because it determines how much weight is given to previous information. Using this interpretation, our model may be seen as containing two kinds of memory: an “active” memory - the recent sample - used for decision making, and a “passive” memory which equally weights all previous samples in the limit $\epsilon \rightarrow 0$ and only uses the most recent when $\epsilon = 1$. We note also that our rule, in common with fictitious play [@Bro51], implicitly assumes that agent’s samples are drawn from a stationary distribution.

Simulation Results
==================

Single memory length
--------------------

![ Probability weights of rock (open circles), scissors (dots) and paper (squares) agents in a group of size $L=100$ with $\alpha=2, \beta= \gamma=1$. All agents have memory $m=100$ and update rate $\epsilon = 5 \times 10^{-4}$. Dashed lines show solutions to delay equations (\[rDelay\]) and (\[sDelay\]) using the same parameter values. \[stable\]](HopfRSPm100eps0005L100.eps){width="8cm"}

![Probability weights of rock (open circles), scissors (dots) and paper (squares) agents in a group of size $L=100$ with $\alpha=2, \beta=\gamma=1$. All agents have memory $m=100$ and update rate $\epsilon = 10^{-3}$. Dashed lines show solutions to delay equations (\[rDelay\]) and (\[sDelay\]) using the same parameter values. \[hopf\]](stableRSPm100eps001L100.eps){width="8cm"}

We begin by considering the behaviour of 