Computation of Cournot-Nash equilibria by entropic regularization
CComputation of Cournot-Nash equilibria byentropic regularization
Adrien Blanchet ∗ , Guillaume Carlier † , Luca Nenna ‡ June 13, 2018
Abstract
We consider a class of games with continuum of players whereequilibria can be obtained by the minimization of a certain functionalrelated to optimal transport as emphasized in [7]. We then use thepowerful entropic regularization technique to approximate the prob-lem and solve it numerically in various cases. We also consider theextension to some models with several populations of players.
Keywords:
Cournot-Nash equilibria, optimal transport, entropic regu-larization.
AMS Subject Classifications: . There is a long tradition in economics and game theory, since the seminalworks of Aumann [2], [3], of considering equilibria in games with a continuumof players, each of whom having a negligible influence on the output of theothers. In particular, Schmeidler [18] introduced a notion of non-cooperativeequilibrium in games with a continuum of agents and, Mas-Colell [15] refor-mulated Schmeidler’s analysis in terms of joint distributions over players’actions and characteristics and emphasized the concept of Cournot-Nash ∗ GREMAQ-TSE, Universit´e de Toulouse, Manufacture des Tabacs 21 all´ee de Brienne, 31000 Toulouse,FRANCE
[email protected] . † Universit´e Paris-Dauphine, PSL Research University, CNRS, CEREMADE, 75016, Paris, France andInria-Paris, MOKAPLAN [email protected] . ‡ Inria-Paris, MOKAPLAN, 2, rue Simone Iff, 75012, Paris, France and CEREMADE, UMR CNRS7534, Universit´e Paris IX Dauphine [email protected] . a r X i v : . [ m a t h . O C ] S e p quilibrium distributions. There are many examples where such conceptsare relevant such as strategic route use in road traffic or networks, socialinteractions....The problem can be described as follows: heterogeneous players each haveto choose a strategy (or a probability over strategies, i.e. mixed strategiesare allowed) so as to minimize a cost, the latter depending on the choiceof the whole population of players only through the distribution of theirstrategies. In other words, on the one hand, each player, has a negligibleinfluence on the cost. On the other hand, the interactions between playersare of mean-field type: it does not matter who plays such and such strategybut rather how many players chose it. There are different mean-field effects,of different nature and which can be either repulsive (i.e. favoring dispersionof strategies) or attractive (favoring concentration of strategies). Congestion(the cost of a strategy is higher if it is frequently played) is a typical exampleof dispersive effect. In realistic models however, there are also attractiveeffects: choosing a strategy which is ”far” from the strategies played by theother players may be risky or result in some cost.. It should then comeas no surprise that, due to such opposite effects, the analysis of equilibriais complex in general. This explains why, in general, one cannot go muchfurther than proving an existence result, as for instance, following the veryelegant approach of Mas-Colell [15].More recently, the first two authors [7] (also see [6]) emphasized the factthat for a separable class of costs, Cournot-Nash equilibria can be obtainedby the minimization of a certain functional on the set of measures on thespace of strategies. This functional typically involves two terms: an optimaltransport cost and a more standard integral functional which may captureboth congestion and attractive effects (as in [14]). Interestingly, this kind ofminimization problem is very close to the semi-implicit Euler scheme intro-duced in the seminal work of [13] for Wasserstein gradient flows (for whichwe refer to [1]).The variational approach of [7] is somehow more constructive and in-formative (but less general since it requires a separable cost) than the onerelying on fixed-point arguments as in [15] but the optimal transport termcost is delicate to handle. It is indeed costly in general to solve an op-timal transport problem and compute an element of the subdifferential ofthe optimal cost as a function of its marginals. In recent years, however ithas been realized that a powerful way to approximate optimal transport isby adding an entropic penalization term. Doing so, the problem becomesprojecting for the Kullback-Leibler divergence a given joint measure on theset of measures with fixed marginals, a task that can be achieved very effi-2iently by alternate projections (see e.g. Bauschke and Lewis [4], Dysktra[10]) as shown by Cuturi [9]. This powerful method is intimately related toSinkhorn algorithm and the Iterated Proportional Fitting Procedure (IPFP),well-known to statisticians and recently remise au goˆut du jour by Galichonand Salani´e [12] for the estimation of matching models (we also refer to therecent book of Galichon [11] for a broader perspective on optimal transportmethods, with or without regularization, in economics and econometrics).Various applications of the IPFP/Sinkhorn algorithm to optimal transportcan be found in [5]. In order to take advantage of the power of entropic regu-larization on Wasserstein gradient flows, Peyr´e [16] introduced an extensionof Dykstra’s algorithm which he called Dykstra proximal splitting. It turnsout that Peyr´e’s algorithm, recently extended by Chizat et al. [8], is per-fectly well-suited to the computation of Cournot-Nash equilibria as we try toexplain in the sequel of the paper and illustrate by various numerical exam-ples. We would also like to emphasize that, in the context of Cournot-Nashequilibrium, entropic regularization is also natural from a theoretical point,it amounts to replace exact cost minimization by some Gibbs-like measureor, equivalently to assume that the cost involves some random term.The paper is organized as follows. In section 2, we recall the conceptof Cournot-Nash equilibria, its variational counterpart and the entropic reg-ularization of the latter. In Section 3, we describe the proximal splittingalgorithm and the semi-implicit approach. In section 4, we present variousnumerical results both in dimension one and two and emphasize the influenceof the transport cost on the structure of equilibria. Section 5 extends theprevious analysis to some models with several populations for which we alsopresent numerical results. We will restrict ourselves here to the following finite
Cournot-Nash setting.Not only this will simplify the exposition and enable us to give a simpleand self-contained exposition of the variational approach but this will alsobe consistent with our numerical scheme which anyway considers a finitenumber of agents’ types and a finite number of strategies. We refer to [7]for the analysis of the continuum case. We consider a population of players,each of whom is characterized by a type which takes values in the typeset X := { x i } i ∈ I where I is finite. The frequencies of the players’ typein the population is given by a probability µ := { µ i } i ∈ I with µ i ≥ Ni =1 µ i = 1. Each agent has to choose a strategy y from the strategyset Y := { y j } j ∈ J with J finite. The unknown of the problem is a matrix γ := { γ ij } i ∈ I, j ∈ J where γ ij is the probability that a player of type x i choosesstrategy y j , there is an obvious feasibility constraint on this matrix, obviouslyit should have nonnegative entries and its first marginal should match thegiven distribution of players µ i.e.: (cid:88) j ∈ J γ ij = µ i , ∀ i ∈ I. (2.1)The matrix γ induces a probability ν = Λ ( γ ) = { ν j } j ∈ J on the set ofstrategies given by its second marginal : ν j := (cid:88) i ∈ I γ ij , ∀ j ∈ J. (2.2)Agents of type x i who play strategy y j incur a cost that not only dependson x i and y j but also on the whole probability ν := { ν j } j ∈ J on the strategyspace induced by the behavior of the whole population of players, and wedenote this cost by Ψ ij [ ν ]. An equilibrium is then a probability matrix γ which is feasible and which is consistent with the cost minimizing behaviorof players, which is summarized in the next definition: Definition 2.1.
A Cournot-Nash equilibrium is a matrix γ = { γ ij } i ∈ I, j ∈ J ∈ R I × J + which satisfies the feasibility constraint (2.1) and such that, definingthe strategy marginal ν = Λ ( γ ) by (2.2), one has γ ij > ⇒ Ψ ij [ ν ] = min k ∈ J Ψ ik [ ν ] . Provided Ψ ij depends continuously on ν , the existence of an equilibriumcan easily be proven by Kakutanis’ fixed-point theorem, but not much morecan be said, at this level of generality. If one further specifies the form ofthe cost, as we shall do now, following [7], one may obtain equilibria byminimizing a certain cost functional. We now suppose that the cost Ψ ij [ ν ] takes the following separable formΨ ij [ ν ] := c ij + f j ( ν j ) + (cid:88) k ∈ J φ kj ν k where c := { c ij } i ∈ I, j ∈ J ∈ R I × J , each function f j is nondecreasing and con-tinuous, the matrix φ := { φ kj } ∈ R J × J is symmetric, i.e. φ kj = φ jk ,4 possible interpretation of this model is the following: the players rep-resent a population of doctors, their type x represent their region of originand their y strategy represent the location where they chose to dwell, thetotal cost of x i -type doctors is the sum of • a transport cost c ij = c ( x i , y j ), • a congestion cost f j ( ν j ): if location y j is very crowded i.e. if ν j is large,the doctors settling at y j will see their benefit decrease, • an interaction cost with the rest of the population of doctors, one canthink that φ kj is an increasing function of some distance between y k and y j so that (cid:80) k ∈ J φ kj ν k represents the average distance to the restof the population.The variational approach of [7] relies on optimal transport, and we shallgive a self-contained and simple presentation in the present discrete setting.Firstly it is useful to introduce the marginal maps: γ ∈ R I × J (cid:55)→ Λ ( γ ) = α ∈ R I , α i := (cid:88) j ∈ J γ ij , and γ ∈ R I × J (cid:55)→ Λ ( γ ) = ν ∈ R J , ν j := (cid:88) i ∈ I γ ij , as well as C µ := { γ = { γ ij } i ∈ I, j ∈ J ∈ R I × J + : Λ ( γ ) = µ } which is the set of probabilities on X × Y having µ as first marginal (recallthat µ is fixed). For ν = { ν j } j ∈ J ∈ R J + such that (cid:80) j ∈ J ν j = 1, let us alsodefine C ν := { γ = { γ ij } i ∈ I, j ∈ J ∈ R I × J + : Λ ( γ ) = ν } as the set of probabilities on X × Y having ν as second marginal. Let usthen also define the set of transport plans between µ and ν asΠ( µ, ν ) := C µ ∩ C ν . (2.3)Given ν a probability on Y , let us defineMK( ν ) := inf γ ∈ Π( µ,ν ) (cid:110) c · γ := (cid:88) i,j ∈ I × J c ij γ ij (cid:111) (2.4)that is the value of the optimal transport problem between µ and ν for thecost c . Setting P ( Y ) := { ν ∈ R J + : (cid:88) j ∈ J ν j = 1 } ν ∈P ( Y ) MK( ν ) + E ( ν ) (2.5)where the energy E is given by E ( ν ) := (cid:88) j ∈ J F j ( ν j ) + 12 (cid:88) k,j ∈ J × J φ kj ν k ν j (2.6)and F j is a primitive of the congestion function f j : F j ( t ) := (cid:90) t f j ( s )d s. We then have
Theorem 2.2.
Let ν solve (2.5) and γ ∈ Π( µ, ν ) be such that c · γ = MK( ν ) ,then γ is a Cournot-Nash equilibrium. This implies in particular that thereexists Cournot-Nash equilibria.Proof. We have to prove that whenever γ ij > c ij + f j ( ν j ) + (cid:88) k ∈ j φ kj ν k = u i (2.7)with u i := min j ∈ J { c ij + f j ( ν j ) + (cid:88) k ∈ j φ kj ν k } . First observe that E is of class C and by construction ∂E∂ν j = f j ( ν j ) + (cid:88) k ∈ j φ kj ν k . (2.8)To treat the transport term, MK, we shall recall the classical Kantorovichduality (see [19], [17]) as follows. Firstly for v ∈ R J let us defineK( v ) := − (cid:88) i ∈ I min j ∈ J ( c ij − v j ) µ i note that K is a convex and Lipschitz function whose conjugate, thanks toKantorovich duality, can be expressed as K ∗ ( ν ) = MK( ν ) := (cid:40) MK( ν ) if ν ∈ P ( Y ) , + ∞ otherwise.6ince ν minimizes MK + E , one has 0 ∈ ∂ MK( ν ) + ∇ E ( ν ), setting v := −∇ E ( ν ), this can be rewritten as ν ∈ ∂ MK ∗ ( v ) = ∂K ( v ) and since MK( ν ) = c · γ this givesMK( ν ) = c · γ = (cid:88) j ∈J v j ν j + (cid:88) i ∈ I min j ∈ J ( c ij − v j ) µ i = (cid:88) j ∈J v j ν j + (cid:88) i ∈ I u i µ i = (cid:88) i,j ∈ I × J ( u i + v j ) γ ij . which, since u i + v j ≤ c ij implies that whenever γ ij >
0, one has c ij − v j = u i which is exactly (2.7). This clearly implies the existence of Cournot-Nashequilibria since P ( Y ) is compact and both MK and E are continuous.Note that if E is convex then the optimality condition 0 ∈ ∂ MK( ν ) + ∇ E ( ν ) is necessary and sufficient and there is actually an equivalence betweenbeing an equilibrium and being a minimizer in this case. Solving (2.5) in practice (even if E is convex) might be difficult because ofthe transport cost term MK for which it is expensive to compute a subgra-dient. There is however a simple regularization of MK which is much moreconvenient to handle: the entropic regularization (see [5, 9, 12]). Given aregularization parameter ε >
0, let us define for every ν ∈ P ( Y ):MK ε ( ν ) := inf γ ∈ Π( µ,ν ) (cid:110) c · γ + ε (cid:88) i,j ∈ I × J γ ij (ln( γ ij ) − (cid:111) . We then consider the regularization of (2.5)inf ν ∈P ( Y ) MK ε ( ν ) + E ( ν ) (2.9)where E is again given by (2.6). Thanks to the entropic regularization term,(2.9) is a smooth minimization problem which consists in minimizing withrespect to γ and ν the objective c · γ + ε (cid:88) i,j ∈ I × J γ ij (ln( γ ij ) −
1) + E ( ν )subject to γ ij ≥ γ ∈ Π( µ, ν ). The first-order optimality conditions give the following Gibbs form for γ ij : γ ij = a i exp (cid:16) − ε ( c ij + f j ( ν j ) + (cid:88) k ∈ J φ kj ν k ) (cid:17) (2.10)7or some a i > a i = µ i (cid:80) j ∈ J exp (cid:16) − ε ( c ij + f j ( ν j ) + (cid:80) k ∈ J φ kj ν k ) (cid:17) . Note that these conditions can also be interpreted as a regularized form of aCournot-Nash equilibrium since they mean that the conditional probabilitieson the set of strategies given the players type { γ ij µ i } j ∈ J are proportional toexp( − Ψ ij ( ν ) ε ) where Ψ ij [ ν ] = c ij + f j ( ν j )+ (cid:80) k ∈ J φ kj ν k is the total cost incurredby players x i when choosing strategy y j . Another equilibrium interpretation(which is customary in economics and econometrics in the framework ofdiscrete choice models) is to consider that the total cost actually containsa random component that is of the form εX ij where the X ij are i.i.d. logisticrandom variables (see [11]).Of course, again when E is convex, since MK ε is strictly convex, thereis a unique minimizer and the first-order optimality condition for (2.9) isnecessary and sufficient so that there is again equivalence between being aminimizer and a (regularized) Cournot-Nash equilibrium. To solve (2.9), we shall use a proximal splitting scheme using the Kullback-Leibler divergence that was recently introduced by Peyr´e [16] in the contextof entropic regularization of Wasserstein gradient flows and extended recentlyby Chizat et al. [8]. First, let us observe that (2.9) can be rewritten as aspecial instance of a Bregman proximal problem. To see this, let us firstrewrite c · γ + ε (cid:88) i,j ∈ I × J γ ij (ln( γ ij ) −
1) = ε (cid:88) i,j ∈ I × J γ ij (ln (cid:16) γ ij e − cijε (cid:17) − ε KL( γ | γ ) where γ ij = e − cijε and KL is the Kullback-Leibler divergenceKL( γ | θ ) := (cid:88) i,j ∈ I × J γ ij (cid:16) ln (cid:16) γ ij θ ij (cid:17) − (cid:17) , γ ∈ R I × J + , θ ∈ R I × J + . Note that KL is the Bregman divergence associated to the entropy. Solving(2.9) then amounts to the proximal problemprox KL G ( γ ) = argmin γ ∈ R I × J + (cid:110) KL( γ | γ ) + G ( γ ) (cid:111) (3.1)8ith G ( γ ) := χ { Λ ( γ )= µ } + 1 ε E (Λ ( γ )) . Computing directly prox KL G ( γ ) may be an involved task, but the idea ofPeyr´e’s splitting algorithm is to express G as a sum of more elementaryfunctionals: G := L (cid:88) l =1 G l each of whom being simple in the sense that computing prox KL G l can be doneeasily (ideally in close form). The algorithm proposed by Peyr´e generalizesDykstras’ algorithm for KL projections on the intersection of convex setsand can be described as follows. First extend the sequence of functions G , · · · , G L by periodicity: G l + nL = G l , l = { , · · · , L } , n ∈ N initialize the algorithm by setting the following values for the I × J matrices γ (0) = γ, z (0) = z ( − = · · · = z ( − L +1) = e, e ij = 1 , ( i, j ) ∈ I × J, and then iteratively define for n ≥ γ ( n ) = prox KL G n (cid:16) γ n − (cid:12) z ( n − L ) (cid:17) (3.2)and z ( n ) = z ( n − (cid:12) (cid:16) γ ( n − (cid:11) γ ( n ) (cid:17) (3.3)where (cid:12) and (cid:11) stand for entry-wise multiplication/division operations:( γ (cid:12) θ ) ij = γ ij θ ij , ( γ (cid:11) θ ) ij = γ ij θ ij . We refer to [16] and [8] for the convergence of this algorithm under suitableassumptions (convexity of the functions G l and a certain qualification condi-tion), the idea being that at the level of the dual problem, which is smooth,this algorithm amounts to perform an alternate block minimization. Note that the congestion term (cid:80) j ∈ J F j ( ν j ) is convex because f j is nonde-creasing, but the quadratic interaction energy ν (cid:55)→ (cid:80) j,k ∈ J × J φ kj ν k ν j is ingeneral not convex. However, using Cauchy-Schwarz inequality, it satisfies (cid:88) j,k ∈ J × J φ kj ν k ν j ≥ − (cid:16) (cid:88) j,k ∈ J × J φ kj (cid:17) (cid:88) j ∈ J ν j
9o that if F j is 1-strongly convex: F j ( t ) = 12 t + H j ( t )with H j convex and (cid:88) j,k ∈ J × J φ kj < , (3.4)then E is convex as the sum E = E + E of the convex quadratic term E ( ν ) := 12 (cid:88) j ∈ J ν j + 12 (cid:88) k,j ∈ J × J φ kj ν k ν j and the remaining convex congestion term E ( ν ) := (cid:88) j ∈ J H j ( ν j ) . In this setting one can write (2.9) asinf γ ∈ R I × J + (cid:110) KL( γ | γ ) + G ( γ ) + G ( γ ) + G ( γ ) (cid:111) where G ( γ ) = χ { Λ ( γ )= µ } = (cid:40) ( γ ) = µ + ∞ otherwiseand G = 1 ε E ◦ Λ , G = 1 ε E ◦ Λ . To implement the proximal splitting scheme (3.2)-(3.3) in this case, one hasto be able to compute the three proximal maps prox KL G l with l = 1 , ,
3. Theproximal map of G corresponds to the fixed marginal constraint Λ ( γ ) = µ ,it is well-known and it is given in closed form as: (cid:16) prox KL G ( θ ) (cid:17) ij = µ i θ ij (cid:80) k ∈ J θ ik . Given θ ∈ R I × J + , γ := prox KL G ( θ ) is of the form γ ij = θ ij exp (cid:16) − ν j + (cid:80) k ∈ J φ kj ν k ε (cid:17) ν denotes the second marginal of γ , so that summing over i , ν isobtained by solving the system: ν j = (cid:16) (cid:88) i ∈ I θ ij (cid:17) exp (cid:16) − ν j + (cid:80) k ∈ J φ kj ν k ε (cid:17) which, when (3.4) holds, can be solved in practice in a few Newton’s steps.The computation of γ := prox KL G ( θ ) is simpler, setting h j := H (cid:48) j the first-orderequation first leads to γ ij = θ ij exp (cid:16) − h j ( ν j ) ε (cid:17) and the ν j ’s are obtained by solving ν j = (cid:16) (cid:88) i ∈ I θ ij (cid:17) exp (cid:16) − h j ( ν j ) ε (cid:17) (3.5)which is a separable system of monotone equations, which we shall againsolve by Newton’s method. We now go back to the general case where E is not necessarily convex becauseof the interaction term given by the symmetric matrix φ kj . Even thoughthere is no theoretical convergence guarantee (but if the following schemeconverges, it converges to an equilibrium), the semi-implicit scheme which wenow describe gives good results in practice. The idea is simple and consists inreplacing the nonconvex interaction term by its linearization. More precisely,we will approximate our initial problem (2.9):inf ν ∈P ( Y ) MK ε ( ν ) + E ( ν ) (3.6)where E is the sum of the convex congestion cost and the nonconvex quadraticinteraction cost, by a succession of convex problems, starting from ν ∈ P ( Y ),iteratively solve for n ≥ ν ( n +1) = argmin ν ∈P ( Y ) MK ε ( ν ) + E ( n ) ( ν ) (3.7)where in E ( n ) we have linearized the interaction term: E ( n ) ( ν ) = (cid:88) j ∈ J F j ( ν j ) + (cid:88) j ∈ J V ( n ) j ν j , V ( n ) j := (cid:88) k ∈ J φ kj ν ( n ) k .
11f course, we can solve (3.7) by the Dykstra proximal-splitting scheme de-scribed in the previous paragraph. More precisely, the linear term can beabsorbed by the KL term so that we only have two proximal steps: onecorresponding to the (explicit) projection fixed marginal constraint and onecorresponding to the congestion cost (corresponding to (3.5) using f j insteadof h j ). We now present some numerical results in dimension d = 1 and d = 2. As wehave pointed out in section 2.2, the strength of the entropic regularization,and consequently of the Dykstra’s algorithm, lies in the fact that we can treatoptimal transportation problems with any transport cost, in particular bothconcave and convex cost functions can be considered. Thus, if we consider thecost c ij = | x i − y j | p with p > p > ν changes by varyingthe exponent p . Before showing the results, we want to focus on an otheraspect of the entropic regularization, namely diffusion. Indeed, once we addthe entropic term to the optimal transport term, then this regularizationspreads the support of the plan γ and defines a strongly convex problem witha unique solution. So it is interesting to see how the support of the optimal γ varies by decreasing the parameter ε . Let us consider the standard quadraticcost c ij = | x i − y j | and the following energy E ( ν ) E ( ν ) = (cid:88) j ∈ J ν j + 12 (cid:88) k,j ∈ J × J φ kj ν k ν j + (cid:88) | y j − | , (4.1)where φ kj = 10 − | y k − y j | and the third term is a confinement potential. Wenotice that there is no need to compute a proximal step for the potential,indeed it can be absorbed by the KL term. We know that in this case theoptimal γ (for instance see [7]) is a pure Cournot-Nash equilibrium, whichactually means that γ has the form γ T = (id , T ) µ where T is the optimalmap. In Figure 1 we plot the support of the optimal γ and its marginal ν for different values of ε . As expected the support of the regularized γ concentrates on the graph of T as ε decreases.In Section 3.2, we have pointed out that a semi-implicit approach can beapplied in order to treat an energy E which is not convex. We want, now, tocompare the performances of the implicit and the semi-implicit approach interms of CPU time and number of iterations when ε varies. By looking atFigure 2, we notice the number of iterations, as well as the CPU time, of the12 = 0 . ε = 0 . ε = 0 . ε = 10 ε = 0 . ε = 0 . ε = 0 . ε = 10Figure 1: Top: The initial distribution µ (blue solid line) and the solution ν (red solid line) for ε ∈ { . , . , , . , } . Bottom: The support of γ for ε ∈ { . , . , , . , } . semi-implicit approach are smaller than the ones for the implicit approach.This is quite obvious as in the semi-implicit scheme, the interaction term canbe absorbed by the KL term so that one has to compute only two proximalsteps instead of three. Let us first consider the one-dimensional case. One of the main advantagesof the scheme we have proposed is that we can consider any kind of cost func-tion. Thus, take c ij = | x i − y j | p and the energy E given by (4.1), then we wantto visualize the optimal ν as p ∈ (0 , M ] with M large. For the simulations inFigure 3, we have used a N = 500 grid points discretization of [0 ,
16] and wehave treated the interaction term with a semi-implicit approach. Then, wehave chosen the smallest ε possible for each cost function tested. As one cannotice for p ≤ ν has a connected support whereas for p > ν is closer to the one of µ . Finally, we obtain an optimal ν which tends to be concentrated near y = 9 due to the external potential,except for large p where the optimal transport term becomes dominant sothat the second marginal ν tends to be close to the initial distribution µ .Let us now consider an energy E given by E ( ν ) = (cid:88) j ∈ J ln( ν j ) + (cid:88) k,j ∈ J × J φ kj ν k ν j + (cid:88) j ∈ J ( y j − (4.2)13terations CPU time in secondsFigure 2: Left: the number of iterations for the semi-implicit (blue) and forthe implicit (red). Right: CPU time for the semi-implicit (blue) and for theimplicit (red). where φ kj is a cubic interaction φ kj = 10 − | x i − y j | . The simulations are pre-sented in Figures 4 and 5 for different initial distribution: a uniform densityon [0 ,
1] and the sum of two translated Gaussians, respectively. For both thenumerical experiments we have used N = 500 grid points discretization of[0 ,
10] and treated the interaction term with a semi-implicit approach. Onecan observe, as in the previous case, that the structure of the optimal ν becomes close to the one of the initial ditribution as p increases. For the 2 d case, we always take c ( x, y ) = (cid:107) x − y (cid:107) p , a congestion of the form F j ( ν j ) = ν j , quadratic interactions φ kj = 10 − (cid:107) y k − y j (cid:107) and a potential v j = (cid:107) y j − (cid:107) . The simulations in Figure 6 are obtained by using a N × N ,with N = 80, discretization of [0 , and by treating the interaction termwith a semi-implicit approach. As in the 1 − dimensional case, we notice thesame effect on the support of ν when we make p vary. We end the paper by briefly explaining how our approach can easily beextended to the case of several populations of players. For the sake of sim-14 = 0 . p = 1 p = 2 p = 3 p = 4 p = 8 p = 16 p = 32 p = 64Figure 3: The initial distribution µ , a sum of two translated Gaus-sian, (blue solid line) and the solution ν (red solid line) for p ∈{ . , , , , , , , , } . = 0 . p = 1 p = 2 p = 3 p = 4 p = 8 p = 16 p = 32 p = 64Figure 4: The initial distribution µ , a uniform density on [0 , , (blue solidline) and the solution ν (red solid line) for p ∈ { . , , , , , , , , } . = 0 . p = 1 p = 2 p = 3 p = 4 p = 8 p = 16 p = 32 p = 64Figure 5: The initial distribution µ , a sum of two translated Gaus-sians, (blue solid line) and the solution ν (red solid line) for p ∈{ . , , , , , , , , } . µ support of µ surface plot of ν for p = 0 . ν for p = 0 . ν for p = 1 support of ν for p = 1 surface plot of ν for p = 2 support of ν for p = 2surface plot of ν for p = 4 support of ν for p = 4Figure 6: The initial distribution µ , a sum of two translated Gaussian, andthe solution ν for different values of p . X = { x i } i ∈ I and X = { x i } i ∈ I , a common strat-egy space Y = { y j } j ∈ J , given distributions of the players types µ ∈ P ( X ), µ ∈ P ( X ), two transport cost matrices c ∈ R I × J , c ∈ R I × J , and considerthe minimization problem:inf ( ν ,ν ) ∈P ( Y ) ×P ( Y ) (cid:110) MK ε ( ν )+MK ε ( ν )+ E ( ν )+ E ( ν )+ F ( ν + ν ) (cid:111) (5.1)where for l = 1 , ε l > lε l ( ν l )represents the regularized transport cost:MK lε l ( ν l ) := inf γ ∈ Π( µ l ,ν l ) (cid:110) c l · γ + ε l (cid:88) i,j ∈ I l × J γ ij (ln( γ ij ) − (cid:111) ,E l ( ν l ) represents an individual cost for population k , for instance, an inter-action cost: E l ( ν l ) := (cid:88) j,k ∈ J × J φ lkj ν lj ν lk and F is a total congestion cost F ( ν + ν ) := (cid:88) j ∈ J F j ( ν j + ν j )where F j is convex. Remark . The proximal step related to F can be computed as in (3.5) bytaking ν j = ν j + ν j . For the two populations case, we take the following energies E l E l ( ν l ) = (cid:88) j ∈ J ( ν lj ) + (cid:88) k,j ∈ J × J φ lkj ν lj ν lk + (cid:88) j ∈ J | y j − | , where φ lkj = 10 − | y k − y j | and the total congestion F j is given by F j ( ν j + ν j ) = ( ν j + ν j ) . As usual, we consider cost functions of the form c ij = | x i − y j | p and we want toanalyze the support of ν l as p varies. For the simulations in Figure 7 we haveused N = 500 grid points discretization of [0 ,
16] and treated the interaction19erm with a semi-implicit approach. As we can notice in Figure 7 there isa competition between the confinement potential and the total congestion:the two populations tend to concentrated near y = 10 by the potential, butthe effect of the congestion term makes it costly. This becomes clear if wecompare (for instance, the case with p = 2) ν with the optimal one inFigure 3; even if the energies are the same, the effect of congestion makesthe support of the optimal solutions quite different. p = 0 . p = 1 p = 1 . p = 2Figure 7: The initial distributions µ and µ (blue solid line and blue dottedline) and the solutions ν and ν (red solid line and red dotted line) fordifferent values of p . Thus, let us now consider the following case: let E l be as above and p = 2,then we take the total congestion given by F j ( ν j + ν j ) = ( ν j + ν j ) r and we compute the optimal ν l for different values of r . In Figure 8 we cansee that the congestion term becomes more dominant as r increases so that20 = 4 r = 8 r = 32Figure 8: The initial distributions µ and µ (blue solid line and blue dottedline) and the solutions ν and ν (red solid line and red dotted line) fordifferent values of r . the two populations try to be as far as possible, despite the effect of theconfinement potential which is minimal at y = 10. Acknowledgements:
G.C. and L.N. gratefully acknowledge the supportfrom the ANR, through the project ISOTACE (ANR-12- MONU-0013).
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