Autocratic strategies for iterated games with arbitrary action spaces

Alex McAvoy, Christoph Hauert

The recent discovery of zero-determinant strategies for the repeated Prisoner’s Dilemma sparked a surge of interest in the surprising fact that a player can exert control over iterated interactions regardless of the opponent’s response. These remarkable strategies, however, are known to exist only in games in which players choose between two alternative actions such as “cooperate” and “defect.” Here we introduce a broader class of autocratic strategies by extending zero-determinant strategies to iterated games with more general action spaces. We use the continuous Donation Game as an example, which represents an instance of the Prisoner’s Dilemma that intuitively extends to a continuous range of cooperation levels. Surprisingly, despite the fact that the opponent has infinitely many donation levels from which to choose, a player can devise an autocratic strategy to enforce a linear relationship between his or her payoff and that of the opponent even when restricting his or her actions to merely two discrete levels of cooperation throughout the course of the interaction. In particular, a player can use such a strategy to extort an unfair share of the payoffs from the opponent. Therefore, although the action space for the continuous Donation Game dwarfs that of the classical Prisoner’s Dilemma, players can still devise relatively simple autocratic and, in particular, extortionate strategies.