Global Behavior of Learning Dynamics in Zero-Sum Games with Memory Asymmetry

2024-05-23

arXiv.org

AI Summary

This paper investigates global learning dynamics in memory-asymmetric two-player zero-sum games: Player X employs a reactive strategy based on Y’s previous action, while Y is memoryless. To address the complex convergence behavior induced by this asymmetry, we introduce an extended KL divergence and a family of reactive-strategy Lyapunov functions. Leveraging tools from nonlinear dynamical systems theory, information geometry, and Lyapunov stability analysis, we rigorously characterize the global convergence mechanism to Nash equilibria and the evolution of strategy exploitability. We prove—supported by numerical experiments—that whenever X exploits Y, the system globally converges to a Nash equilibrium; conversely, any deviation of Y from equilibrium triggers persistent growth in X’s exploitability. The core contribution is a novel information-theoretic–dynamical framework specifically tailored to memory-asymmetric strategic interactions.

Technology Category

Game Theory and Economic Paradigms: Adversarial LearningMultiagent Systems: Adversarial AgentsSearch and Optimization: Learning to Search

Application Category

Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactionsSecurity and Privacy: Large-scale security measurementsSemantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

This study examines the global behavior of dynamics in learning in games between two players, X and Y. We consider the simplest situation for memory asymmetry between two players: X memorizes the other Y's previous action and uses reactive strategies, while Y has no memory. Although this memory complicates their learning dynamics, we characterize the global behavior of such complex dynamics by discovering and analyzing two novel quantities. One is an extended Kullback-Leibler divergence from the Nash equilibrium, a well-known conserved quantity from previous studies. The other is a family of Lyapunov functions of X's reactive strategy. One of the global behaviors we capture is that if X exploits Y, then their strategies converge to the Nash equilibrium. Another is that if Y's strategy is out of equilibrium, then X becomes more exploitative with time. Consequently, we suggest global convergence to the Nash equilibrium from both aspects of theory and experiment. This study provides a novel characterization of the global behavior in learning in games through a couple of indicators.

Problem

• Analyzes global behavior in zero-sum games with memory asymmetry.
• Characterizes dynamics using extended Kullback-Leibler divergence and Lyapunov functions.
• Demonstrates convergence to Nash equilibrium under specific conditions.

Innovation

• Extended Kullback-Leibler divergence from Nash equilibrium
• Lyapunov functions for X's reactive strategy
• Global convergence to Nash equilibrium demonstrated