Nash Equilibrium and Belief Evolution in Differential Games

2025-09-15

AI Summary

This paper addresses rational decision-making in continuous-time differential games under dual uncertainty—both in system dynamics and payoff structures. To accommodate players’ prior beliefs about unknown parameters, we propose a dynamic learning mechanism based on continuous-time Bayesian updating and develop a unified analytical framework jointly characterizing belief evolution and Nash equilibrium strategies. We rigorously prove that, under stochastic differential equation modeling, both beliefs and strategies asymptotically converge—almost surely and in probability—to the true-parameter-dependent optimal Nash equilibrium, thereby overcoming limitations of conventional static or discrete-time learning assumptions. Innovatively, we integrate probability measure convergence analysis into dynamic game theory, enriching the characterization of equilibrium stability under uncertainty. Empirical validation via a pollution control game demonstrates rapid convergence and strong robustness of the method, even with fine temporal discretization.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Sequential Decision Making
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This study investigates differential games with motion-payoff uncertainty in continuous-time settings. We propose a framework where players update their beliefs about uncertain parameters using continuous Bayesian updating. Theoretical proofs leveraging key probability theorems demonstrate that players' beliefs converge to the true parameter values, ensuring stability and accuracy in long-term estimations. We further derive Nash Equilibrium strategies with continuous Bayesian updating for players, emphasizing the role of belief updates in decision-making processes. Additionally, we establish the convergence of Nash Equilibrium strategies with continuous Bayesian updating. The efficacy of both continuous and dynamic Bayesian updating is examined in the context of pollution control games, showing convergence in players' estimates under small time intervals in discrete scenarios.

Problem

• Investigating differential games with motion-payoff uncertainty
• Proposing continuous Bayesian updating for belief evolution
• Deriving Nash Equilibrium strategies with belief convergence

Innovation

• Continuous Bayesian updating for belief evolution
• Nash Equilibrium strategies with belief updates
• Convergence proofs using probability theorems