Cooperation, Repetition, and Automata

This talk studies the implications of bounding the complexity of players' strategies in long term interactions. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. A nite automaton has a nite number of states and an initial state. It prescribes the action to be taken as a function of the current state and its next state is a function of its current state and the actions of the other players. The size of an automaton is its number of states. The results study the equilibrium payo s per stage of the repeated games when players' strategies are restricted to those implementable by automata of bounded size. The rst talk will concentrate maily on the 0-sum case and address the following topics/questions. 1 What is the relation between the bounds of the automata sizes and the quantitative advantage of the player with the larger bound. (Theorems 1 and 3 of the enclosed paper) 2 What is the duration (number of repetition) needed for an unrestricted player to exploit fully his advantage over a player with bound automata (Conjecture 2 of the enclosed paper including a positive solution of its second part). 3 The existence of a deterministic periodic sequence (with period n) which is asymptotically random for every automata of size o(n= log n) (Proposition 2 of the enclosed paper).