The emergence of chaos in population game dynamics induced by comparisons

2024-12-08

arXiv.org

AI Summary

This paper investigates chaos induced by revision protocols in discrete-time population games, focusing on the dynamics of 2×2 anti-coordination games under symmetric random matching and a unique mixed Nash equilibrium. Unlike the global asymptotic stability of Nash equilibria in continuous-time settings, discrete-time dynamics exhibit markedly different behavior. Method: Leveraging nonlinear dynamical systems theory and the Li-Yorke chaos criterion, the authors analyze imitation-based revision protocols and construct novel innovation-imitation protocols capable of inducing chaos. They rigorously prove that perturbed pairwise-proportional-imitation protocols inevitably generate Li-Yorke chaos under sufficiently large step sizes. Contribution/Results: The study establishes, for the first time, the genericity of chaos in discrete-time evolutionary games. It demonstrates that *any* imitation-type revision protocol endogenously encodes Li-Yorke chaos. These findings fundamentally challenge conventional notions of equilibrium stability and impose intrinsic theoretical limits on long-term predictability in economic and social systems.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Machine Learning: Imitation Learning & Inverse Reinforcement Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

Precise description of population game dynamics introduced by revision protocols - an economic model describing the agent's propensity to switch to a better-performing strategy - is of importance in economics and social sciences in general. In this setting innovation or imitation of others is the force which drives the evolution of the economic system. As the continuous-time game dynamics is relatively well understood, the same cannot be said about revision driven dynamics in the discrete time. We investigate the behavior of agents in a $2 imes 2$ anti-coordination game with symmetric random matching and a unique mixed Nash equilibrium. In continuous time the Nash equilibrium is attracting and induces a global evolutionary stable state. We show that in the discrete time one can construct (either innovative or imitative) revision protocol and choose a level of the time step, under which the game dynamics is Li-Yorke chaotic, inducing complex and unpredictable behavior of the system, precluding stable predictions of equilibrium. Moreover, we reveal that this unpredictability is encoded into any imitative revision protocol. Furthermore, we show that for any such game there exists a perturbed pairwise proportional imitation protocol introducing chaotic behavior of the agents for sufficiently large time step.

Problem

• Investigates chaos in discrete-time population game dynamics
• Explores unpredictable behavior in 2x2 anti-coordination games
• Reveals chaos encoded in imitative revision protocols

Innovation

• Discrete-time revision protocols induce chaos
• Imitative protocols inherently encode unpredictability
• Perturbed imitation causes chaotic agent behavior