# Finitely Repeated Games with Finite Automata

The paper studies the implication of bounding the complexity of the strategies players may select on the set of equilibrium payoffs in repeated games. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. A finite automaton is an automated machine that implements a strategy; it has a finite number of states and an initial state. It prescribes the action to be taken as a function of the current state and a transition function changing the states of the automaton as a function of its current state and the present actions of the other players. The size of an automaton is its number of states. The Main results imply in particular that in two person repeated games, the equilibrium payoffs of a sequence of such games, G(n), n = 1,2,..., converges as n goes to infinity to the individual rational and feasible payoffs of the one shot game, whenever the bound on one of the two automata sizes is polynomial or subexponential in n and the length of the game and the bounds of the automata sizes are at least n. A special case of such result justifies cooperation in the finitely repeated prisoner's dilemma, without departure from strict utility maximization or complete information, but under the assumption that there are bounds (possibly very large) to the complexity of the strategies that the players may use.