Ising-type suppergames on planar lattices

In this paper a broad class of large Ising-type supergames, i.e., infinitely repeated games played by many players are studied. The players located on the vertex set of some planar lattice play several basic games with his neighbors. The basic game is two-person Ising-type game with symmetric payoffs. Three planar lattice models are discussed. Under the conditions of the pre-specified updating rules and the transition probabilities i.e., these relevant stochastic process of strategy configuration given, the formula of invariant measures which represent the long-run equilibrium plays are obtained. The phase transition phenomena is discovered.