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

Abstract

Zero-determinant strategies are a class of strategies in repeated games which unilaterally control payoffs. Zero-determinant strategies have attracted much attention in studies of social dilemma, particularly in the context of evolution of cooperation. So far, not only general properties of zerodeterminant strategies have been investigated, but zero-determinant strategies have been applied to control in the fields of information and communications technology and analysis of imitation. Here, we provide another example of application of zero-determinant strategies: control of a particle on a lattice. We first prove that zero-determinant strategies, if exist, can be implemented by some one-dimensional transition probability. Next, we prove that, if a two-player game has a non-trivial potential function, a zero-determinant strategy exists in its repeated version. These two results enable us to apply the concept of zero-determinant strategies to control the expected potential energies of two coordinates of a particle on a two-dimensional lattice.

Unexploitable Games and Unbeatable Strategies

“…we conclude that this ZD strategy unilaterally enforces S j ≥ max k̸ =j S k , that is, it is unbeatable. □ We remark that the converse of Theorem 1 does not hold, even in the case of N = 2 [12].…”

“…In terms of unbeatable strategies, the author proved that the unbeatable TFT is a ZD strategy in two-player symmetric games [29]. In addition, for two-player symmetric games where the unbeatable imitation strategy exists, unbeatable ZD strategies also exist [12]. These results suggest that there may be some relation between the existence of unbeatable imitation strategies and the existence of unbeatable ZD strategies, even in multi-player symmetric games.…”

“…for all strategies T k (∀k ̸ = j). Several examples of unbeatable strategies have been found in the prisoner's dilemma game [13], [38], [39], in two-player symmetric potential games [9], [29], in two-player symmetric games with no generalized rock-paper-scissors cycles [8], [12], and in the public goods game [36], [37]. In addition, a uniform strategy in the repeated rock-paper-scissors game is also an unbeatable strategy.…”

“…Recently, it has been pointed out that the existence of unbeatable imitation strategies may be related to the existence of unbeatable zero-determinant (ZD) strategies [12]. ZD strategies are a class of memory-one strategies in infinitely repeated games, which unilaterally enforce linear relationships between payoffs of players [13].…”