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An Adaptive Collaborative Neurodynamic Approach to Compute Nash Equilibrium in Normal-Form Games

Jianing Chen

TL;DR

The paper addresses the problem of computing exact Nash equilibria in general N-player normal-form games with mixed strategies, where payoffs are non-convex and the pseudo-gradient is non-monotone. It reformulates NE as a global minimization of a nonnegative cost tilde{Q}(x) over a convex, bounded feasible set and introduces an Adaptive Neurodynamic Approach (ANA) with an adaptive penalty to ensure finite-time constraint entry, proving convergence to the critical point set. To achieve global optimality, it augments ANA with a Collaborative Neurodynamic Approach (ACNA) by incorporating particle swarm optimization, guaranteeing global convergence to a NE with probability one. A numerical example on a three-player rock-paper-scissors game demonstrates convergence of strategies to the NE, zeroing of the cost tilde{Q}, and feasibility via $G(x) \to 0$ and $H(x) \to 0$, validating the method’s effectiveness for large-scale, non-convex multi-player games.

Abstract

The Nash Equilibrium (NE), one of the elegant and fundamental concepts in game theory, plays a crucial part within various fields, including engineering and computer science. However, efficiently computing an NE in normal-form games remains a significant challenge, particularly for large-scale problems. In contrast to widely applied simplicial and homotopy methods, this paper designs a novel Adaptive Collaborative Neurodynamic Approach (ACNA), which for the first time guarantees both exact and global NE computation for general $N$-player normal-form games with mixed strategies, where the payoff functions are non-convex and the pseudo-gradient is non-monotone. Additionally, leveraging the adaptive penalty method, the ACNA ensures its state enters the constraint set in finite time, which avoids the second-order sufficiency conditions required by Lagrangian methods, and the computationally complicated penalty parameter estimation needed by exact penalty methods. Furthermore, by incorporating the particle swarm algorithm, it is demonstrated that the ACNA achieves global convergence to an exact NE with probability one. At last, a simulation is conducted to validate the effectiveness of the proposed approach.

An Adaptive Collaborative Neurodynamic Approach to Compute Nash Equilibrium in Normal-Form Games

TL;DR

The paper addresses the problem of computing exact Nash equilibria in general N-player normal-form games with mixed strategies, where payoffs are non-convex and the pseudo-gradient is non-monotone. It reformulates NE as a global minimization of a nonnegative cost tilde{Q}(x) over a convex, bounded feasible set and introduces an Adaptive Neurodynamic Approach (ANA) with an adaptive penalty to ensure finite-time constraint entry, proving convergence to the critical point set. To achieve global optimality, it augments ANA with a Collaborative Neurodynamic Approach (ACNA) by incorporating particle swarm optimization, guaranteeing global convergence to a NE with probability one. A numerical example on a three-player rock-paper-scissors game demonstrates convergence of strategies to the NE, zeroing of the cost tilde{Q}, and feasibility via and , validating the method’s effectiveness for large-scale, non-convex multi-player games.

Abstract

The Nash Equilibrium (NE), one of the elegant and fundamental concepts in game theory, plays a crucial part within various fields, including engineering and computer science. However, efficiently computing an NE in normal-form games remains a significant challenge, particularly for large-scale problems. In contrast to widely applied simplicial and homotopy methods, this paper designs a novel Adaptive Collaborative Neurodynamic Approach (ACNA), which for the first time guarantees both exact and global NE computation for general -player normal-form games with mixed strategies, where the payoff functions are non-convex and the pseudo-gradient is non-monotone. Additionally, leveraging the adaptive penalty method, the ACNA ensures its state enters the constraint set in finite time, which avoids the second-order sufficiency conditions required by Lagrangian methods, and the computationally complicated penalty parameter estimation needed by exact penalty methods. Furthermore, by incorporating the particle swarm algorithm, it is demonstrated that the ACNA achieves global convergence to an exact NE with probability one. At last, a simulation is conducted to validate the effectiveness of the proposed approach.

Paper Structure

This paper contains 12 sections, 6 theorems, 51 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary and Problem Statement
  3. Notations
  4. Normal-Form Game with Mixed Strategies
  5. Optimization Problem Transformation
  6. Main Results
  7. Design of Adaptive Neurodynamic Approach
  8. Finite-Time Convergence to the Constraint Set
  9. Convergence Analysis
  10. An Adaptive Collaborative Neurodynamic Approach for Global Optimal Solutions
  11. Numerical Examples
  12. Conclusion

Key Result

Lemma II.1

$x^*$ is an NE of game wenti1, if and only if, it is a global minima of problem wenti2.

Figures (1)

  • Figure :

Theorems & Definitions (16)

  • Definition II.1
  • Remark II.1
  • Lemma II.1
  • proof
  • Remark III.1
  • Theorem III.1
  • proof
  • Theorem III.2
  • proof
  • Remark III.2
  • ...and 6 more