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

TL;DR

This paper demonstrates the existence of Nash equilibria outside the scope of the folk theorem in repeated games, revealing a larger set of stable states through coordinated strategies among players.

Contribution

It proves that Nash equilibria not characterized by the folk theorem exist and introduces the concept of type-k equilibria involving coordinated strategies.

Findings

Nash equilibria outside the folk theorem exist.

Type-k equilibria involve coordinated strategies among groups.

The set of stable states is larger than previously predicted.

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.