memory values. In particular we study the symmetric game, allowing us to discover the analytical conditions for stability and the period of oscillation.

Individual estimates of optimal strategy
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In order to construct equations which describe the dynamics of the strategy weights $r(t), s(t), p(t)$, we require expressions for the probabilities that randomly selected agents will perceive each of the three strategies as optimal. If the strategy weights are constant then the numbers $R,S$ and $P=m-R-S$ of strategy types in a sample of size $m$ will be trinomially distributed with probability mass function $$\label{tri}
f(x,y;r,s)= \frac{m! r^x s^y (1-r-s)^{m-x-y}}{x! y! (m-x-y)!}.$$ In reality, the weights will change while the sample is being taken, producing a sequence of non-identical trivariate Bernoulli trials. Our aim is to approximate this non-stationary distribution with the stationary version (\[tri\]) having appropriately chosen values of $r$ and $s$. We will discover, once our analysis is complete, that the period, $T$, of oscillations at the transition to instability satisfies $T \propto m^{3/2}$. Letting the magnitude of the change in $r(t)$ over a typical agent’s sample be $\delta r$ then we see that $ \delta r = \mathcal{O} (m^{-1/2})$ as $m \rightarrow \infty$, with the same behaviour holding 