Zero-determinant strategies in finitely repeated games

TL;DR

This paper analytically explores zero-determinant strategies in finitely repeated prisoner's dilemma games, extending known results from infinite games and identifying conditions for their existence.

Contribution

It generalizes ZD strategies to finitely repeated games with arbitrary payoff matrices and establishes that only ZD or unconditional strategies enforce linear payoff relationships.

Findings

Extended ZD strategies to finitely repeated games with 0<w<1.

Derived threshold values of w for key ZD strategies.

Proved only ZD or unconditional strategies enforce linear payoff relationships.

Abstract

Direct reciprocity is a mechanism for sustaining mutual cooperation in repeated social dilemma games, where a player would keep cooperation to avoid being retaliated by a co-player in the future. So-called zero-determinant (ZD) strategies enable a player to unilaterally set a linear relationship between the player's own payoff and the co-player's payoff regardless of the strategy of the co-player. In the present study, we analytically study zero-determinant strategies in finitely repeated (two-person) prisoner's dilemma games with a general payoff matrix. Our results are as follows. First, we present the forms of solutions that extend the known results for infinitely repeated games (with a discount factor w of unity) to the case of finitely repeated games (0 < w < 1). Second, for the three most prominent ZD strategies, the equalizers, extortioners, and generous strategies, we derive the threshold value of w above which the ZD strategies exist. Third, we show that the only strategies that enforce a linear relationship between the two players' payoffs are either the ZD strategies or unconditional strategies, where the latter independently cooperates with a fixed probability in each round of the game, proving a conjecture previously made for infinitely repeated games.