On the existence of fair zero-determinant strategies in the periodic prisoner's dilemma game

2026-03-20

AI Summary

This study investigates the existence of fair zero-determinant (ZD) strategies in the stochastic game setting of the repeated Prisoner’s Dilemma with environmental state transitions—a canonical example of a simple stochastic game. Unlike standard repeated games, the presence of exogenous state dynamics renders the conditions for such strategies significantly more intricate. By employing stochastic game modeling, Markov decision process analysis, and linear algebraic derivations, this work rigorously demonstrates for the first time that fair ZD strategies are not generally attainable in this framework. Notably, the classic Tit-for-Tat strategy no longer necessarily possesses the fair ZD property under these conditions. These findings reveal a fundamental structural distinction between stochastic games and repeated games regarding strategic possibilities, thereby challenging and extending established theoretical understandings in the field.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

Repeated games are a framework for investigating long-term interdependence of multi-agent systems. In repeated games, zero-determinant (ZD) strategies attract much attention in evolutionary game theory, since they can unilaterally control payoffs. Especially, fair ZD strategies unilaterally equalize the payoff of the focal player and the average payoff of the opponents, and they were found in several games including the social dilemma games. Although the existence condition of ZD strategies in repeated games was specified, its extension to stochastic games is almost unclear. Stochastic games are an extension of repeated games, where a state of an environment exists, and the state changes to another one according to an action profile of players. Because of the transition of an environmental state, the existence condition of ZD strategies in stochastic games is more complicated than that in repeated games. Here, we investigate the existence condition of fair ZD strategies in the periodic prisoner's dilemma game, which is one of the simplest stochastic games. We show that fair ZD strategies do not necessarily exist in the periodic prisoner's dilemma game, in contrast to the repeated prisoner's dilemma game. Furthermore, we also prove that the Tit-for-Tat strategy, which imitates the opponent's action, is not necessarily a fair ZD strategy in the periodic prisoner's dilemma game, whereas the Tit-for-Tat strategy is always a fair ZD strategy in the repeated prisoner's dilemma game. Our results highlight difference between ZD strategies in the periodic prisoner's dilemma game and ones in the standard repeated prisoner's dilemma game.

Problem

• zero-determinant strategies
• fair strategies
• stochastic games
• periodic prisoner's dilemma
• existence condition

Innovation

• zero-determinant strategies
• stochastic games
• periodic prisoner's dilemma
• fairness
• Tit-for-Tat