  Learning in games considers how multiple agents maximize their own rewards
through repeated games. Memory, an ability that an agent changes his/her action
depending on the history of actions in previous games, is often introduced into
learning to explore more clever strategies and discuss the decision-making of
real agents like humans. However, such games with memory are hard to analyze
because they exhibit complex phenomena like chaotic dynamics or divergence from
Nash equilibrium. In particular, how asymmetry in memory capacities between
agents affects learning in games is still unclear. In response, this study
formulates a gradient ascent algorithm in games with asymmetry memory
capacities. To obtain theoretical insights into learning dynamics, we first
consider a simple case of zero-sum games. We observe complex behavior, where
learning dynamics draw a heteroclinic connection from unstable fixed points to
stable ones. Despite this complexity, we analyze learning dynamics and prove
local convergence to these stable fixed points, i.e., the Nash equilibria. We
identify the mechanism driving this convergence: an agent with a longer memory
learns to exploit the other, which in turn endows the other's utility function
with strict concavity. We further numerically observe such convergence in
various initial strategies, action numbers, and memory lengths. This study
reveals a novel phenomenon due to memory asymmetry, providing fundamental
strides in learning in games and new insights into computing equilibria.
