On Nash Equilibrium and Evolutionarily Stable States That Are Not Characterised by the Folk Theorem.

Jiawei Li,Graham Kendall

doi:10.1371/journal.pone.0136032

Jiawei Li, Graham Kendall

Open Access

https://doi.org/10.1371/journal.pone.0136032

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Abstract

In evolutionary game theory, evolutionarily stable states are characterised by the folk theorem because exact solutions to the replicator equation are difficult to obtain. It is generally assumed that the folk theorem, which is the fundamental theory for non-cooperative games, defines all Nash equilibria in infinitely repeated games. Here, we prove that Nash equilibria that are not characterised by the folk theorem do exist. By adopting specific reactive strategies, a group of players can be better off by coordinating their actions in repeated games. We call it a type-k equilibrium when a group of k players coordinate their actions and they have no incentive to deviate from their strategies simultaneously. The existence and stability of the type-k equilibrium in general games is discussed. This study shows that the sets of Nash equilibria and evolutionarily stable states have greater cardinality than classic game theory has predicted in many repeated games.

Highlights

A population is considered to be in an evolutionarily stable state if its genetic composition is restored by selection after a disturbance [1]
An evolutionarily stable state has a close relationship with the concept of Nash equilibrium (NE) and the folk theorem for infinitely repeated games [2, 3]
The folk theorem proves the existence of the type-n equilibrium in repeated n-player games

Summary

Introduction

A population is considered to be in an evolutionarily stable state if its genetic composition is restored by selection after a disturbance [1]. Reactive strategies are the reason why a payoff profile that is not NE in the stage game can be NE in an infinitely repeated game. Coordination among a group of players can be formed and maintained when specific reactive strategies are adopted by those players, which leads to equilibrium that does not exist in one-shot games. Reactive strategies provide a way of coordination among a group of players in repeated games.

Results

Conclusion