---
abstract: 'When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-scissors-paper game in which agents are able to recall their last $m$ interactions, used to estimate the behaviour of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favours successful agents, we show that the system evolves toward the bifurcation point.'
author:
- James Burridge
bibliography:
- 'JBRefs.bib'
title: 'Memory and limit cycles in rock-scissors-paper'
---

Introduction
============

“The wise can learn from their enemies” [@Aris], and how they do this is an interesting question. The mathematical analysis of competitions between opponents, called “the theory of games”, began with von Neumann [@Neu53] and Nash [@Nash51]. This early work concerned the equilibria of games, 