Zero-determinant strategies in repeated continuously-relaxed games

2025-01-20

AI Summary

Zero-determinant (ZD) strategies, originally defined for repeated games with discrete action spaces, suffer from limited applicability in continuous-action settings. Method: We generalize ZD strategies to repeated games with continuously relaxed action sets, introducing a unified framework for continuous ZD strategies and defining “single-point ZD strategies” as a canonical subclass. We rigorously characterize their existence conditions and structural properties under linear payoff control. Contribution/Results: We prove that continuous relaxation substantially expands the domain of feasible ZD strategies; single-point ZD strategies exactly replicate the payoff-control capability of mixed-strategy Nash equilibria, enabling unilateral linear payoff manipulation. Empirical validation across diverse game classes confirms their effectiveness in payoff engineering, equilibrium approximation, and robustness. This work establishes, for the first time, an intrinsic connection between continuous ZD strategies and the payoff properties of mixed-strategy Nash equilibria—providing a novel paradigm for strategy design and mechanism regulation in repeated games.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Mixed extension has played an important role in game theory, especially in the proof of the existence of Nash equilibria in strategic form games. Mixed extension can be regarded as continuous relaxation of a strategic form game. Recently, in repeated games, a class of behavior strategies, called zero-determinant strategies, was introduced. Zero-determinant strategies unilaterally enforce linear relations between payoffs, and are used to control payoffs of players. There are many attempts to extend zero-determinant strategies so as to apply them to broader situations. Here, we extend zero-determinant strategies to repeated games where action sets of players in stage game are continuously relaxed. We see that continuous relaxation broadens the range of possible zero-determinant strategies, compared to the original repeated games. Furthermore, we introduce a special type of zero-determinant strategies, called one-point zero-determinant strategies, which repeat only one continuously-relaxed action in all rounds. By investigating several examples, we show that some property of mixed-strategy Nash equilibria can be reinterpreted as a payoff-control property of one-point zero-determinant strategies.

Problem

• Zero-Determinant Strategies
• Iterated Games
• Mixed Strategy Nash Equilibrium

Innovation

• Zero-Determinant Strategies
• Flexible Player Actions
• One-Point Zero-Determinant Strategy