Learning in Multi-Memory Games Triggers Complex Dynamics Diverging from Nash Equilibrium

TL;DR

This paper extends key learning algorithms to multi-memory zero-sum games, revealing that their dynamics diverge from Nash equilibrium and tend to form heteroclinic cycles, highlighting complex behaviors in strategic learning.

Contribution

It introduces a novel extension of replicator dynamics and gradient ascent to multi-memory games and proves their equivalence, showing divergence from Nash equilibrium.

Findings

Dynamics diverge from Nash equilibrium in multi-memory zero-sum games.

Learning trajectories tend to form heteroclinic cycles.

Theoretical and experimental validation of divergence and cycle formation.

Abstract

Repeated games consider a situation where multiple agents are motivated by their independent rewards throughout learning. In general, the dynamics of their learning become complex. Especially when their rewards compete with each other like zero-sum games, the dynamics often do not converge to their optimum, i.e., the Nash equilibrium. To tackle such complexity, many studies have understood various learning algorithms as dynamical systems and discovered qualitative insights among the algorithms. However, such studies have yet to handle multi-memory games (where agents can memorize actions they played in the past and choose their actions based on their memories), even though memorization plays a pivotal role in artificial intelligence and interpersonal relationship. This study extends two major learning algorithms in games, i.e., replicator dynamics and gradient ascent, into multi-memory games. Then, we prove their dynamics are identical. Furthermore, theoretically and experimentally, we clarify that the learning dynamics diverge from the Nash equilibrium in multi-memory zero-sum games and reach heteroclinic cycles (sojourn longer around the boundary of the strategy space), providing a fundamental advance in learning in games.