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

Ken Nakamura; Masahiko Ueda

physics.soc-phMar 20, 2026

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

Authors: Ken Nakamura, Masahiko Ueda

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

Abstract

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.

Figures & tables

	$C$	$D$
$C$	$R^{(1)},R^{(2)}$	$S^{(1)},T^{(2)}$
$D$	$T^{(1)},S^{(2)}$	$P^{(1)},P^{(2)}$

Table 1: Payoffs in state

\sigma_{1}

Panel of Figure 1 — Figure 1: Relations between $\left\langle s_{1}\right\rangle^{*}$ and $\left\langle s_{2}\right\rangle^{*}$ when player $1$ uses the fair ZD strategy ( 155 ) and ( 156 ) with $c_{1}$ satisfying the equality condition of Eq. ( 154 ) and player $2$ uses $200$ randomly generated memory-one strategies. The payoffs in the left panel are set to $\left(\bar{R},\bar{S},\bar{T},\bar{P}\right)=(3,0,5,1)$ and $\left(\delta_{R},\delta_{S},\delta_{T},\delta_{P}\right)=(1,1,1,1)$ (satisfying $\delta_{R}+\delta_{P}=\delta_{S}+\delta_{T}$ ). The payoffs in the right panel are set to $\left(\bar{R},\bar{S},\bar{T},\bar{P}\right)=(3,0,5,1)$ and $\left(\delta_{R},\delta_{S},\delta_{T},\delta_{P}\right)=(2,1,1,1)$ (not satisfying $\delta_{R}+\delta_{P}=\delta_{S}+\delta_{T}$ ). Each $\left\langle s_{j}\right\rangle^{*}$ is calculated by time average over $10^{6}$ time steps. The fair ZD strategy indeed enforces a linear relation $\left\langle s_{1}\right\rangle^{*}=\left\langle s_{2}\right\rangle^{*}$ .

	$C$	$D$
$C$	$2\delta_{R},-2\delta_{R}$	$\bar{S}-\bar{T}+\delta_{S}+\delta_{T},\bar{T}-\bar{S}-\delta_{S}-\delta_{T}$
$D$	$\bar{T}-\bar{S}+\delta_{S}+\delta_{T},\bar{S}-\bar{T}-\delta_{S}-\delta_{T}$	$2\delta_{P},-2\delta_{P}$

Table 3: Relative payoffs in state

\sigma_{1}

Panel of Figure 2 — Figure 2: Relations between $\left\langle s_{1}\right\rangle^{*}$ and $\left\langle s_{2}\right\rangle^{*}$ when player $1$ uses TFT and player $2$ uses $200$ randomly generated memory-one strategies. The payoffs in the left panel are set to $\left(\bar{R},\bar{S},\bar{T},\bar{P}\right)=(3,0,5,1)$ and $\left(\delta_{R},\delta_{S},\delta_{T},\delta_{P}\right)=(1,1,1,1)$ (satisfying $\delta_{R}+\delta_{P}=\delta_{S}+\delta_{T}$ ). The payoffs in the right panel are set to $\left(\bar{R},\bar{S},\bar{T},\bar{P}\right)=(3,0,5,1)$ and $\left(\delta_{R},\delta_{S},\delta_{T},\delta_{P}\right)=(2,1,1,1)$ (not satisfying $\delta_{R}+\delta_{P}=\delta_{S}+\delta_{T}$ ). Each $\left\langle s_{j}\right\rangle^{*}$ is calculated by time average over $10^{6}$ time steps. TFT enforces a linear relation $\left\langle s_{1}\right\rangle^{*}=\left\langle s_{2}\right\rangle^{*}$ in the left panel, whereas TFT does not enforce a linear relation $\left\langle s_{1}\right\rangle^{*}=\left\langle s_{2}\right\rangle^{*}$ in the right panel.

The condition for the existence of fair ZD strategies	$\left\{\begin{array}[]{c}\bar{S}-\bar{T}+\delta_{S}+\delta_{T}\leq 2\min\left\{\delta_{R},\delta_{P}\right\}\\ \bar{T}-\bar{S}+\delta_{S}+\delta_{T}\geq 2\max\left\{\delta_{R},\delta_{P}\right\}\end{array}\right.$
	or
	$\left\{\begin{array}[]{c}\bar{S}-\bar{T}+\delta_{S}+\delta_{T}\geq 2\min\left\{\delta_{R},\delta_{P}\right\}\\ \bar{T}-\bar{S}+\delta_{S}+\delta_{T}\leq 2\max\left\{\delta_{R},\delta_{P}\right\}\end{array}\right.$
The condition under which TFT is a fair ZD strategy	$\delta_{R}+\delta_{P}=\delta_{S}+\delta_{T}$
The condition for the existence of fair partial ZD strategies	$\delta_{R}=0$ , $\delta_{P}=0$ , $\left\|\delta_{S}+\delta_{T}\right\|\leq\bar{T}-\bar{S}$

Table 5: Summary of this study.

	$C$	$D$
$C$	$R_{1},R_{1}$	$S_{1},T_{1}$
$D$	$T_{1},S_{1}$	$P_{1},P_{1}$

Table 6: Payoffs in state

\sigma_{1}

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 1: A linear relation between $\left\langle s_{1}\right\rangle^{*}$ and $\left\langle s_{2}\right\rangle^{*}$ when player $1$ uses the fair ZD strategy ( 158 ) and $T_{1}(C|\sigma_{1},\bm{a}^{\prime},\sigma_{2})=0$ and player $2$ uses $200$ randomly generated memory-one strategies. The payoffs are set to $\left(\bar{R},\bar{S},\bar{T},\bar{P}\right)=(3,0,5,1)$ and $\left(\delta_{R},\delta_{S},\delta_{T},\delta_{P}\right)=(3.5,1,1,-1.5)$ . Each $\left\langle s_{j}\right\rangle^{*}$ is calculated by time average over $10^{6}$ time steps. The fair ZD strategy indeed enforces a linear relation $\left\langle s_{1}\right\rangle^{*}=\left\langle s_{2}\right\rangle^{*}$ .

Figure 2: A linear relation between $\left\langle s_{1}\right\rangle^{*}$ and $\left\langle s_{2}\right\rangle^{*}$ when player $1$ uses the fair ZD strategy ( 167 ) and ( 168 ) with $c_{1}$ satisfying the equality condition of Eq. ( 166 ) and player $2$ uses $200$ randomly generated memory-one strategies. The payoffs are set to $\left(\bar{R},\bar{S},\bar{T},\bar{P}\right)=(3,0,5,1)$ and $\left(\delta_{R},\delta_{S},\delta_{T},\delta_{P}\right)=(-3,1,1,4)$ . Each $\left\langle s_{j}\right\rangle^{*}$ is calculated by time average over $10^{6}$ time steps. The fair ZD strategy indeed enforces a linear relation $\left\langle s_{1}\right\rangle^{*}=\left\langle s_{2}\right\rangle^{*}$ .

Figure 3: A linear relation between $\left\langle s_{1}\right\rangle^{*}$ and $\left\langle s_{2}\right\rangle^{*}$ when player $1$ uses the fair ZD strategy ( 176 ) and ( 177 ) with $c_{1}$ satisfying the equality condition of Eq. ( 175 ) and player $2$ uses $200$ randomly generated memory-one strategies. The payoffs are set to $\left(\bar{R},\bar{S},\bar{T},\bar{P}\right)=(3,0,5,1)$ and $\left(\delta_{R},\delta_{S},\delta_{T},\delta_{P}\right)=(4,1,1,-3)$ . Each $\left\langle s_{j}\right\rangle^{*}$ is calculated by time average over $10^{6}$ time steps. The fair ZD strategy indeed enforces a linear relation $\left\langle s_{1}\right\rangle^{*}=\left\langle s_{2}\right\rangle^{*}$ .

Game Theory

Game theory studies how interacting decision-makers choose strategies, form agreements, and shape outcomes under competition and cooperation.

See all 236 papers

Performance of Zero-determinant Strategies in Repeated Games without Discounting
Oct 2, 2026Masahiko UedaRepeated Games
Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
Feedback autocratic strategies in stochastic games
Oct 27, 2026Masahiko UedaFeedback Control
Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Fairness in two-player zero-sum games with bandit feedback
May 31, 2026S Akash, Pratik GajaneNash Equilibrium Zero-Sum Games
LatentForce.ai Bengaluru, India · Laboratoire d’Informatique Fondamentale d’Orléans University of Orléans, France

Oct 2, 2026J/K move · Enter open · S save

Performance of Zero-determinant Strategies in Repeated Games without Discounting

Masahiko Ueda

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

Repeated Games

Open paper AskPDF arXiv

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.

Abstract

Game Theory

Explore similar work

Performance of Zero-determinant Strategies in Repeated Games without Discounting

Feedback autocratic strategies in stochastic games

Fairness in two-player zero-sum games with bandit feedback