for $s(t)$. Thus the approximation of the sample distribution with a stationary distribution (\[tri\]), becomes increasingly accurate as $m$ becomes large and our approximation is self consistent.

To calculate the probabilities that each of the strategies $\omega \in \{\text{Rock, Scissors, Paper}\}$ will appear optimal to a randomly selected agent, given recent behaviour of $r(t)$ and $s(t)$ we define indicator functions for the domains defined in (\[RDom\]), (\[SDom\]) and (\[PDom\]) $$I_\omega (R,S) =
\begin{cases}
1 & \text{ if } (R,S) \in \mathcal{D}_\omega\\
0 & \text{ otherwise. }
\end{cases}$$ In the limit $L \rightarrow \infty$ the interactions of an individual agent have negligible effect on the weights $r(t),s(t)$, which evolve deterministically. If we select an agent at random at time $t$, then conditional on the history of the weights $\{r(u),s(u)|u \leq t\}$, the probability that this agent will perceive $\omega$ as optimal is a functional of this history $$\label{pomega}
\tilde{p}_\omega[r,s](t) := {\mathbb{E}}[I_\omega(R,S)](t).$$ Here the expectation is taken over the set of interaction times and types in the agent’s memory. Letting $\tau_k$ be the time elapsed since the $k$th interaction back from time $t$, we define the following average $${\langle}r {\rangle}_m := \frac{1}{m} \sum_{k=1}^{m} r(t-\tau_k),$$ which is a random variable depending on $\{ 