a population of agents, all of whom have the same memory length and update rate. In order to explore the dynamics of the population, we observe the average probability weights associated with each strategy. For example, the weight associated with rock is $$r(t) := \frac{1}{L} \sum_{i=1}^L r_i(t)$$ with $s(t)$ and $p(t)$ defined similarly. The evolution of the total probability weights in a population with update rate $\epsilon = 5 \times 10^{-4}$ and memory $m=100$ is shown in Figure \[stable\]. Initial oscillations decay, eventually leaving the system in a stable state. The strategies of individuals in the population track the curves in Figure \[stable\], but with a greater stochastic component. In Figure \[hopf\] we have increased the rate to $\epsilon=10^{-3}$, and we see that stable oscillations have formed. For given memory length, these appear when the update rate is sufficiently large. Conversely, for given update rate, stable oscillations emerge when the memory length is sufficiently large. We will show later that it is the product $\epsilon m^2$ which must exceed a critical value for stable oscillations to appear. This transition from stable equilibrium to stable oscillations at critical parameter values is known as a “Hopf Bifurcation” [@Ern09].

Agents as statisticians
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