Performance of Zero-determinant Strategies in Repeated Games without Discounting

Masahiko Ueda

physics.soc-phOct 2, 2026

Performance of Zero-determinant Strategies in Repeated Games without Discounting

Authors: Masahiko Ueda

Organizations: Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

Abstract

Zero-determinant (ZD) strategies are a class of strategies in repeated games, which unilaterally control payoffs. It has been shown that several ZD strategies promote cooperation in social dilemma games. It has been widely believed that the performance of ZD strategies is expressed as ``unilaterally enforce linear relations between payoffs''. However, in order to interpret properties of ZD strategies in repeated games without discounting, previous studies assumed that the limit of the Cesaro average of the probability distribution of the action profile exists. Here, we explain the performance of ZD strategies without this assumption, which leads to a stronger result than previous ones.

Figures & tables

Figure 1: Numerical verification of our theoretical result ( 26 ). The horizontal axis is $T$ , and the vertical axis is $\sum_{t=1}^{T}\sum_{\bm{a}^{\prime}}B\left(\bm{a}^{\prime}\right)P_{t}\left(\bm{a}^{\prime}\right)$ with $B=s_{2}-r$ , where the ensemble average is calculated from $10^{6}$ samples. The parameters are $(R,S,T,P)=(3,0,5,1)$ and $r=(R+P)/2=2$ . $W$ is calculated as $W=3$ .

Repeated Games

Repeated games study how strategies, cooperation, and competition develop when players interact repeatedly and respond to past actions.

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

Ken Nakamura, Masahiko Ueda

Graduate School of Sciences and Technology for Innovation, Yamaguchi University, Yamaguchi 753-8512, Japan · Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan

Game Theory Repeated Games

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Stochastic games are a framework for investigating long-term interdependence of multi-agent systems with environmental feedback. When the number of environmental states is one, they are reduced to repeated games. 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 remains largely unclear. 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. The periodic prisoner's dilemma game consists of two environmental states and the two states alternate deterministically. Whereas each stage game is not necessarily the prisoner's dilemma game, the whole game can be regarded as the prisoner's dilemma game on average. 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 those in the standard repeated prisoner's dilemma game.

Abstract

Repeated Games

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