Properties of zero-determinant strategies in multichannel games

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.