Controlling conditional expectations by zero-determinant strategies

Abstract

Zero-determinant strategies are memory-one strategies in repeated games which unilaterally enforce linear relations between expected payoffs of players. Recently, the concept of zero-determinant strategies was extended to the class of memory-$n$ strategies with $n\ge 1$, which enables more complicated control of payoffs by one player. However, what we can do by memory-$n$ zero-determinant strategies is still not clear. Here, we show that memory-$n$ zero-determinant strategies in repeated games can be used to control conditional expectations of payoffs. Equivalently, they can be used to control expected payoffs in biased ensembles, where a history of action profiles with large value of bias function is more weighted. Controlling conditional expectations of payoffs is useful for strengthening zero-determinant strategies, because players can choose conditions in such a way that only unfavorable action profiles to one player are contained in the conditions. We provide several examples of memory-$n$ zero-determinant strategies in the repeated prisoner's dilemma game. We also explain that a deformed version of zero-determinant strategies is easily extended to the memory-$n$ case.

22 pages

References in corpus (10)

• Dynamic first-order phase transition in kinetically constrained models of glasses
• Zero-determinant alliances in multiplayer social dilemmas
• A minimal model of dynamical phase transition
• Extortion under Uncertainty: Zero-Determinant Strategies in Noisy Games
• Zero-determinant strategies in finitely repeated games
• Combination with anti-tit-for-tat remedies problems of tit-for-tat
• Memory-two zero-determinant strategies in repeated games
• Symmetric equilibrium of multi-agent reinforcement learning in repeated prisoner's dilemma
• Tit-for-Tat Strategy as a Deformed Zero-Determinant Strategy in Repeated Games
• Unbeatable Tit-for-Tat as a Zero-Determinant Strategy

Cited by in corpus (4)

• On the implementation of zero-determinant strategies in repeated games
• Unexploitable games and unbeatable strategies
• Properties of zero-determinant strategies in multichannel games
• Zero-determinant strategies in repeated continuously-relaxed games