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On the implementation of zero-determinant strategies in repeated games

Masahiko Ueda

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 zero-determinant 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 further deepen our understanding on general mathematical properties of zero-determinant strategies. 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 results assist us to implement zero-determinant strategies in broader situations.

On the implementation 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 zero-determinant 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 further deepen our understanding on general mathematical properties of zero-determinant strategies. 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 results assist us to implement zero-determinant strategies in broader situations.
Paper Structure (13 sections, 6 theorems, 60 equations)

This paper contains 13 sections, 6 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Repeated games
  4. Zero-determinant strategies
  5. Potential games
  6. Realization of ZD strategies by one-dimensional transition probability
  7. Main result
  8. Continuum limit
  9. Existence of ZD strategies in two-player potential games
  10. Main result
  11. Asymmetric prisoner's dilemma
  12. Discussion
  13. Concluding remarks

Key Result

Proposition 1

A ZD strategy of player $j$ controlling $B$ exists if and only if there exist two different actions $\overline{a}_j, \underline{a}_j \in A_j$ of player $j$ such that and $B$ is not identically zero.

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1: Ued2022b
  • Definition 2
  • Proposition 2: Ui2000
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3: McAHau2016
  • Theorem 3