Stochastic Automata Modeling of Certain Problems of Collective Behavior

The behavior of a collective of interacting stochastic automata in a stationary random environment is considered. Each automaton is a goal-seeking element which manipulates its strategy only as a function of the environmental response. The goals of the automata as well as the nature of their interactions are assumed to satisfy certain qualitative properties which ensure the existence and uniqueness of a Nash strategy. It is shown that, without need of any a priori information, the collective behavior of the automata converges in probability to the Nash strategy. Two applications are presented. The first concerns the process of market price formation in a competitive economy. The second concerns the optimal allocation of a unidimensional resource in the process of system operation.