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From Nash to Cournot--Nash equilibria via $Γ$-convergence

João Miguel Machado, Guilherme Mazanti, Laurent Pfeiffer

Abstract

This work addresses the issue of the convergence of an $N$-player game towards a limit model involving a continuum of players, as the number of agents $N$ goes to infinity. More precisely, we investigate the convergence of Nash equilibria to a Cournot--Nash equilibrium of the limit model. When the cost function of the players is the first variation of some potential function, equilibria can be characterized by a stationarity condition, satisfied in particular by the minimizers of the potential. We demonstrate such a characterization under low regularity assumptions. Then we focus on the case where the players interact in a pairwise fashion; in this case we show that the original sequence of $N$-player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of $Γ$-convergence to the potential function of the limit game.

From Nash to Cournot--Nash equilibria via $Γ$-convergence

Abstract

This work addresses the issue of the convergence of an -player game towards a limit model involving a continuum of players, as the number of agents goes to infinity. More precisely, we investigate the convergence of Nash equilibria to a Cournot--Nash equilibrium of the limit model. When the cost function of the players is the first variation of some potential function, equilibria can be characterized by a stationarity condition, satisfied in particular by the minimizers of the potential. We demonstrate such a characterization under low regularity assumptions. Then we focus on the case where the players interact in a pairwise fashion; in this case we show that the original sequence of -player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of -convergence to the potential function of the limit game.

Paper Structure

This paper contains 22 sections, 20 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Contributions of this work
  3. Related work
  4. Examples
  5. Organization of this paper
  6. Preliminaries in measure theory and Gamma-convergence
  7. Topologies on spaces of Radon measures
  8. Random probability measures and their weak topologies
  9. Gamma-convergence
  10. Potential structure and stability of the value function
  11. Potential structure for Cournot--Nash equilibria
  12. On the finiteness of the infimum
  13. Stability of the value function
  14. Convergence of Nash to Cournot--Nash equilibria
  15. Convergence in the open-loop formulation
  16. ...and 7 more sections

Key Result

Theorem 2.2

Let $\mathcal{F} \subset \mathscr{P}(\mathcal{X})$ be a family of probability measures over $\mathcal{X}$. Then $\mathcal{F}$ is compact for the narrow topology if, and only if, it is a tight family, i.e., for all $\varepsilon>0$, there is a compact set $K$ such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2: Prokhorov's theorem, ambrosio2021lectures
  • Proposition 2.3: Ambrosio2008GigliSavare
  • Lemma 2.4
  • Definition 2.5
  • Theorem 2.6: Disintegration theorem
  • Lemma 2.7: Ambrosio2008GigliSavare
  • proof
  • Definition 2.8
  • ...and 42 more