Backward induction for repeated games

Abstract

We present a method of backward induction for computing approximate subgame perfect Nash equilibria of infinitely repeated games with discounted payoffs. This uses the selection monad transformer, combined with the searchable set monad viewed as a notion of 'topologically compact' nondeterminism, and a simple model of computable real numbers. This is the first application of Escardó and Oliva's theory of higher-order sequential games to games of imperfect information, in which (as well as its mathematical elegance) lazy evaluation does nontrivial work for us compared with a traditional game-theoretic analysis. Since a full theoretical understanding of this method is lacking (and appears to be very hard), we consider this an 'experimental' paper heavily inspired by theoretical ideas. We use the famous Iterated Prisoner's Dilemma as a worked example.