Dynamic supergames on trees

A class of discrete-time, dynamic supergames played by infinitely many players on tree is studied. Each player plays a number of two-person games with her neighbors simultaneously and updates her strategy stochastically based on the information from her neighbors' strategies and the payoff she gets. Under the conditions of the pre-specified updating rules and the transition probabilities, the relevant strategy evolution process of the supersgame is proved to be a reversible Markov chain. The invariant Gibbsian measures which represent the long-run equilibrium plays with binary strategy and symmetric payoffs are obtained. They are well known Ising models on trees which exhibit phase transition phenomena in certain cases.