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Repeated games consider a situation where multiple agents are motivated bytheir independent rewards throughout learning. In general, the dynamics oftheir learning become complex. Especially when their rewards compete with eachother like zero-sum games, the dynamics often do not converge to their optimum,i.e., the Nash equilibrium. To tackle such complexity, many studies haveunderstood various learning algorithms as dynamical systems and discoveredqualitative insights among the algorithms. However, such studies have yet tohandle multi-memory games (where agents can memorize actions they played in thepast and choose their actions based on their memories), even thoughmemorization plays a pivotal role in artificial intelligence and interpersonalrelationship. This study extends two major learning algorithms in games, i.e.,replicator dynamics and gradient ascent, into multi-memory games. Then, weprove their dynamics are identical. Furthermore, theoretically andexperimentally, we clarify that the learning dynamics diverge from the Nashequilibrium in multi-memory zero-sum games and reach heteroclinic cycles(sojourn longer around the boundary of the strategy space), providing afundamental advance in learning in games.

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