Linear algebraic structure of zero-determinant strategies in repeated games

TL;DR

This paper explores the mathematical structure of zero-determinant strategies in repeated games, revealing their linear relations, independence, and conditions for multiple strategies to coexist, using linear algebra.

Contribution

It provides a comprehensive linear algebra framework for understanding ZD strategies when multiple players employ them, including existence, independence, and control of payoff relations.

Findings

Solutions always exist for the linear relations enforced by ZD strategies.

Linear payoff relations are independent under certain conditions.

An example shows a player can enforce multiple payoff relations simultaneously.

Abstract

Zero-determinant (ZD) strategies, a recently found novel class of strategies in repeated games, has attracted much attention in evolutionary game theory. A ZD strategy unilaterally enforces a linear relation between average payoffs of players. Although existence and evolutional stability of ZD strategies have been studied in simple games, their mathematical properties have not been well-known yet. For example, what happens when more than one players employ ZD strategies have not been clarified. In this paper, we provide a general framework for investigating situations where more than one players employ ZD strategies in terms of linear algebra. First, we theoretically prove that a set of linear relations of average payoffs enforced by ZD strategies always has solutions, which implies that incompatible linear relations are impossible. Second, we prove that linear payoff relations are independent of each other under some conditions. These results hold for general games with public monitoring including perfect-monitoring games. Furthermore, we provide a simple example of a two-player game in which one player can simultaneously enforce two linear relations, that is, simultaneously control her and her opponent's average payoffs. All of these results elucidate general mathematical properties of ZD strategies.