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On the tractability of Nash equilibrium

Ioannis Avramopoulos

Abstract

In this paper, we propose a method for solving a PPAD-complete problem [Papadimitriou, 1994]. Given is the payoff matrix $C$ of a symmetric bimatrix game $(C, C^T)$ and our goal is to compute a Nash equilibrium of $(C, C^T)$. In this paper, we devise a nonlinear replicator dynamic (whose right-hand-side can be obtained by solving a pair of convex optimization problems) with the following property: Under any invertible $0 < C \leq 1$, every orbit of our dynamic starting at an interior strategy of the standard simplex approaches a set of strategies of $(C, C^T)$ such that, for each strategy in this set, a symmetric Nash equilibrium strategy can be computed by solving the aforementioned convex mathematical programs. We prove convergence using results in analysis (the analytic implicit function theorem), nonlinear optimization theory (duality theory, Berge's maximum principle, and a theorem of Robinson [1980] on the Lipschitz continuity of parametric nonlinear programs), and dynamical systems theory (such as the LaSalle invariance principle).

On the tractability of Nash equilibrium

Abstract

In this paper, we propose a method for solving a PPAD-complete problem [Papadimitriou, 1994]. Given is the payoff matrix of a symmetric bimatrix game and our goal is to compute a Nash equilibrium of . In this paper, we devise a nonlinear replicator dynamic (whose right-hand-side can be obtained by solving a pair of convex optimization problems) with the following property: Under any invertible , every orbit of our dynamic starting at an interior strategy of the standard simplex approaches a set of strategies of such that, for each strategy in this set, a symmetric Nash equilibrium strategy can be computed by solving the aforementioned convex mathematical programs. We prove convergence using results in analysis (the analytic implicit function theorem), nonlinear optimization theory (duality theory, Berge's maximum principle, and a theorem of Robinson [1980] on the Lipschitz continuity of parametric nonlinear programs), and dynamical systems theory (such as the LaSalle invariance principle).
Paper Structure (7 sections, 11 theorems, 71 equations)

This paper contains 7 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries in game theory
  3. Our techniques and main result
  4. Other related work
  5. On optimization problem \ref{['optproblem']}
  6. On optimization problem \ref{['first_expression']}
  7. Proof of Theorem \ref{['main_theorem']}

Key Result

Theorem 1

Let $0 < C \leq 1$ be the payoff matrix of a symmetric bimatrix game $(C, C^T)$ and let $S \equiv CC^T$. Furthermore, let $RE(\cdot, \cdot)$ denote the relative entropy function. If $C$ is invertible, starting at any $X^0 \in \mathbb{\mathring{X}}(C)$, every limit point of the orbit $X_t, t \in [0, as $t \rightarrow \infty$, is a strategy, say $X$, whose multiplier $Z_X$ is a symmetric Nash equil

Theorems & Definitions (24)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 14 more