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Properties of zero-determinant strategies in multichannel games

Masahiko Ueda

TL;DR

This work extends zero-determinant (ZD) payoff-control theory to multichannel repeated games, showing that the existence of ZD strategies in a multichannel game is tightly constrained by the structure of each channel. The authors derive an autocratic condition that decomposes additively across channels, revealing that MC ZD strategies exist only when channel-wise conditions can be satisfied (and may fail even if some channels do not admit ZD. They prove that nontrivial equalizer strategies in MC require their existence in at least one channel, and that fair ZD strategies in MC necessitate fairness in every channel. The paper provides concrete examples, including multichannel prisoner’s dilemma and combined games, to illustrate how equalizers can emerge through channel interactions and to establish limits on the prevalence of MC ZD strategies. These results have implications for understanding and engineering payoff control and cooperation in systems where agents participate in multiple parallel interactions.

Abstract

Controlling payoffs in repeated games is one of the important topics in control theory of multi-agent systems. Recently proposed zero-determinant strategies enable players to unilaterally enforce linear relations between payoffs. Furthermore, based on the mathematics of zero-determinant strategies, regional payoff control, in which payoffs are enforced into some feasible regions, has been discovered in social dilemma situations. More recently, theory of payoff control was extended to multichannel games, where players parallelly interact with each other in multiple channels. However, the existence of payoff-controlling strategies in multichannel games seems to require the existence of payoff-controlling strategies in some channels, and properties of zero-determinant strategies specific to multichannel games are still not clear. In this paper, we elucidate properties of zero-determinant strategies in multichannel games. First, we relate the existence condition of zero-determinant strategies in multichannel games to that of zero-determinant strategies in each channel. We then show that the existence of zero-determinant strategies in multichannel games requires the existence of zero-determinant strategies in some channels. This result implies that the existence of zero-determinant strategies in multichannel games is tightly restricted by structure of games played in each channel.

Properties of zero-determinant strategies in multichannel games

TL;DR

This work extends zero-determinant (ZD) payoff-control theory to multichannel repeated games, showing that the existence of ZD strategies in a multichannel game is tightly constrained by the structure of each channel. The authors derive an autocratic condition that decomposes additively across channels, revealing that MC ZD strategies exist only when channel-wise conditions can be satisfied (and may fail even if some channels do not admit ZD. They prove that nontrivial equalizer strategies in MC require their existence in at least one channel, and that fair ZD strategies in MC necessitate fairness in every channel. The paper provides concrete examples, including multichannel prisoner’s dilemma and combined games, to illustrate how equalizers can emerge through channel interactions and to establish limits on the prevalence of MC ZD strategies. These results have implications for understanding and engineering payoff control and cooperation in systems where agents participate in multiple parallel interactions.

Abstract

Controlling payoffs in repeated games is one of the important topics in control theory of multi-agent systems. Recently proposed zero-determinant strategies enable players to unilaterally enforce linear relations between payoffs. Furthermore, based on the mathematics of zero-determinant strategies, regional payoff control, in which payoffs are enforced into some feasible regions, has been discovered in social dilemma situations. More recently, theory of payoff control was extended to multichannel games, where players parallelly interact with each other in multiple channels. However, the existence of payoff-controlling strategies in multichannel games seems to require the existence of payoff-controlling strategies in some channels, and properties of zero-determinant strategies specific to multichannel games are still not clear. In this paper, we elucidate properties of zero-determinant strategies in multichannel games. First, we relate the existence condition of zero-determinant strategies in multichannel games to that of zero-determinant strategies in each channel. We then show that the existence of zero-determinant strategies in multichannel games requires the existence of zero-determinant strategies in some channels. This result implies that the existence of zero-determinant strategies in multichannel games is tightly restricted by structure of games played in each channel.

Paper Structure

This paper contains 14 sections, 7 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multichannel game
  4. Zero-determinant strategies
  5. Results
  6. Existence condition
  7. Equalizer strategies in multichannel prisoner's dilemma game
  8. Combining two different games
  9. Absence of nontrivial equalizer strategies in multichannel games
  10. Absence of nontrivial fair ZD strategies in multichannel two-player symmetric games
  11. More complicated payoff control
  12. Discussion and Conclusion
  13. Derivation of Eq. (\ref{['eq:condition_exsitence_mod']})
  14. Construction of ZD strategies in numerical simulation

Key Result

Lemma 1

For memory-one strategies of player $j$, when we define then the equality holds for all $\vec{a}_j \in \mathcal{A}_j$.

Figures (1)

  • Figure 1: A linear relation between $\left\langle \tilde{s}_1 \right\rangle^{*}$ and $\left\langle \tilde{s}_2 \right\rangle^{*}$ when player $1$ uses the equalizer strategy with $r=2$ and player $2$ uses $1000$ randomly generated memory-one strategies. The payoffs in channel $1$ are set to $\left( R^{(1)}, S^{(1)}, T^{(1)}, P^{(1)} \right)=(3, 0, 5, 1)$, and the payoffs in channel $2$ are set as in Eq. (\ref{['eq:MP_2']}). Each $\left\langle \tilde{s}_j \right\rangle^{*}$ is calculated by time average over $10^6$ time steps. The equalizer strategy indeed enforces a linear relation $\left\langle \tilde{s}_2 \right\rangle^{*}=2$.

Theorems & Definitions (12)

  • Lemma 1: SCFet2025
  • Definition 1
  • Proposition 1: Ued2022b
  • Proposition 2
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 2 more