Characterisation of reactive Nash equilibria in repeated additive games

2026-06-25

AI Summary

This study investigates symmetric Nash equilibria in two-player repeated additive games with finite action sets, focusing on reactive strategies that depend solely on the opponent’s previous move. By reformulating equilibrium conditions as a system of linear equalities and inequalities in strategy parameters, the work provides the first complete characterization of all such symmetric Nash equilibria and establishes a one-to-one correspondence between nonempty subsets of actions and equilibrium classes. Introducing the novel concept of “S-supported equilibria,” it proves that equilibria supported on the full action set coincide with the classical equalizer strategies. Integrating theoretical analysis with evolutionary simulations, the study further demonstrates that the evolutionary success of an equilibrium class is jointly determined by its emergence probability and robustness against invasion.

Abstract

In this paper, we study reactive strategies in repeated additive games between two players with finitely many actions. Reactive strategies condition only on the opponent's previous action, making them one of the simplest ways players can respond to past interactions. Additive games include important models of cooperation, such as the donation game and games with a punishment option. We show that, for this class of games and strategies, the conditions for symmetric Nash equilibria reduce to a system of linear equalities and inequalities in the strategy parameters, allowing us to characterise all such equilibria. We establish a one-to-one correspondence between non-empty subsets S of the action set and equilibrium classes, which we call S-supporting equilibria. These are equilibria that use exactly the actions in S when playing against themselves. As a special case, we recover the well-known equalizer strategies as the equilibria supported on the entire action set. To assess which equilibrium classes are most evolutionarily relevant, we complement our analytical characterisation with simulations of social learning dynamics. We find that their prevalence is determined by two factors: how likely they are to be generated and how robust they are against invasion.

Problem

• reactive strategies
• repeated games
• additive games
• Nash equilibria
• symmetric equilibrium

Innovation

• reactive strategies
• additive games
• Nash equilibria
• S-supporting equilibria
• social learning dynamics