Memory and limit cycles in rock-scissors-paper

TL;DR

This paper investigates how memory length affects strategy dynamics in a rock-scissors-paper game, revealing bifurcations and the impact of memory on agent advantage and population evolution.

Contribution

It introduces a model where agents recall past interactions, analyzes bifurcations at critical memory lengths, and explores how population dynamics influence strategy success.

Findings

Longer memory agents have an advantage under certain conditions.

Limit cycles emerge at critical memory lengths due to Hopf bifurcation.

Population dynamics tend to evolve toward the bifurcation point.

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.