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Zero-determinant strategies in finitely repeated games
Genki Ichinose1, Naoki Masuda2
1Department of Mathematical and Systems Engineering, Shizuoka University, 3-5- 1 Johoku, Naka-ku, Hamamatsu, 432-8561, Japan.
Journal of Theoretical Biology
|November 21, 2017
Summary
Zero-determinant (ZD) strategies in repeated prisoner's dilemma games allow players to control payoff relationships. This study extends ZD strategy analysis to finite games, defining conditions for their existence and identifying strategies that enforce linear payoff relationships.
Area of Science:
- Game Theory
- Evolutionary Game Theory
- Computational Game Theory
Background:
- Direct reciprocity sustains cooperation in repeated social dilemmas by deterring defection.
- Zero-determinant (ZD) strategies offer players unilateral control over payoff relationships in games.
- Previous research primarily focused on ZD strategies in infinitely repeated games.
Purpose of the Study:
- To analytically investigate Zero-Determinant (ZD) strategies in finitely repeated prisoner's dilemma games.
- To extend existing knowledge of ZD strategies from infinite to finite game scenarios.
- To identify conditions for the existence of prominent ZD strategies and characterize payoff-enforcing strategies.
Main Methods:
- Analytical study of Zero-Determinant (ZD) strategies within a general payoff matrix for finitely repeated prisoner's dilemma games.
- Derivation of solutions applicable to finite repetitions, extending results from infinite games.
- Calculation of threshold discount factors (w) for the existence of specific ZD strategies (equalizers, extortioners, generous).
Main Results:
- Solutions for ZD strategies were successfully extended from infinitely repeated games to finitely repeated games (0 < w < 1).
- Threshold values of the discount factor (w) were derived, determining the existence of prominent ZD strategies.
- It was proven that only ZD strategies or unconditional strategies enforce linear payoff relationships between players, confirming a prior conjecture.
Conclusions:
- ZD strategies can be effectively analyzed and applied within the framework of finitely repeated prisoner's dilemma games.
- The study establishes clear conditions for the viability of various ZD strategies based on the discount factor.
- The findings confirm that ZD strategies are key mechanisms for enforcing linear payoff relationships, alongside unconditional strategies.
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