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Learning in games considers how multiple agents maximize their own rewardsthrough repeated games. Memory, an ability that an agent changes his/her actiondepending on the history of actions in previous games, is often introduced intolearning to explore more clever strategies and discuss the decision-making ofreal agents like humans. However, such games with memory are hard to analyzebecause they exhibit complex phenomena like chaotic dynamics or divergence fromNash equilibrium. In particular, how asymmetry in memory capacities betweenagents affects learning in games is still unclear. In response, this studyformulates a gradient ascent algorithm in games with asymmetry memorycapacities. To obtain theoretical insights into learning dynamics, we firstconsider a simple case of zero-sum games. We observe complex behavior, wherelearning dynamics draw a heteroclinic connection from unstable fixed points tostable ones. Despite this complexity, we analyze learning dynamics and provelocal convergence to these stable fixed points, i.e., the Nash equilibria. Weidentify the mechanism driving this convergence: an agent with a longer memorylearns to exploit the other, which in turn endows the other's utility functionwith strict concavity. We further numerically observe such convergence invarious initial strategies, action numbers, and memory lengths. This studyreveals a novel phenomenon due to memory asymmetry, providing fundamentalstrides in learning in games and new insights into computing equilibria.

Memory Asymmetry Creates Heteroclinic Orbits to Nash Equilibrium in Learning in Zero-Sum Games