the payoffs of the three pure strategies are not equal is essential in order for long memory players to have an advantage. Simulations similar to that illustrated in Figure \[popDyn\] show that as the payoffs $\alpha, \beta, \gamma$ approach equality, the long memory players do not prosper and the self organizing Hopf bifurcation is no longer present. An intuitive understanding of this may be obtained by considering the effect of shortening the memory length, which causes individual agents’ samples to perform higher variance random walks around the simplex of Figure \[rspDoms\]. As the variance of the walk increases, agents’ predictions of the best strategy are subject to greater variability, driving their strategies toward the symmetric case $[\tfrac{1}{3},\tfrac{1}{3}]$. If the game is not symmetric then a small shift towards the symmetric strategy for short memory players gives an advantage for longer memory players. In the symmetric case, this advantage disappears because the symmetric strategy is optimal.

Theory
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Here we investigate the dynamics of a system of identical agents as $m \rightarrow \infty$ and $L \rightarrow \infty$. The results we obtain by considering these limits provide excellent approximations to the behaviour of smaller groups of agents for a wide range of 