Optimal Transport and Cournot-Nash Equilibria

Abstract

We study a class of games with a continuum of players for which Cournot-Nash equilibria can be obtained by the minimisation of some cost, related to optimal transport. This cost is not convex in the usual sense in general but it turns out to have hidden strict convexity properties in many relevant cases. This enables us to obtain new uniqueness results and a characterisation of equilibria in terms of some partial differential equations, a simple numerical scheme in dimension one as well as an analysis of the inefficiency of equilibria.

From Nash to Cournot–Nash equilibria via the Monge–Kantorovich problem

“…Here, we give a brief and informal account of our recent works [11,12] and the computation of Cournot-Nash equilibria in the separable case, i.e. the case where, using the notations of §3 with φ minimal when x = y.…”

“…Hence, equilibria can be computed numerically, see [12], as shown in figure 1. Let us mention that in higher dimensions, for a quadratic c and a logarithmic f , finding an equilibrium amounts to solve the following Monge-Ampère equation, see [11, section 4.3]: In [11], we observe that whenever V takes the form (5.2) with a symmetric kernel φ then it is the first variation of the energy…”

“…In §4, we give asymptotic results concerning the convergence of Nash equilibria to Cournot-Nash equilibria as the number of players tends to infinity. In §5, we give an account on recent results from [11,12] which enable one to find Cournot-Nash equilibria using tools from optimal transport and perform numerical simulations.…”

Population Games and Discrete Optimal Transport

“…The stability of the dynamics is governed by optimal transport metric, entropy and Fisher information.Recently, a new viewpoint has been brought into the realm of population games based on optimal transport, see Villani's book [3,38] and mean field games in the series work of Larsy, Lions [6,13,19]. The mean field games have continuous strategy sets and infinite players [4,5]. Each player is assumed to make decisions according to a stochastic process instead of making a one-shot decision.…”

Computation of Cournot–Nash Equilibria by Entropic Regularization

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We consider a class of games with continuum of players where equilibria can be obtained by the minimization of a certain functional related to optimal transport as emphasized in [7]. We then use the powerful entropic regularization technique to approximate the problem and solve it numerically in various cases.

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