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The collective behavior of finite state stochastic automata is considered, which is of interest in view of the possibility of modeling group behavior of subjects in terms of these automata. The natural language for considering the collective behavior is that of game theory. After a brief introduction to a class of deterministic automata, the stochastic automaton is formulated and a nonlinear reinforcement specified. The finite state stochastic automaton is first considered in a game with nature, and conditions under which the automaton's winnings reach the Von Neumann value of the game are established. Next, two stochastic automata with an arbitrary number of states for each are considered in a game, the game matrix being specified. Performance of the automata for various conditions on the elements of the game matrix is considered. In a comparison of performance with deterministic automata, it is established that, for performance comparable to that of the finite state stochastic automaton, the deterministic automaton needs an infinite number of states. Finally, some games are simulated on a computer which verifies the general analysis and further sheds light on the details of the game.