Non-computable strategies and discounted repeated games

Summary A number of authors have used formal models of computation to capture the idea of bounded rationality in repeated games. Most of this literature has used computability by a finite automaton as the standard. A conceptual difficulty with this standard is that the decision problem is not closed. That is, for every strategy implementable by an automaton, there is some best response implementable by an automaton, but there may not exist any algorithm forfinding such a best response that can be implemented by an automaton. However, such algorithms can always be implemented by a Turing machine, the most powerful formal model of computation. In this paper, we investigate whether the decision problem can be closed by adopting Turing machines as the standard of computability. The answer we offer is negative. Indeed, for a large class of discounted repeated games (including the repeated Prisoner's Dilemma) there exist strategies implementable by a Turing machine for whichno best response is implementable by a Turing machine.The work was begun while Nachbar was a visitor at The Center for Mathematical studies in Economics and Management Science at Northwestern University; he is grateful for their hospitality. We are also grateful to Robert Anderson and Neil Gretsky and to seminar audiences at UCLA for useful comments, and to the National Science Foundation and the UCLA Academic Senate Committee on Research for financial support. This paper is an outgrowth of work reported in Learning and Computability in Discounted Supergames.