Memory-two strategies forming symmetric mutual reinforcement learning equilibrium in repeated prisoners' dilemma game

TL;DR

This paper explores how two players can develop symmetric mutual reinforcement learning strategies with memory-two in the repeated prisoners' dilemma, establishing conditions and examples for such equilibria.

Contribution

It provides a necessary condition for memory-two deterministic strategies to form symmetric equilibria and demonstrates their stability under higher-memory reinforcement learning.

Findings

Identified necessary conditions for symmetric memory-two strategies to form equilibria.

Presented three explicit examples of such strategies.

Proved stability of these equilibria under higher-memory reinforcement learning.

Abstract

We investigate symmetric equilibria of mutual reinforcement learning when both players alternately learn the optimal memory-two strategies against the opponent in the repeated prisoners' dilemma game. We provide a necessary condition for memory-two deterministic strategies to form symmetric equilibria. We then provide three examples of memory-two deterministic strategies which form symmetric mutual reinforcement learning equilibria. We also prove that mutual reinforcement learning equilibria formed by memory-two strategies are also mutual reinforcement learning equilibria when both players use reinforcement learning of memory-$n$ strategies with $n>2$.