Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

TL;DR

This paper investigates whether zero-determinant strategies in repeated prisoner's dilemma games can still enforce unconditional cooperation when future payoffs are discounted, extending their applicability to real-world scenarios.

Contribution

It mathematically demonstrates that ZD strategies can induce unconditional cooperation even with payoff discounting, broadening their practical relevance.

Findings

ZD strategies can enforce unconditional cooperation under discounting.

Mathematical proof using determinant representation of expected payoffs.

Numerical calculations confirm the theoretical results.

Abstract

Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.