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

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.