The basic variable describing the strategic interaction is thus a profile x= {xi, i∈I}, where each xi ={xip, p∈Si} is an element of the simplexXi= ∆(Si) onSi

American Institute of Mathematical Sciencesc

Volume3, Number1, January2016 pp.101–120

FINITE COMPOSITE GAMES: EQUILIBRIA AND DYNAMICS

Sylvain Sorin

Sorbonne Universit´es, UPMC Univ Paris 06 Institut de Math´ematiques de Jussieu-Paris Rive Gauche UMR 7586, CNRS, Univ Paris Diderot, Sorbonne Paris Cit´e

F-75005, Paris, France

Cheng Wan^∗

Department of Economics, University of Oxford Nuffield College, New Road, Oxford, OX1 1NF, United Kingdom

(Communicated by Josef Hofbauer)

Abstract. We study games with finitely many participants, each having finitely many choices. We consider the following categories of participants:

(I) populations: sets of nonatomic agents, (II) atomic splittable players,

(III) atomic non splittable players.

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

Then we consider composite games where the three categories of partici- pants 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 strategic interaction is thus a profile x= {xⁱ, i∈I}, where each xⁱ ={xⁱ_p, p∈Sⁱ} is an element of the simplexXⁱ= ∆(Sⁱ) onSⁱ. LetX =Q

i∈IXⁱ.

We consider three frameworks (or categories):

(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 “typep” in populationi.

The two others correspond to two kinds ofI-player games where each participant i∈I stands for an atomic player:

(II) Splittable case: xⁱ_p is the proportion that player iallocates to choice p. (The set of pure moves of playeri isXⁱ.)

2010Mathematics Subject Classification. Primary: 91A10, 91A22; Secondary: 91A06, 91A13.

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

The first author was partially supported by PGMO 2014-COGLED. We thank the referee for his or her very precise reading and comments.

∗Corresponding author: Cheng Wan.

101

(III) Non splittable case: xⁱ_p is the probability that playeri choosesp. (The set of pure moves isSⁱ andxⁱ 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 connectingoto d.

o d

arc 1: m

arc 2: 1

First, in a population game, assume two populations of agents going fromotod.

Then a proportionx¹₁of population 1 is taking arc 1 while the rest of the population (x¹₂= 1−x¹₁) uses arc 2, and similarly for population 2.

Next, in the splittable case, consider two players who both have a stock to send fromotodand that they can divide. Thus player 1 sends a fractionx¹₁of his stock by arc 1 and the remaining (x¹₂) by arc 2, and similarly for player 2.

Finally, in the non splittable case, still assume two players having to send their stock fromotod. However, they can no longer divide their stock, but have to send it entirely by one arc. Then with probabilityx¹₁ player 1 sends it by arc 1 and with probabilityx¹₂ by arc 2, and similarly for player 2.

In all cases, the basic variable isx∈X, defined by (x¹₁, x²₁)∈[0,1]².

Assume, more specifically, that the two participants are of size ¹₂ each and that the congestion is mon arc 1 and 1 on arc 2, if the quantity of users is m∈[0,1].

Then it is easy to check that the equilibria are given by:

– in frameworkI,x¹₁=x²₁= 1: all the agents choose arc 1;

– in frameworkII,x¹₁ =x²₁= ²₃: both players send ²₃ of their stock on arc 1 where m= ²₃;

– in frameworkIII, the set of equilibria has one player choosing arc 1 (sayx¹₁ = 1) and the other choosing at random with anyx²₁∈[0,1].

2. Description of the models.

2.1. Framework I: Population games. We consider here the nonatomic frame- work where each participanti∈Icorresponds to a population of nonatomic agents.

The payoff (fitness) is defined by a family of continuous functions{F_pⁱ, i∈I, p∈ Sⁱ}from X toR, whereF_pⁱ(x) is the outcome of an agent in populationichoosing p, when the environment is given by the basic variablex.

Anequilibriumis a pointx∈X satisfying:

xⁱ_p>0⇒F_pⁱ(x)≥F_qⁱ(x), ∀p, q∈Sⁱ, ∀i∈I. (1) This corresponds to aWardrop equilibrium[35].

Proposition 2.1 (Smith [26], 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 interac- tion where eachFⁱ depends only on x⁻ⁱ.

2.2. Framework II: Atomic splittable players. In this case, each participant i ∈I corresponds to an atomic player with action set Xⁱ. Given functionsF_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 fractionsxⁱ_pallocated to different choicesp.

Anequilibriumis 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 classC¹on a neighborhood Ω ofX, so that

∂ Hⁱ

∂ xⁱ_p(x) =F_pⁱ(x) + X

q∈Sⁱ

xⁱ_q∂ F_qⁱ

∂ xⁱ_p(x).

Let∇ⁱHⁱ(x) stand for the gradient ofHⁱ(xⁱ, x⁻ⁱ) with respect to xⁱ. Then by a classical optimization criteria [14] one has:

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 eachHⁱ is concave with respect to xⁱ, there is equivalence.

Variational inequalities characterizing Nash equilibrium in atomic splittable games (Haurie and Marcotte [10]) and those characterizing Wardrop equilibrium in nonatomic games have different origins. Inequalities in (5) for a Nash equilib- rium are obtained as first order conditions, while inequalities in (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}, from S =Q

i∈ISⁱ toR. We still denote byGthe multilinear extension toX where eachXⁱ= ∆(Sⁱ) is considered as the set of mixed actions.

An equilibrium is a profilex∈X satisfying:

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

An equilibrium is thus a profilex∈X satisfying:

hVG(x), x−yi=X

i∈I

hVGⁱ(x⁻ⁱ), xⁱ−yⁱi ≥0, ∀y∈X. (7) 2.4. Remarks. FrameworksIandIIIhave been extensively studied. FrameworkIII corresponds to games with external interaction, but the multilinearity ofVGⁱ(x⁻ⁱ) will not be used in this paper.

Note thatF, ∇H and VGplay similar roles in the three frameworks. This can be seen from the three variational characterizations of equilibrium: (2), (5) and (7).

We callF, ∇H andVG 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) pop- ulation games whereF_pⁱ are continuous on X for alli andp, (ii) atomic splittable games whereHⁱ is concave and of classC¹ on a neighborhood of Xⁱ for alli, and (iii) atomic non splittable games. A typical game in this class is denoted by Γ(Φ), where Φ is its evaluation function.

Definition 2.3. NE(Φ) is the set ofx∈X satisfying:

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

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

Theorem 2.4. Let C⊂R^d be a closed convex set andΨa map fromC toR^d. Consider the variational inequality:

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

Ψ(x)∈NC(x), (10)

whereN_C(x)is the normal cˆone toC atx;

Ψ(x)∈[TC(x)]^⊥, (11)

whereTC(x) is the tangent cˆone toC atxand[TC(x)]^⊥ its polar;

Π_TC(x)Ψ(x) = 0, (12)

whereΠ 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 ΠT_C(x)Ψ(x) = 0 by Moreau’s decomposition [17].

Finally the characterization of the projection gives:

x+ Ψ(x)−Π_C[x+ Ψ(x)], y−Π_C[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 Ω of X, is a potential for Φ if for eachi ∈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) whereX₀ⁱ ={y∈R^|S

i|, P

p∈Sⁱy_p = 0} is the tangent space toXⁱ and∇ⁱ denotes the gradient w.r.t. xⁱ.

The game Γ(Φ) is then called apotential gameand one says that Φderivesfrom W.

Some alternative definitions of potential games have been used such as:

∂W(x)

∂xⁱ_p −∂W(x)

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

∂W(x)

∂xⁱ_p =µⁱ(x)Φⁱ_p(x), ∀x∈Ω, ∀p∈Sⁱ. (16) Remark that (16) yields (15), and (15) implies that the vector{^∂W_∂x^(x)i

p −µⁱΦⁱ_p(x)}_p∈Si

is proportional to (1, . . . ,1), hence is orthogonal toX₀ⁱ, thus (14) holds.

Sandholm [21] defines a population potential game by (16) withµ_i≡1 for alli.

Monderer and Shapley [16] define potential games for finite games, which is equivalent to our definition (15) in frameworkIII.

Proposition 3.2. Let Γ(Φ)be a game with potential W. 1. Every local maximum ofW is an equilibrium ofΓ(Φ).

2. IfW is concave onX, then any equilibrium ofΓ(Φ) is a global maximum ofW onX.

Proof. Since a local maximumxofW on the convex setX satisfies:

h∇W(x), x−yi ≥0, ∀y∈X, (17)

it follows from (14) thathµⁱ(x)Φⁱ(x), xⁱ−yⁱi ≥0 for alliand ally∈X. This further yields (8). On the other hand, ifW is concave onX, a solutionxof (17) is a global maximum ofW onX.

3.2. Dissipative games.

Definition 3.3. The game Γ(Φ) isdissipative if Φ satisfies:

hΦ(x)−Φ(y), x−yi ≤0, ∀(x, y)∈X×X.

It isstrictly dissipative if

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

In the framework of population games, Hofbauer and Sandholm [12] introduce this class of games and call them “stable games”.

Notice that if Φ is dissipative and derives from a potentialW, thenW is concave.

The set of equilibria in dissipative games has a specific structure (see Hofbauer and Sandholm [12, Proposition 3.1, Theorem 3.2]) described as follows.

Definition 3.4. SNE(Φ) is the set ofx∈X satisfying:

hΦ(y), x−yi ≥0, ∀y∈X. (18) Proposition 3.5.

SNE(Φ)⊂NE(Φ).

If Γ(Φ)is dissipative, then

SNE(Φ) =NE(Φ).

Corollary 3.6. If Φis 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 3.7. Suppose thatΦis of classC¹on a neighborhoodΩofX. Denote by JΦ(x) the Jacobian matrix of Φ atx, i.e. JΦ(x) = ^∂Φ

i p

∂x^j_q

q∈S^j

p∈Sⁱ. Then, Φ dissipative implies that JΦ(x)is negative semidefinite on TX(x), the tangent cˆone toX atx.

Proof. Given x ∈ X and z ∈ TX(x), there exists > 0 such that x+tz ∈ X for all t ∈]0, ]. 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 classC¹, letting t go to 0 yieldshJ_Φ(x)z, zi ≤ 0.

Definition 3.8. Suppose that Φ is of class C¹ on a neighborhood Ω of X. The game Γ(Φ) isstrongly dissipativeifJ_Φ(x) is negative definite onT_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 given by:

˙

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

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

(1)Replicator dynamics (RD) (Taylor and Jonker [31])

˙

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.

(2) Brown-von-Neumann-Nash dynamics (BNN) (Brown and von Neumann [2], Smith [27,29], Hofbauer [11])

˙

xⁱ_p= ˆΦⁱ_p(x)−xⁱ_p X

q∈Sⁱ

Φˆⁱ_q(x), p∈Sⁱ, i∈I,

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

(3)Smith dynamics (Smith) (Smith [28])

˙

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 [24].

(4) Local/direct projection dynamics (LP) (Dupuis and Nagurney [5], Lahkar and Sandholm [15])

˙ xⁱ= Π_T

Xi(xⁱ)[Φⁱ(x)], i∈I, whereT_Xi(xⁱ) denotes the tangent cˆone toXⁱ atxⁱ.

(5)Global/target projection dynamics(GP) (Friesz et al. [6], Tsakas and Voorneveld [32])

˙

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.

(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 BΦ satisfies:

i)positive correlation (PC)(Sandholm [21]) if:

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

(This corresponds to MAD (myopic adjustment dynamics) (Swinkels [30]): given a configuration, any unilateral change should increase the evaluation);

ii) Nash stationarityif: for x∈X,BΦ(x) = 0 if and only ifxis an equilibrium of Γ(Φ).

The next proposition collect results that have been obtained in different frame- works. We provide a unified treatment with simple and short proofs.

Proposition 4.2. (RD), (BNN), (Smith), (LP), (GP) and (BR) satisfy (PC).

Proof. (1) RD (Sandholm [23,24]):

hBⁱ_Φ(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 allp∈Sⁱ,xⁱ_p[Φⁱ(x)−Φⁱ(x)]²= 0 or, equivalently xⁱ_p[Φⁱ(x)−Φⁱ(x)] = 0, henceB_Φⁱ(x) = 0.

(2) BNN (Sandholm [21,22,24], Hofbauer [11]):

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 allp∈Sⁱ, ˆΦⁱ_p(x) = 0, in which caseBⁱ_Φ(x) = 0.

(3) Smith (Sandholm [23,24]):

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 allq∈Sⁱ, eitherxⁱ_q = 0 or Φⁱ_q(x)≥Φⁱ_p(x) for allp∈Sⁱ, in which caseB_Φⁱ(x) = 0.

(4) LP (Lahkar and Sandholm [15], Sandholm [24]): Recall that, ifN_Xi(xⁱ) denotes the normal cˆone to Xⁱ at xⁱ (the polar of T_Xi(xⁱ)), then for any v ∈ R^S

i: v = ΠT_Xi(xⁱ)v + ΠN_Xi(xⁱ)v and hΠT_Xi(xⁱ)v,ΠN_Xi(xⁱ)vi = 0 (Moreau’s decomposition, [17]).

Thus,

hBⁱ_Φ(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 Π_T

Xi(xⁱ)[Φⁱ(x)] = 0, i.e. Bⁱ_Φ(x) = 0.

(5) GP (Tsakas and Voorneveld [32]): Letz=x+ Φ(x). Then, hBⁱ_Φ(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), in which case Bⁱ_Φ(x) = 0.

(6) BR (Sandholm [24]):

hBⁱ_Φ(x),Φⁱ(x)i=hyⁱ−xⁱ,Φⁱ(x)i ≥0

sinceyⁱ ∈BRⁱ(x). The equality holds if and only if xⁱ ∈BRⁱ(x), hence Bⁱ_Φ(x) = 0.

Proposition 4.3. (BNN), (Smith), (LP), (GP) and (BR) satisfy Nash stationarity onX.

(RD) satisfy Nash stationarity onintX.

Proof. (1) BNN (Sandholm [21, 22, 24]): x∈ NE(Φ) is equivalent to: ˆΦp(x) = 0 for allp∈S. Hence BΦ(x) = 0.

Reciprocally, assume the existence ofi∈Iandp∈Sⁱsuch that ˆΦⁱ_p(x)>0. Since there existsqwithxⁱ_q >0 and ˆΦⁱ_q(x) = 0, one obtainsBⁱ_Φ,q(x)<0, contradiction.

(2) Smith (Sandholm [23,24]): Assumex∈NE(Φ). Then for all i∈I andq∈Sⁱ, eitherxⁱ_q = 0 or Φⁱ_q(x)≥Φⁱ_p(x) for allp∈Sⁱ. HenceBΦ(x) = 0.

Reciprocally, assume the existence of i ∈ I and p ∈ Sⁱ such that xⁱ_p > 0 and [Φⁱ_q(x)−Φⁱ_p(x)]⁺ > 0. Choose such a p with smallest Φⁱ_p(x) then one obtains Bⁱ_Φ,p(x)<0.

(3) LP (Lahkar and Sandholm [15], Sandholm [24]): The result follows from (12).

(4) GP (Tsakas and Voorneveld [32]): The result follows from (13).

(5) BR (Sandholm [24]): By definitionBΦ(x) = 0 if and only ifxⁱ ∈BRⁱ(x) hence x∈NE(Φ).

(6) RD (Sandholm [23,24]): Assumex∈intX∩NE(Φ). Then, Φⁱ_p(x) = Φⁱ(x) for alli∈I andp∈Sⁱ and thusBΦ(x) = 0.

Reciprocally, ifx∈intX andBΦ(x) = 0, xis equalizing hence inNE(Φ).

4.3. Potential games. We establish here results that are valid for all three frame- works of games.

Proposition 4.4. Consider a potential game Γ(Φ) with potential function W. If the dynamics x˙ =BΦ(x) satisfies (PC), then W is a strict Lyapunov function for BΦ. Besides, all ω-limit points are rest points of BΦ.

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

V˙_t=h∇W(x_t),x˙_ti=X

i∈I

h∇ⁱW(x_t),x˙ⁱ_ti=X

i∈I

µⁱ(x)hΦⁱ(x_t),x˙ⁱ_ti ≥0.

(Recall that ˙x_t ∈ X₀.) Moreover, hΦⁱ(x_t),x˙ⁱ_ti = 0 holds for all i if and only if

˙

x=BΦ(xt) = 0.

One concludes by using Lyapunov’s theorem (e.g. [13, Theorem 2.6.1]).

This result is proved by Sandholm [21] for the version of potential games in frameworkIdefined by (16).

It follows that, with the appropriate definitions, the convergence results estab- lished for several dynamics and potential games in frameworkIorIIIextend to all dynamics and frameworks. Explicitly:

Proposition 4.5. 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 forBΦ.

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

4.4. Dissipative games. We apply also for this class the previous “dictionary”

used for potential games and dynamics.

Proposition 4.6. Consider a dissipative game Γ(Φ).

(1) RD: Letx^∗∈NE(Φ). Define [12]:

H(x) =X

i∈I

X

p∈supp(x^i∗)

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

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

(2) BNN: Assume ΦC¹ on a neighborhoodΩofX. Define [27,29,11]:

H(x) = 1 2

X

i∈I

X

p∈Sⁱ

Φˆⁱ_p(x)².

ThenH is a strict Lyapunov function which is minimal onNE(Φ).

(3) Smith: AssumeΦC¹ on a neighborhoodΩ ofX. Define[28]:

H(x) =X

i∈I

X

p,q∈Sⁱ

xⁱ_p

[Φⁱ_q(x)−Φⁱ_p(x)]^{+ 2}. ThenH is a strict Lyapunov function which is minimal onNE(Φ).

(4) LP: Letx^∗∈NE(Φ). Define[18,37, 19]:

H(x) =1

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

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

(5) GP: Assume ΦC¹ on a neighborhoodΩofX. Define [20]:

H(x) = sup

y∈X

hy−x,Φ(x)i −1

2ky−xk². ThenH is a Lyapunov function.

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

(6) BR: AssumeΦC¹ on a neighborhoodΩ ofX. Define[12]:

H(x) = sup

y∈X

hy−x,Φ(x)i.

ThenH is a strict Lyapunov function which is minimal onNE(Φ).

The proof is given in the Appendix.

5. Example: Congestion games. An eminent example of the games studied in this paper is a network congestion game, or routing game. The underlying network is a finite directed graph G= (V, A), where V is the set of nodes and Athe set of links. The vector l= (la)a∈A denotes a family of cost functions fromR to R⁺: if the aggregate weight on arcaism, the cost per unit (of weight) isl_a(m).

The setI of participants is finite. A participantiis characterized by hisweight 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 linkingoⁱ to dⁱ and available toi. LetP =∪_i∈ISⁱ.

Assume that, for all arcs a ∈ A, the function l_a is continuous and finite on a neighborhoodU of the interval [0, M] and positive onU∩R⁺, whereM =P

i∈Imⁱ is the aggregate weight of the players.

In each of the three frameworks considered in this paper, a participant is respec- tively a population of nonatomic agents (I), an atomic splittable player (II) and an atomic non splittable player (III). Thus, in frameworkI, a fractionxⁱ_p of population itakes pathp; in frameworkII,xⁱ_pis the proportion of the weightmⁱ sent on path p by player i; in framework III, xⁱ_p is the probability with which player i chooses path p. The basic variable x is a profile of strategies of the participants and the corresponding set isX =Q

i∈IXⁱ.

In frameworks Iand II, a strategy xⁱ induces a flow fⁱ on the arcs (or simply flow) for each participanti. Explicitly, the weight on arcaisf_aⁱ=P

p∈Sⁱ, p3amⁱxⁱ_p. Define the aggregate configuration by z = (z_p)_p∈P, where z_p = P

i∈I,Sⁱ3pmⁱxⁱ_p. Theaggregate flowisf = (fa)a∈A,withfa =P

i∈If_aⁱ being the aggregate weight on arca. Notice thatf can also be induced by the aggregate configurationz. Denote f_a⁻ⁱ=f_a−f_aⁱ.

Given an aggregate configurationzand aggregate flowf, thevector of congestion on the arcsisl(f) ={la(fa)}a∈A. This specifies now thecost of a pathpbycp(z) = P

a∈pla(fa). The corresponding vectors are cⁱ(z) = (cp(z))_p∈Si for i ∈ I, c(z) = (cⁱ(z))_i∈I, and lⁱ(f) = l(f) for all i∈I. In particular, path costsc_p and arc costs la 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(z).

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

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

uⁱ(x) =hxⁱ, cⁱ(z)i= X

p∈Sⁱ

xⁱ_pcp(z) = 1

mⁱhfⁱ,l(f)i= 1 mⁱ

X

a∈A

f_aⁱla(fa).

In frameworkIII, first consider the arc flowf and aggregate arc flowf 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(f_a(s)).

Congestion games are thus natural settings where each kind of participants oc- curs. However one can even consider a game where participants of different natures coexist: some of them being of category I, IIor III. This leads to the notion of composite game which is introduced in full generality in Section 6. Consider here a simple example where there are three participants in the network: population 1 of nonatomic agents of weight m¹, atomic splittable player 2 of weight m², and atomic non splittable player 3 of weight m³. Suppose that their basic variables arex¹ = (x¹_p)_p∈S1, x² = (x²_q)_q∈S2 andx³= (x³_s)_s∈S3 respectively. Here x¹ and x² describe pure strategies whilex³ specifies a mixed strategy.

Let us first look at the pure-strategy profiles. Given a pure strategy s ∈ S³, letx= (x¹, x², s),f(x) be the induced aggregate flow: f_a(x) =P

p∈S¹, p3am¹x¹_p+ P

q∈S², q3am²x²_q+m³1{a∈s} and zthe induced configuration. The corresponding cost of a path p is cp(z) = P

a∈pla(fa(x)). Therefore the cost to an agent in population 1 using path p (if x¹_p > 0) is cp(z) for all p∈ S¹, the cost to atomic splittable player 2 isu²(x) =P

q∈S²x²_qc_q(z), and the cost to atomic non splittable player 3 isu³(x) =c_s(z).

Consider now a strategy profile x = (x¹, x², x³). This induces a distribution on pure strategy profiles (x¹, x², s) hence a distribution on the sets of aggregate distributions z_s. The cost to each player is then obtained by taking the relevant expectation. Explicitly, for population 1, the cost to an agent in the population using pathp(ifx¹_p >0) isP

s∈S³x³_scp(zs), while its evaluation function is Φ¹(x) = (Φ¹_p(x))_p∈S1 where Φ¹_p(x) =−P

s∈S³x³_sc_p(z_s). For atomic splittable player 2, his cost isu²(x) =P

s∈S³x³_sP

q∈S²x²_qcq(zs), while his evaluation function is Φ²(x) = (Φ²_q(x))_q∈S2 where Φ²_q(x) = −^∂u_∂x^2(x)2

q

. For atomic non splittable player 3, his cost is u³(x) = P

s∈S³x³_scs(zs), while his evaluation function is Φ³(x) = (Φ³_s(x))_s∈S3

where Φ³_s(x) =−cs(zs).

Through this example, one can further see that congestion games are a natural example of an aggregative game (see [25]) where the payoff of a participantidepends only on xⁱ ∈ Xⁱ and on some fixed dimensional function αⁱ({x^j}j6=i) ∈ ∆(R^P)

(here the aggregate distribution induced by the participants −i). Because of the aggregative property of congestion games, one can show that accumulation points of flows induced by Nash equilibria for a sequence of composite congestion games, when the atomic players split into identical players with vanishing weights, are Wardrop equilibria of the ‘limit’ nonatomic game, i.e. the nonatomic game obtained where the atomic splittable players in the previous sequences games are replaced by populations of nonatomic agents. This result shows intrinsic link between the different frameworks discussed in this paper. The reader is referred to Haurie and Marcotte [10] or Wan [33] for details.

In frameworkII,uⁱ is of classC¹ and convex when the arc cost functions satisfy a mild condition, as the following lemma shows [34, Lemma 29].

Proposition 5.1. In Γ(Φ), if each cost function l_a is of class C¹, nondecreasing and convex onU, for all arc a∈A, thenuⁱ(xⁱ, x⁻ⁱ)is convex with respect toxⁱ on a neighborhood ofXⁱ for all fixed x⁻ⁱ∈X⁻ⁱ.

6. Composite games.

6.1. Composite games and variational inequalities. We have seen that the properties of equilibrium and dynamics in the three frameworks all depend on the evaluation function Φ and the variational inequalities associated to it. Based upon this idea, let us define a more general class of games calledcomposite games, which exhibit different categories of players. Composite congestion games with partici- pants of categories Iand II have been studied by Harker [8], Boulogne et al. [1], Yang and Zhang [36] and Cominetti et al. [3], among others.

Consider a finite setI1of populations composed of nonatomic agents, a finite set I2 of atomic splittable players and a finite set I3 of atomic non splittable players.

LetI=I1∪I2∪I3.

All the analysis of Sections 3 and 4 extend to this setting wherex={xⁱ}i∈I₁∪I₂∪I₃

and Φⁱ(x) depends upon the category of participanti.

Explicitly, there are finitely many “participants” i ∈ I and each of them has finitely many “choices” p ∈ Sⁱ. Vector xⁱ = {xⁱ_p, p ∈ Sⁱ} belongs to simplex Xⁱ= ∆(Sⁱ) onSⁱ andX =Q

i∈IXⁱ.

Fori∈I₁, the payoffF_pⁱ, p∈Sⁱ is a continuous function onX and Φⁱ=Fⁱ. For i ∈ I2, F_pⁱ, p ∈ Sⁱ is a continuous function on X and the payoff Hⁱ(x) = hxⁱ, Fⁱ(x)iis concave and of classC¹on a neighborhood ofXⁱ. Then Φⁱ=∇ⁱHⁱ.

Fori∈I₃,VGⁱ_p is continuous onX⁻ⁱ, the payoff isGⁱ(x) =hxⁱ, VGⁱ(x⁻ⁱ)iand Φⁱ=VGⁱ.

Let this composite game be denoted by Γ(Φ).

The main point to note is that the evaluation by participantiis independent of the category of his/her opponent (their evaluation functions Φ⁻ⁱ) hence the analysis of Section 2 implies the following.

Proposition 6.1. x∈X is a composite equilibrium of a composite game Γ(Φ) if and only if

hΦ(x), x−yi ≥0, ∀y∈X. (19) An example of such a composite game is a congestion game with the three cate- gories of participants in a network (see Section 5).

6.2. Composite potential games and composite dynamics. Once the equi- libria of a composite game are formulated in terms of solutions of variational in- equalities, those properties of equilibria and dynamics based on such a formulation in the three frameworks discussed so far are naturally inherited in the composite setting.

The general form of a dynamics in a composite game Γ(Φ) is again

˙

x=BΦ(x).

The definitions of positive correlation and Nash stationarity forBΦare exactly the same as in Definition4.1.

Arguments similar to those for Propositions4.2and4.3show the following.

Proposition 6.2. In composite gameΓ(Φ), composite dynamics (Smith), (BNN), (LP), (GP) and (RD) satisfy (PC) and Nash stationarity onX.

(RD) satisfies (PC) and Nash stationarity on intX.

The definition of a potential game and that of a dissipative game for composite games are analogous to those for each of the three frameworks. Note that the condition for each participant is independent of the others. Hence the corresponding properties of composite dynamics for these specific classes of composite games can be proved in the same way as for Propositions4.5 and4.6.

Definition 6.3. A composite game Γ(Φ) is a composite potential game if there is a real-valued functionW of class C¹ defined on a neighborhood Ω of X, called potential function, and strictly positive functions µⁱ, i∈ I on X such that for all x∈X,

∇ⁱW(x)−µⁱ(x)Φⁱ(x), yⁱ

= 0, ∀x∈X,∀yⁱ∈X₀ⁱ, ∀i∈I, (20) whereX₀ⁱ={y∈R^|S

i|, P

p∈Sⁱy_p= 0} for alli∈I.

Proposition 6.4. In a composite potential game Γ(Φ)with potential function W, if the dynamicsx˙ =BΦ(x)satisfies Nash stationarity and (PC), thenW is a global strict Lyapunov function forBΦ. Besides, all ω-limit points are rest points of BΦ. Corollary 6.5. A potential function W of a composite potential game Γ(Φ) is a global strict Lyapunov function for (RD), (Smith), (BNN), (LP), (GP) and (BR).

Definition 6.6. A composite game Γ(Φ) is acomposite dissipative gameif−Φ is a monotone operator onX. The game isstrictly dissipativeif−Φ is strictly monotone onX.

Proposition 6.7. If a composite congestion game Γ(Φ) is dissipative, then one can find Lyapunov functions for (RD), (Smith), (BNN), (LP), (GP) and (BR) as defined in Proposition 4.6 (with the assumption that Φ is of class C¹ for (Smith), (BNN), (GP) and (BR)).

As a matter of fact, we can obtain results stronger than Proposition6.1, Corollary 6.5 and Proposition 6.4 which consider only homogeneous dynamical models, i.e.

where all the participants follow the same dynamics. Looking more closely into the proofs for the results in Sections 4.2-4.4, one can see that the results extend naturally to heterogeneous dynamical models where the participants follow specific dynamics.

Basically note that the condition for positive correlation can be written com- ponent by component, hence it corresponds to a unilateral property: it depends

only for each participanti on the evaluation Φⁱ and the dynamics B_Φⁱ =Bⁱ_Φi. For example, in a composite potential game, as long as each of the dynamics followed by different participants satisfies (PC), the potential function is a strict Lyapunov function. Because of this unilateral property, the dynamics in this paper belong to the class ofuncoupled dynamics studied by Hart and Mas-Colell [9].

6.3. One example of a composite potential game. Consider a composite con- gestion game, with three categories of participants i∈I =I₁∪I₂∪I₃, of weight mⁱ each, taking place in a network composed of two nodesoanddconnected by a finite setAof parallel arcs.

Figure 1. Example of a composite potential game

O D

l1(·) l2(·)

lA−1(·) lA(·)

Denote by s = (s^k)_k∈I3 ∈ S3 = A^I3 a pure strategy profile of participants in I3 and let z = ((xⁱ)_i∈I1,(x^j)_j∈I2,(s^k)_k∈I3). Let f(z) be the aggregate flow in- duced by the pure-strategy profilez. Namely: fa(z) =P

i∈I1mⁱxⁱ_a+P

j∈I2m^jx^j_a+ P

k∈I3m^k1{s^k=a}.

Theorem 6.8. Assume that for all a∈A, the per-unit cost function is affine, i.e.

la(m) =bam+da, with ba >0 and da≥0. Then a composite congestion game in this network is a potential game.

A potential function defined onX is given by:

W(x) =− X

s∈S3

Y

k∈I3

x^k_sk

n1 2

X

a∈A

ba

(fa(z)²+X

j∈I2

(m^jx^j_a)²

+ X

k∈I3

(m^k)²1{s^k=a}

+X

a∈A

d_af_a(z)o ,

(21)

withµⁱ(x)≡mⁱ for alli∈I=I₁∪I₂∪I₃ and allx∈X.

Proof. First notice that functionW defined in (21) is the multilinear extension of the following function defined onZ, the set of pure-strategy profiles:

W(z) =−1 2

X

a∈A

ba

(fa(z))²+X

j∈I2

(m^jx^j_a)²+X

k∈I3

(m^k)²1{s^k=a}

+X

a∈A

dafa(z).

The per-unit cost to take arc a when the pure-strategy profile is z is ca(z) = bafa(z) +da, for all arca∈A. Recall that this is the opposite of the evaluation for the nonatomic players in populationi∈I1: Φⁱ_a(z) =−ca(z). On the other hand,

−∂W(z)

∂xⁱ_a = 1

2ba·2fa(z)·mⁱ+damⁱ=mⁱ

bafa(z) +da

=mⁱca(z). (22) For an atomic splittable player j ∈ I2, when the pure-strategy profile is z, the cost isu^j(z) =P

a∈Ax^j_a

bafa(z) +da

. Therefore,

∂u^j(z)

∂x^ja

=b_af_a(z) +d_a+b_am^jx^j_a =−Φ^j_a(z).

Moreover,

−∂W(z)

∂x^ja

= 1 2ba

2m^jfa(z) + 2(m^j)²x^j_a

+dam^j

=m^j

bafa(z) +bam^jx^j_a+da

=m^j∂u^j(z)

∂x^ja

.

(23)

Finally, for an atomic non splittable playerk∈I₃, the cost to take arc awhen the other players playz^−k isu^k(z^−k, a) =b_af_a(z^−k, a) +d_a = Φ^k_a. On the one hand,

u^k(z^−k, a)−u^k(z^−k, r) = [bafa(z^−k, a) +da]−[brfr(z^−k, r) +dr].

On the other hand, W(z^−k, r)−W(z^−k, a)

=X

p∈A

b_p 2

(f_p(z^−k, a))²+X

j∈I2

(m^jx^j_p)²

+X

p∈A

d_pf_p(z^−k, a) +1

2[b_a(m^k)²+ X

l∈I3\{k}

b_sl(m^l)²]−X

p∈A

bp

2

(f_p(z^−k, r))²+X

j∈I2

(m^jx^j_p)²

−X

p∈A

dpfp(z^−k, r)−1

2[br(m^k)²+ X

l∈I3\{k}

b_sl(m^l)²]

= X

p∈{a,r}

bp

2(fp(z^−k, a))²+dpfp(z^−k, a)−bp

2(fp(z^−k, r))²−dpfp(z^−k, r) +ba(m^k)²

2 −br(m^k)² 2

=ba

2 [fa(z^−k, a)]²+dafa(z^−k, a)−ba

2[fa(z^−k, a)−m^k]²−da[fa(z^−k, a)−m^k] +ba(m^k)²

2 +br

2[fr(z^−k, r)−m^k]²+dr[fr(z^−k, r)−m^k]−br

2[fr(z^−k, r)]²

−drfr(z^−k, r)−br(m^k)² 2

=m^k[bafa(z^−k, a) +da]−m^k[brfr(z^−k, r) +dr].

Thus

W(z^−k, r)−W(z^−k, a) =m^k[u^k(z^−k, a)−u^k(z^−k, r)]. (24) By a multilinear extension with respect toz, (22)–(24) imply (20).

One can verify that the potential functionW is concave onX. Therefore,x∈X is a composite equilibrium of this composite congestion game if and only if it is a global maximizer of the potential functionW.

Concerning the dynamics in this congestion game, note that Proposition 6.4 applies.

7. Conclusion. This paper first describes three frameworks with distinct cate- gories of participants: nonatomic populations, atomic splittable and atomic non splittable players, then introduces a class of games called composite games where the three categories coexist. We show that the static properties of the equilibria such as its characterization via variational inequalities, some conditions for the dy- namics studied such as positive correlation, and the notion of potential games and