Properties of zero-determinant strategies in multichannel games

TL;DR

This paper investigates the properties and existence conditions of zero-determinant strategies in multichannel repeated games, revealing that such strategies require their existence in individual channels and are structurally constrained.

Contribution

It extends the understanding of zero-determinant strategies to multichannel games and clarifies the conditions under which these strategies can exist.

Findings

Existence of zero-determinant strategies in multichannel games depends on their existence in individual channels.

Zero-determinant strategies in multichannel games require strategies in some channels, not necessarily all.

Existence conditions are tightly linked to the structure of the game in each channel.

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.