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Evolutionary Algorithms for Computing Nash Equilibria in Dynamic Games

Alireza Rezaee

TL;DR

This paper tackles the challenge of computing Nash equilibria in dynamic, non-cooperative N-player games with nonlinear dynamics and long horizons, where classical methods often fail or require strong assumptions. It introduces two population-based approaches: a co-evolutionary genetic algorithm that searches joint strategy spaces by evolving per-player strategy subpopulations, and a hybrid PSO method augmented with local search refinement. Through numerical experiments on a three-player linear-quadratic game and a Kydland two-player game with non-quadratic objectives, the methods demonstrate robust convergence to Nash equilibria and resilience to local optima, outperforming traditional dynamic programming in certain scenarios. The proposed algorithms offer flexible, scalable tools for multi-agent decision making in control, economics, and networked systems, with potential extensions to more advanced co-evolutionary schemes and hybrid optimization paradigms.

Abstract

Dynamic nonzero sum games are widely used to model multi agent decision making in control, economics, and related fields. Classical methods for computing Nash equilibria, especially in linear quadratic settings, rely on strong structural assumptions and become impractical for nonlinear dynamics, many players, or long horizons, where multiple local equilibria may exist. We show through examples that such methods can fail to reach the true global Nash equilibrium even in relatively small games. To address this, we propose two population based evolutionary algorithms for general dynamic games with linear or nonlinear dynamics and arbitrary objective functions: a co evolutionary genetic algorithm and a hybrid genetic algorithm particle swarm optimization scheme. Both approaches search directly over joint strategy spaces without restrictive assumptions and are less prone to getting trapped in local Nash equilibria, providing more reliable approximations to global Nash solutions.

Evolutionary Algorithms for Computing Nash Equilibria in Dynamic Games

TL;DR

This paper tackles the challenge of computing Nash equilibria in dynamic, non-cooperative N-player games with nonlinear dynamics and long horizons, where classical methods often fail or require strong assumptions. It introduces two population-based approaches: a co-evolutionary genetic algorithm that searches joint strategy spaces by evolving per-player strategy subpopulations, and a hybrid PSO method augmented with local search refinement. Through numerical experiments on a three-player linear-quadratic game and a Kydland two-player game with non-quadratic objectives, the methods demonstrate robust convergence to Nash equilibria and resilience to local optima, outperforming traditional dynamic programming in certain scenarios. The proposed algorithms offer flexible, scalable tools for multi-agent decision making in control, economics, and networked systems, with potential extensions to more advanced co-evolutionary schemes and hybrid optimization paradigms.

Abstract

Dynamic nonzero sum games are widely used to model multi agent decision making in control, economics, and related fields. Classical methods for computing Nash equilibria, especially in linear quadratic settings, rely on strong structural assumptions and become impractical for nonlinear dynamics, many players, or long horizons, where multiple local equilibria may exist. We show through examples that such methods can fail to reach the true global Nash equilibrium even in relatively small games. To address this, we propose two population based evolutionary algorithms for general dynamic games with linear or nonlinear dynamics and arbitrary objective functions: a co evolutionary genetic algorithm and a hybrid genetic algorithm particle swarm optimization scheme. Both approaches search directly over joint strategy spaces without restrictive assumptions and are less prone to getting trapped in local Nash equilibria, providing more reliable approximations to global Nash solutions.
Paper Structure (29 sections, 7 equations, 5 figures)

This paper contains 29 sections, 7 equations, 5 figures.

Table of Contents

  1. Introduction to Game Theory
  2. Definition of an $N$-Player Dynamic Game
  3. Nash Equilibrium
  4. Evolutionary Approaches to Dynamic Games
  5. Genetic Algorithm for Nash Equilibrium Search
  6. Chromosome Encoding
  7. Co-Evolutionary Structure
  8. Fitness Evaluation
  9. Selection, Crossover, and Mutation
  10. Selection.
  11. Crossover.
  12. Mutation.
  13. Stopping Criteria
  14. Co-Evolutionary GA Algorithm
  15. Particle Swarm Optimization and Hybrid PSO
  16. ...and 14 more sections

Figures (5)

  • Figure 1: Convergence of the fitness function for the three-player, three-stage LQ game using the co-evolutionary GA (complete information). The smooth decrease is due to elitism, which preserves the best chromosome in each generation.
  • Figure 2: Convergence of the PSO algorithm for the three-player LQ game: fitness versus iteration.
  • Figure 3: Trajectories of particle positions (strategies) for the three players in the search space as PSO converges to the Nash equilibrium in the three-player LQ game.
  • Figure 4: Convergence of the GA fitness function for the two-player Kydland dynamic game with complete information.
  • Figure 5: Evolution of the strategies for the two players in the Kydland game under the GA: trajectories of decision variables across generations.