Reactive learning strategies for iterated games

Abstract

In an iterated game between two players, there is much interest in characterizing the set of feasible pay-offs for both players when one player uses a fixed strategy and the other player is free to switch. Such characterizations have led to extortionists, equalizers, partners and rivals. Most of those studies use memory-one strategies, which specify the probabilities to take actions depending on the outcome of the previous round. Here, we consider ‘reactive learning strategies’, which gradually modify their propensity to take certain actions based on past actions of the opponent. Every linear reactive learning strategy, p *, corresponds to a memory one-strategy, p , and vice versa. We prove that for evaluating the region of feasible pay-offs against a memory-one strategy, C ( p ) , we need to check its performance against at most 11 other strategies. Thus, C ( p ) is the convex hull in R 2 of at most 11 points. Furthermore, if p is a memory-one strategy, with feasible pay-off region C ( p ) , and p * is the corresponding reactive learning strategy, with feasible pay-off region C ( p ∗ ) , then C ( p ∗ ) is a subset of C ( p ) . Reactive learning strategies are therefore powerful tools in restricting the outcomes of iterated games.