\tau_k\}_{k=1}^m$. Conditional on the interaction times, we approximate the non stationary distribution of types with the stationary distribution $f(x,y;{\langle}r {\rangle}_m, {\langle}s {\rangle}_m )$. We note that differences between second moments of this distribution and the true distribution will be of order $\delta r$ and $\delta s$, and the means will be identical. In order to average over the interaction times, we note that for a particular agent, the time intervals between interactions are exponentially distributed. The distribution function for $\tau_k$ is therefore the Gamma density $${\mathbb{P}}(\tau_k \in [t,t+\delta t]) = \frac{t^{k-1} e^{-t}}{(k-1)!} \delta t: = \gamma_k(t) \delta t$$ so that $$\begin{aligned}
{\mathbb{E}}[{\langle}r {\rangle}_m] &= \frac{1}{m} \sum_{k=1}^m \int_0^\infty \gamma_k(u) r(t-u) du \\
&= \int_0^\infty \left(\sum_{k=1}^m \frac{\gamma_k(u)}{m} \right) r(t-u) du \\
&= \int_0^\infty \frac{\Gamma(m,u)}{m!} r(t-u) du \\
\label{gdef}
&:= \int_0^\infty g_m(u) r(t-u) ds\end{aligned}$$ where $\Gamma$ is the incomplete gamma function, defined as $$\Gamma(m,u) = \int_u^\infty x^{m-1} e^{-x} dx.$$ Equation (\[gdef\]) defines a probability density $g_m(s)$ having the shape of a smoothed top hat (uniform) distribution on $[0,m]$, and may be thought of as representing the strength of the collective memory of all agents $u$ time units before the present. As $m \rightarrow \infty$, the distribution of ${\langle}r {\rangle}_m$ becomes increasingly sharply peaked 