Zero-determinant strategies in repeated continuously-relaxed games

TL;DR

This paper extends zero-determinant strategies to repeated games with continuously relaxed action sets, broadening their applicability and linking certain strategies to Nash equilibrium properties.

Contribution

It introduces zero-determinant strategies in continuous action spaces and defines one-point strategies, connecting them to Nash equilibrium characteristics.

Findings

Continuous relaxation expands zero-determinant strategies' scope.

One-point strategies can control payoffs with single actions.

Some Nash equilibrium properties are reinterpreted as payoff control.

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 control payoffs of players by unilaterally enforcing linear relations between payoffs. 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.