Necessary and Sufficient Condition for the Existence of Zero-Determinant Strategies in Repeated Games

TL;DR

This paper establishes a precise criterion for when zero-determinant strategies can exist in repeated games, clarifying the conditions under which they can unilaterally control payoff relationships.

Contribution

It provides the first necessary and sufficient condition for the existence of zero-determinant strategies in stage games, linking game structure to strategic enforceability.

Findings

Derived a condition based on actions adjusting linear payoff combinations

Connected the existence of strategies to specific game action properties

Clarified the relationship between game classes and strategy existence

Abstract

Zero-determinant strategies are a class of memory-one strategies in repeated games which unilaterally enforce linear relationships between payoffs. It has long been unclear for what stage games zero-determinant strategies exist. We provide a necessary and sufficient condition for the existence of zero-determinant strategies. This condition can be interpreted as the existence of two different actions which unilaterally adjust the total value of a linear combination of payoffs. A relation between the class of stage games where zero-determinant strategies exist and other class of stage games is also provided.