Zero-determinant strategies in repeated asymmetric games

Abstract

Zero-determinant (ZD) strategies are conditional strategies which allow players adopting them to establish a relation between their expected payoffs and those of their opponents. The ZD strategies were investigated in different models including finite and infinite repeated two players’ games, multiplayer games, continuous action spaces and alternating move games. However, all previous studies considered only symmetric games between players, i.e., players have the same strategies and the same associated payoffs, thus the players’ identities are interchangeable without affecting the game's dynamics. In this study, we analytically model and derive the ZD strategies for asymmetric two players’ games, focusing on one-memory strategies and infinite repeated encounters. We derive the analytical bounds of equalizer and extortionate ZD strategies in 2 × 2 asymmetric games, which differ from the symmetric games case. Furthermore, we derive under what conditions a player using an extortionate ZD strategy will get a higher expected payoff than his/her opponent. Finally using a numerical example, we investigate ZD strategies in 2 × 2 asymmetric prisoner's dilemma game.