱ_Φ(x),Φⁱ(x)i= X

p∈Sⁱ

X

q∈Sⁱ

xⁱ_q[Φⁱ_p(x)−Φⁱ_q(x)]⁺

Φⁱ_p(x)−X

p∈Sⁱ

xⁱ_pΦⁱ_p(x)X

q∈Sⁱ

[Φⁱ_q(x)−Φⁱ_p(x)]⁺

=X

p,q

xⁱ_qΦⁱ_p(x)[Φⁱ_p(x)−Φⁱ_q(x)]⁺−X

q,p

xⁱ_qΦⁱ_q(x)[Φⁱ_p(x)−Φⁱ_q(x)]⁺

=X

p,q

xⁱ_q([Φⁱ_p(x)−Φⁱ_q(x)]⁺)²

≥0.

The equality holds if and only if for all q∈Sⁱ, either xⁱ_q= 0 or Φⁱ_q(x)≥Φⁱ_p(x) for all p∈Sⁱ, in which case B_Φ(x) = 0.

(4) LP: Recall that, ifN_Xi(xⁱ) denotes the normal cˆone toXⁱatxⁱ(the polar ofT_Xi(xⁱ)), then for anyv ∈R^Si: v= Π_T

Xi(xⁱ)v+ Π_N

Xi(xⁱ)v andhΠ_T

Xi(xⁱ)v,Π_N

Xi(xⁱ)vi= 0 (Moreau’s decomposition, [16]). Thus,

hB_Φⁱ(x),Φⁱ(x)i= Π_T

Xi(xⁱ)[Φⁱ(x)],Φⁱ(x)

= Π_T

Xi(xⁱ)[Φⁱ(x)],Π_T

Xi(xⁱ)[Φⁱ(x)] + Π_N

Xi(xⁱ)[Φⁱ(x)]

= Π_T

Xi(xⁱ)[Φⁱ(x)]

2

≥0,

and the equality holds if and only if for all i∈I, Π_T

Xi(xⁱ)[Φⁱ(x)] = 0, i.e. B_Φ(x) = 0.

(5) GP: Let z=x+ Φ(x). Then, hB_Φⁱ(x),Φⁱ(x)i=

Π_Xⁱ(zⁱ)−xⁱ, zⁱ−xⁱ

=

Π_Xⁱ(zⁱ)−xⁱ, zⁱ−Π_Xⁱ(zⁱ) + Π_Xⁱ(zⁱ)−xⁱ

=−

xⁱ−Π_Xⁱ(zⁱ), zⁱ−Π_Xⁱ(zⁱ)

+kΠ_Xⁱ(zⁱ)−xⁱk²

≥ kΠ_Xi(zⁱ)−xⁱk²

≥0.

The second last inequality holds sincexⁱ ∈Xⁱ, thus

xⁱ−Π_Xi(zⁱ), zⁱ−Π_Xi(zⁱ)

≤0. The equality occurs in both inequalities if and only ifxⁱ = Π_Xⁱ(zⁱ) for all i∈I, in which caseBΦ(x) = 0.

(6) BR:

hB_Φⁱ(x),Φⁱ(x)i=hyⁱ−xⁱ,Φⁱ(x)i ≥0

sinceyⁱ ∈BRⁱ(x). The equality holds if and only ifxⁱ∈BRⁱ(x) for alli∈I, henceB_Φ(x) = 0.

Proposition 4.2. (BNN), (Smith), (LP), (GP) and (BR) satisfy Nash stationarity on X.

(RD) satisfy Nash stationarity on intX.

Proof.

(1) BNN: x∈NE(Φ) is equivalent to: ˆΦp(x) = 0 for all p∈S. HenceB_Φ(x) = 0.

Reciprocally, assume the existence of i∈I and p∈Sⁱ such that ˆΦⁱ_p(x)>0. Since there exists q withxⁱ_q>0 and ˆΦⁱ_q(x) = 0, one obtains Bⁱ_Φ,q(x)<0, contradiction.

(2) Smith: Assume x ∈NE(Φ). Then for all i∈I and q ∈ Sⁱ, either xⁱ_q = 0 or Φⁱ_q(x) ≥Φⁱ_p(x) for all p∈Sⁱ. HenceB_Φ(x) = 0.

Reciprocally, assume the existence ofi∈I andp∈Sⁱ such thatxⁱ_p >0 and [Φⁱ_q(x)−Φⁱ_p(x)]⁺>

0. Choose such ap with smallest Φⁱ_p(x) then one obtainsB_Φ,pⁱ (x)<0.

(3) LP: The result follows from (12).

(4) GP: The result follows from (13).

(5) BR: By definitionB_Φ(x) = 0 if and only ifxⁱ∈BRⁱ(x) hencex∈NE(Φ).

(6) RD: Assume x ∈ intX∩NE(Φ). Then, Φⁱ_p(x) = Φⁱ(x) for all i ∈ I and p ∈ Sⁱ and thus B_Φ(x) = 0.

Reciprocally, ifx∈intX and B_Φ(x) = 0,x is equalizing hence in NE(Φ).

4.3. Potential games.

We establish here results that are valid for all three classes of games.

Proposition 4.3. Consider a potential game Γ(Φ) with potential function W. If the dynamics

˙

x = B_Φ(x) satisfies (PC), then W is a strict Lyapunov function for B_Φ. Besides, all ω-limit points are rest points of B_Φ.

Proof. Consider x ∈ X. Let {xt}_t≥0 be the trajectory of B_Φ with initial point x₀ = x, and V_t=W(x_t) for t≥0. Then

V˙t=h∇W(xt),x˙ti=X

i∈I

h∇ⁱW(xt),x˙ⁱ_ti=X

i∈I

µⁱ(x)hΦⁱ(xt),x˙ⁱ_ti ≥0.

(Recall that ˙xt∈X₀.) Moreover,hΦⁱ(xt),x˙ⁱ_ti= 0 holds for alliif and only if ˙x=B_Φ(xt) = 0.

One concludes by using Lyapunov’s theorem (e.g. Th. 2.6.1. in [12]).

This result is proved in Sandholm [20] for the version of potential games in frameworkIdefined by (16).

It follows that, with the appropriate definitions, the convergence results established for several dynamics and potential games in frameworkI extend to all the frameworks. Explicitly:

Proposition 4.4. Consider a potential game Γ(Φ) with potential functionW.

If the dynamics is (RD), (BNN), (Smith), (LP), (GP) or (BR),W is a strict Lyapunov function for B_Φ.

In addition, except for (RD), allω-limit points are equilibria of Γ(Φ).

4.4. Dissipative games.

Proposition 4.5. Consider a dissipative game Γ(Φ).

(1) RD: Let x^∗ ∈NE(Φ). Define [11]:

H(x) =X

i∈I

X

p∈supp(x^i∗)

x^i∗_p lnx^i∗_p xⁱ_p. Then H is a local Lyapunov function.

If Γ(Φ) is strictly dissipative, then H is a local strict Lyapunov function.

(2) BNN: AssumeΦ C¹ on a neighborhood Ω of X. Define [24, 26, 10]:

H(x) = 1 2

X

i∈I

X

p∈Sⁱ

Φˆⁱ_p(x)². Then H is a strict Lyapunov function which is minimal on NE(Φ).

(3) Smith: Assume ΦC¹ on a neighborhood Ωof X. Define [25]:

H(x) =X

i∈I

X

p,q∈Sⁱ

xⁱ_p

[Φⁱ_q(x)−Φⁱ_p(x)]^{+ 2}. Then H is a strict Lyapunov function which is minimal on NE(Φ).

(4) LP: Let x^∗ ∈NE(Φ). Define [17, 33, 18]:

H(x) = 1

2kx−x^∗k². Then H is a Lyapunov function.

If Γ(Φ) is strictly dissipative, then H is a strict Lyapunov function.

(5) GP: AssumeΦ C¹ on a neighborhood Ω of X. Define [19]:

H(x) = sup

y∈X

hy−x,Φ(x)i − 1

2ky−xk². Then H is a Lyapunov function.

If Γ(Φ) is strongly dissipative, then H is a strict Lyapunov function.

(6) BR: Assume Φ C¹ on a neighborhood Ω of X. Define [11]:

H(x) = sup

y∈X

hy−x,Φ(x)i.

Then H is a strict Lyapunov function which is minimal on NE(Φ).

Proof. For a trajectory of dynamics ˙x=B_Φ(x) with initial point x₀,

dH

dt(xt) =h∇H(xt),x˙_ti=h∇H(xt),B_Φ(xt)i=X

i

h∇ⁱH(xt),B_Φⁱ (xt)i.

Hence we focus onh∇H(xt),B_Φ(xt)i. The subscript for timet is omitted.

(1) RD: Given an equilibriumx^∗, defineH(x) =P

i∈Ihⁱ(xⁱ) with hⁱ(xⁱ) =P

p∈supp(x^i∗)x^i∗_p ln^x_x^i∗pi p. H has a strict local minimum atx^∗. In fact, consider the neighborhood ofx^∗ inX defined by V ={x∈X, supp(x^∗)∈supp(x)}. The concavity of lnxand Jensen’s inequality imply:

hⁱ(xⁱ) =− X

p∈supp(x^i∗)

x^i∗_p ln xⁱ_p

x^i∗_p ≥ −ln X

p∈supp(x^i∗)

x^i∗_p xⁱ_p x^i∗_p

≥ −ln X

p∈supp(x)

xⁱ_p

= 0, and the equality in both inequalities holds if and only if xⁱ =x^i∗.

Consider a trajectory of the RD dynamics with initial point x₀ ∈ V. h∇H(x),B_Φ(x)i=−X

i∈I

X

p∈supp(x^i∗)

x^i∗_p xⁱ_p xⁱ_p

Φⁱ_p(x)− hxⁱ,Φⁱ(x)i

=X

i∈I

hxⁱ−x^i∗,Φⁱ(x)i

=hx−x^∗,Φ(x)i.

For x 6= x^∗, since Γ(Φ) is dissipative (resp. strictly dissipative), one has hx − x^∗,Φ(x) − Φ(x^∗)i ≤ (resp. <) 0, which implies that hx −x^∗,Φ(x)i ≤ (resp. <) hx−x^∗,Φ(x^∗)i ≤ 0, i.e. h∇H(x),B_Φ(x)i ≤(resp. <) 0.

Therefore,H is a local Lyapunov function when Γ(Φ) is dissipative. If Γ(Φ) is strictly dissipa- tive thenx^∗ is the unique equilibrium, andH is a strict local Lyapunov function.

(2) BNN: H(x) = ¹₂P

i∈I

P

p∈SⁱΦˆⁱ_p(x)². h∇H(x),B_Φ(x)i=X

j∈I

X

q∈S^j

∂

∂x^jq

h X

i∈I,p∈Sⁱ

Φˆⁱ_p(x)²i

˙ x^j_q.

Fori=j,

∂

∂x^jq

Φˆⁱ_p(x)² = 2 ˆΦⁱ_p(x) ^∂

∂x^jq

Φˆⁱ_p(x) = 2 ˆΦⁱ_p(x)h

∂

∂x^jq

Φⁱ_p(x)− hxⁱ, ^∂

∂x^jq

Φⁱ(x)i −Φ^j_q(x)i , and for i6=j,

∂

∂x^jq

Φˆⁱ_p(x)² = 2 ˆΦⁱ_p(x)h

∂

∂x^jqΦⁱ_p(x)− hxⁱ, ^∂

∂x^jqΦⁱ(x)ii . Thus,

h∇H(x),BΦ(x)i= X

j∈I

X

q∈S^j

X

i∈I

X

p∈Sⁱ

hΦˆⁱ_p(x)−xⁱ_p X

r∈Sⁱ

Φˆⁱ_r(x)i

∂

∂x^jqΦⁱ_p(x) ˙x^j_q

−X

i∈I

X

p∈Sⁱ

Φˆⁱ_p(x) X

q∈Sⁱ

Φⁱ_q(x) ˙xⁱ_q

=hB_Φ(x), J_Φ(x)B_Φ(x)i −X

i∈I

X

p∈Sⁱ

Φˆⁱ_p(x)

hΦⁱ(x),Bⁱ_Φ(x)i.

Since B_Φ(x)∈T_X(x) and Γ(Φ) is dissipative, hB_Φ(x), J_Φ(x)B_Φ(x)i ≤0. Because BNN dynamics satisfies (PC), hΦⁱ(x),B_Φⁱ(x)i >0 for x such thatB_Φⁱ(x)6= 0, henceh∇H(x),B_Φ(x)i ≤0 and the equality holds if and only if BΦ(x) = 0.

Therefore, H is a strict Lyapunov function.

It is clear that H(x) ≥ 0 and the equality holds if and only if for all i ∈ I and all p ∈ Sⁱ, Φˆⁱ_p(x) = 0, i.e. x∈NE(Φ).

(3) Smith: H(x) =P

i∈I

P

p,q∈Sⁱxⁱ_p{(Φⁱ_q(x)−Φⁱ_p(x)]⁺)². One has:

∂H(x)

∂xⁱ_p = X

q∈Sⁱ

[Φⁱ_q−Φⁱ_p]⁺2

+X

j∈I

X

l∈S^j

x^j_l X

q∈S^j

2[Φ^j_q−Φ^j_l]^{+ ∂Φ}_∂xi^jq p − ^∂Φ

j l

∂xⁱ_p)

= X

q∈Sⁱ

([Φⁱ_q−Φⁱ_p]⁺)²+ 2X

j∈I

X

q∈S^j

X

l∈S^j

x^j_l[Φ^j_q−Φ^j_l]⁺−x^j_qX

l∈S^j

[Φ^j_l −Φ^j_q]^{+ ∂Φ}_∂xi^jq p

= X

q∈Sⁱ

([Φⁱ_q−Φⁱ_p]⁺)²+ 2X

j∈I

X

q∈S^j

˙ x^j_q^∂Φ

j q

∂xⁱ_p. It follows that:

h∇H(x),B_Φ(x)i=A+ 2hB_Φ(x), J_Φ(x)B_Φ(x)i,

where

A=X

i∈I

X

p,q∈Sⁱ

([Φⁱ_q−Φⁱ_p]⁺)²x˙ⁱ_p

=X

i∈I

X

p,q∈Sⁱ

[Φⁱ_q−Φⁱ_p]⁺2 X

l∈Sⁱ

xⁱ_l[Φⁱ_p−Φⁱ_l]⁺−xⁱ_pX

l∈Sⁱ

[Φⁱ_l−Φⁱ_p]⁺

=X

i∈I

X

p,l,q∈Sⁱ

xⁱ_p[Φⁱ_l−Φⁱ_p]⁺

([Φⁱ_q−Φⁱ_l]⁺)²−([Φⁱ_q−Φⁱ_p]⁺)² .

Recall thatB_Φ(x)∈TX(x), thushB_Φ(x), JF(x)B_Φ(x)i ≤0 since Γ(P hi) is dissipative. Also notice that if Φⁱ_l−Φⁱ_p >0, then Φⁱ_q−Φⁱ_p>Φⁱ_q−Φⁱ_l and thus [Φⁱ_q−Φⁱ_p]⁺≥[Φⁱ_q−Φⁱ_l]⁺. As a consequence, every term in Ais non positive. By taking only the terms such thatq =l, one obtains:

h∇H(x),B_Φ(x)i ≤ −X

i∈I

X

p,l,q∈Sⁱ

xⁱ_p([Φⁱ_l(x)−Φⁱ_p(x)]⁺)³ ≤0.

In addition h∇H(x),B_Φ(x)i = 0 if and only if for all i ∈ I and all p ∈ Sⁱ, either xⁱ_p = 0 or Φⁱ_p(x)≥Φⁱ_l(x) for all l∈Sⁱ, equivalently BΦ(x) = 0.

Therefore, H is a strict Lyapunov function.

Clearly, H(x) ≥ 0. And the equality holds if and only if for all i ∈ I and all q ∈ Sⁱ, either xⁱ_q = 0 or Φⁱ_q(x)≥Φⁱ_p(x) for all p∈Sⁱ; equivalently, x∈NE(Φ).

(4) LP: Given an equilibrium x^∗, let H(x) = ¹₂kx−x^∗k².

Recall that for allx∈X,x^∗−x∈T_X(x), thushx^∗−x,Π_NX(x)[Φ(x)]i ≤0. Then:

h∇H(x),B_Φ(x)i=X

i∈I

hxⁱ−x^i∗,Π_T

Xi(xⁱ)[Φⁱ(x)]i

=hx−x^∗,Π_TX(x)[Φ(x)]i

=hx−x^∗,Φ(x)−ΠN_X(x)[Φ(x)]i

=hx−x^∗,Φ(x)i − hx−x^∗,Π_NX(x)[Φ(x)]i

≤ hx−x^∗,Φ(x)−Φ(x^∗)i+hx−x^∗,Φ(x^∗)i

≤0,

because Γ(Φ) is dissipative and x^∗ is an equilibrium. Besides, when Γ(Φ) is strictly dissipative, the equality holds if and only if x=x^∗, the unique equilibrium.

Therefore, H is a global Lyapunov function when F is dissipative. In addition, H is a global strict Lyapunov function when Γ(Φ) is strictly dissipative.

(5) GP:H(x) = sup_y∈XL(x, y) withL(x, y) =hy−x,Φ(x)i −¹₂ky−xk², for x, y∈X.

Since:

ky−(x+ Φ(x))k² =ky−xk²+kΦ(x)k²−2hy−x,Φ(x)i, one hasH(x) =L(x, y^∗(x)) with y^∗(x) = Π_X(x+ Φ(x)). By the Envelope theorem:

∇H(x) =∇_xL(x, y^∗(x)) =−Φ(x) + (y^∗(x)−x)(J_Φ^τ(x) +I) so that:

h∇H(x),B_Φ(x)i=h−Φ(x) + (y^∗(x)−x)(J_Φ^τ(x) +I), y^∗(x)−xi

=h(x+ Φ(x))−ΠX(x+ Φ(x)), x−ΠX(x+ Φ(x))i+hB_Φ(x), J_Φ(x)B_Φ(x)i

≤0.

The first term is negative by property of ΠX. The second term is negative because Γ(Φ) is dissipative.

Therefore, H is a global Lyapunov function.

Note that H is a global strict Lyapunov function when Γ(Φ) is strongly dissipative.

Finally,

H(x) =hΠX(x+ Φ(x))−x,Φ(x)i −¹₂kΠX(x+ Φ(x))−xk²

= ¹₂kΦ(x)k²−¹₂kΠX(x+ Φ(x))−(x+ Φ(x))k²

= ¹₂k(x+ Φ(x))−xk²−¹₂k(x+ Φ(x))−ΠX(x+ Φ(x))k²

≥0.

The inequality is due to the definition of projection ΠX, and the equality holds if and only if x= Π_X(x+ Φ(x)), i.e. x∈NE(Φ).

(6) BR: H(x) = sup_y∈XM(x, y) withM(x, y) =hy−x,Φ(x)i, forx, y∈X.

LetH(x) =M(x,y(x)) with ¯¯ y(x)∈BR(x). By the Envelope theorem, for any ¯y(x),

∇H(x) =∇xM(x,y(x))¯

=−Φ(x) + (¯y(x)−x)J_Φ^τ(x) Hence,

h∇H,B_Φ(x)i=h−Φ(x) + (¯y(x)−x)J_Φ^τ(x),y(x)¯ −xi

=−H(x) +hB_Φ(x), J_Φ(x)B_Φ(x)i

≤0.

The second term is negative because Γ(Φ) is dissipative. Then the equality holds if and only if H(x) = 0 or, equivalently,x∈NE(Φ).

ThereforeH is a strict Lyapunov function.

5. Composite Games 5.1. Example: congestion game.

An eminent example of the games studied in this paper is network congestion game, or routing game. The underlying network is a finite directed graphG= (V, A), whereV is the set of nodes, A the set of links. Vector l = (l_a)_a∈A denotes a family of cost functions from R to R⁺: if the aggregate weight on arca ism, the cost per unit (of weight) is l_a(m).

The set I of participants is finite. A participant i is characterized by his weight mⁱ and an origin/destination pair (oⁱ, dⁱ) ∈ V ×V such that the constraint is to send a quantity mⁱ from oⁱ to dⁱ. The set of choices of participant i∈I is Sⁱ: directed acyclic paths linking oⁱ to dⁱ and available toi. LetP =∪i∈ISⁱ.

Assume that, for all arc a∈ A, the function la is continuous and finite on a neighborhood U of interval [0, M] and positive on U ∩R+, where M =P

i∈Imⁱ is the aggregate weight of the players.

In each of the three frameworks considered in this paper, a participant is respectively a pop- ulation of nonatomic players (I), an atomic splittable player (II) and an atomic non splittable player (III). Thus, in framework I, a fraction xⁱ_p of population i takes path p; in framework II, xⁱ_p is the proportion of the weight mⁱ sent on path p by player i; in framework III, xⁱ_p is the probability with which playeritake path p. The basic variable x is also called the configuration of the network, and the feasible configuration set is X=Q

i∈IXⁱ.

In frameworks I and II, a configuration x induces a flow fⁱ on the arcs (or simply flow) for participant i. Explicitly, the weight on arc a from participant iis f_aⁱ =P

p∈Sⁱ, p∋amⁱxⁱ_p. Define the aggregate configuration x = (xp)p∈P, where x_p = P

i∈I,Sⁱ∋pmⁱxⁱ_p. The aggregate flow is f = (fa)_a∈A,withfa=P

i∈If_aⁱ the aggregate weight on arc a. Notice that f can also be induced by the aggregate configuration x. Denote f_a⁻ⁱ =fa−f_aⁱ.

Given a configuration x, the induced flow f and aggregate flow f, thevector of congestion on the arcsisl(f) ={la(fa)}_a∈A. This specifies now thecost of a pathp byc_p(x) =P

a∈pl_a(fa). The corresponding vectors arecⁱ(x) = (c_p(x))_p∈Si fori∈I,c(x) = (cⁱ(x))_i∈I, andc(x) = (c_p(x))_p∈P.

Besides, letl(f) = (lⁱ(f))_i∈I andlⁱ(f) =l(f) for alli∈I. In particular, path costscand arc costs l are determined only by theaggregate configuration or the aggregate flow.

The evaluation functions in the first two frameworks are respectively:

I: population: Φⁱ_p(x) =−cp(x).

II: atomic splittable: Φⁱ_p(x) =−^∂u_∂x^i(x)i

p , where uⁱ(x) is the cost to atomic playeri:

uⁱ(x) =hxⁱ, cⁱ(x)i= X

p∈Sⁱ

xⁱ_pc_p(x) = 1

mⁱhfⁱ,l(f)i= 1 mⁱ

X

a∈A

f_aⁱl_a(f_a).

In framework III, first consider the arc flow f and aggregate arc flow f induced by a pure- strategy profile s: f_a(s) = P

i∈If_aⁱ(s) and f_aⁱ(s) = mⁱ1a∈sⁱ. Then the evaluation function is Φⁱ_p(x) =−VU_pⁱ(p, x⁻ⁱ) =−Uⁱ(p, x⁻ⁱ), where

Uⁱ(x) = X

s∈Q

j∈IS^j

Y

j∈I

x^j_sj

X

a∈sⁱ

l_a(fa(s))

Congestion games are thus natural settings where each kind of participants occurs. However one can even consider a game where participants of different natures coexist: some of them being of typeI,IIor III. This leads to the notion of composite game.

In addition, congestion games are a natural example of an aggregative game (see [22]) where the payoff of a participant i depends only on xⁱ ∈ Xⁱ and on some fixed dimensional function αⁱ({x^j}_j6=i)∈∆(R^P).

Theorem 5.1. x∈X is a composite equilibrium of the composite game Γ(Φ)if and only if hΦ(x), x−yi ≥0, ∀y∈X, (19) In frameworkII,uⁱis of classC¹and convex when the arc cost functions satisfy a mild condition, as the following lemma shows.

Lemma 5.1. In Γ(Φ), if each cost functionla is of classC¹, nondecreasing and convex on U, for all arc a∈A, then uⁱ(xⁱ, x⁻ⁱ) is convex with respect to xⁱ on a neighborhood of Xⁱ for all fixed x⁻ⁱ ∈X⁻ⁱ.

Proof. In order to prove thatuⁱ(xⁱ, x⁻ⁱ) is convex with respect toxⁱ, it is sufficient to show uⁱ(yⁱ, x⁻ⁱ)≥uⁱ(xⁱ, x⁻ⁱ) +h ∇ⁱuⁱ(xⁱ, x⁻ⁱ), yⁱ−xⁱi, ∀xⁱ, yⁱ ∈ Xⁱ. (20) Suppose thatyⁱ induces arc flowgⁱ for playeri. For all arca, the cost functionl_ais convex, which implies

l_a(gⁱ_a+f_a⁻ⁱ) ≥ l_a(f_aⁱ+f_a⁻ⁱ) + (gⁱ_a−f_aⁱ)l^′_a(f_aⁱ+f_a⁻ⁱ) thus

gⁱ_al_a(gⁱ_a+f_a⁻ⁱ)≥ g_aⁱ l_a(f_a) +gⁱ_a(g_aⁱ −f_aⁱ)l^′_a(f_a) and l_a nondecreasing gives

g_aⁱ l_a(g_aⁱ +f_a⁻ⁱ) ≥ gⁱ_al_a(fa) +f_aⁱ(gⁱ_a−f_aⁱ)l^′_a(fa)

= f_aⁱla(fa) + (g_aⁱ −f_aⁱ) [la(fa) +f_aⁱl^′_a(fa)].

Thus 1 mⁱ

X

a∈A

gⁱ_ala(g_aⁱ +f_a⁻ⁱ)≥ 1 mⁱ

X

a∈A

f_aⁱla(fa) + 1 mⁱ

X

a∈A

(g_aⁱ −f_aⁱ) [la(fa) +f_aⁱl_a^′(fa) ].