  Learning in games discusses the processes where multiple players learn their
optimal strategies through the repetition of game plays. The dynamics of
learning between two players in zero-sum games, such as matching pennies, where
their benefits are competitive, have already been well analyzed. However, it is
still unexplored and challenging to analyze the dynamics of learning among
three players. In this study, we formulate a minimalistic game where three
players compete to match their actions with one another. Although interaction
among three players diversifies and complicates the Nash equilibria, we fully
analyze the equilibria. We also discuss the dynamics of learning based on some
famous algorithms categorized into Follow the Regularized Leader. From both
theoretical and experimental aspects, we characterize the dynamics by
categorizing three-player interactions into three forces to synchronize their
actions, switch their actions rotationally, and seek competition.
