Backward Induction for Repeated Games

TL;DR

This paper introduces a novel backward induction method for approximating subgame perfect Nash equilibria in infinitely repeated discounted games, leveraging advanced functional programming concepts and lazy evaluation.

Contribution

It applies higher-order sequential game theory to imperfect information games using the selection monad transformer and computable reals, pioneering this approach.

Findings

First application of the theory to imperfect information games

Uses lazy evaluation for nontrivial computational benefits

Demonstrates method with Iterated Prisoner's Dilemma

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 Escard\'o 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.