Autocratic strategies for alternating games

TL;DR

This paper extends the concept of autocratic strategies, including zero-determinant strategies, to alternating games, revealing how asymmetries and move order influence strategic control in biological and behavioral contexts.

Contribution

It generalizes autocratic strategies to alternating and randomly-alternating games, providing conditions for their existence and illustrating their roles in asymmetric interactions.

Findings

Autocratic strategies exist in alternating move games with different move orders.

Asymmetries can empower subordinate players with more control.

Autocratic strategies can be applied to continuous Donation Games.

Abstract

Repeated games have a long tradition in the behavioral sciences and evolutionary biology. Recently, strategies were discovered that permit an unprecedented level of control over repeated interactions by enabling a player to unilaterally enforce linear constraints on payoffs. Here, we extend this theory of "zero-determinant" (or, more generally, "autocratic") strategies to alternating games, which are often biologically more relevant than traditional synchronous games. Alternating games naturally result in asymmetries between players because the first move matters or because players might not move with equal probabilities. In a strictly-alternating game with two players, $X$ and $Y$, we give conditions for the existence of autocratic strategies for player $X$ when (i) $X$ moves first and (ii) $Y$ moves first. Furthermore, we show that autocratic strategies exist even for (iii) games with randomly-alternating moves. Particularly important categories of autocratic strategies are extortionate and generous strategies, which enforce unfavorable and favorable outcomes for the opponent, respectively. We illustrate these strategies using the continuous Donation Game, in which a player pays a cost to provide a benefit to the opponent according to a continuous cooperative investment level. Asymmetries due to alternating moves could easily arise from dominance hierarchies, and we show that they can endow subordinate players with more autocratic strategies than dominant players.