(dots) agents in system of size $L=200$ with update rate $\epsilon = 0.01$ initially composed of 199 short memory agents and a single long memory agent. Game parameters are $\alpha = 2, \beta = \gamma =  1$ and population dynamics parameters are $\rho = \kappa = 10^{-3}$. Also shown in fraction of paper agents (black line), illustrating how the onset of oscillations coincides with stable equilibrium between memory lengths. \[popDyn\]](popDyn_L200_m10_m100_eps01.eps){width="8cm"}

We now consider the case of a mixed population containing two memory lengths, $m \in \{10,100\}$. Intuitively we expect that agents with longer memories will be at an advantage because they are able to more accurately predict the optimal strategy; their estimates of opponents’ strategies are subject to smaller errors. To investigate this we introduce some simple population dynamics, based upon an exponential moving average of each agent’s payoffs. We let $v_{i k}$ be the payoff to agent $i$ at his $k$th interaction since the start of the game, and define his moving average, $\bar{v}_{ik}$, as follows $$\bar{v}_{i,k+1} := (1-\rho) \bar{v}_{ik} + \rho v_{i,k+1}.$$ At each pairwise interaction in the game, with probability $\kappa \ll 1$, the agent with the lower average payoff is replaced with a copy of the 