memory is not too long, then the game has a stable fixed point, and agents with long memories fare better than their short memory counterparts; their prediction of the best strategy is subject to smaller random errors. However, the limit cycles which appear when long memory agents are in high enough concentration destroy their competitive advantage.

Model Definition
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We study the zero sum rock-scissors-paper game, with payoff matrix $$\label{POM}
\bordermatrix{      &_R& _S& _P  \cr
                _R &0 & \alpha & -\gamma  \cr
                _S & -\alpha & 0 & \beta \cr
                _P &\gamma & -\beta & 0  \cr}.$$ The game is played in continuous time by $L \ge 2$ agents who interact using random pairings which occur at rate $L/2$ per unit time so that each agent experiences, on average, one interaction per unit time. Formally, the probability that a single pairing takes place in time $\delta t$ is $L \delta t/2 + o(\delta t)$. Agents each adopt a probabilistic strategy which, for agent $i \in \{1,\ldots\L\}$, after the $n$th interaction is written $[r_i,s_i]_n$, where $r_i,s_i$ and $1-r_i-s_i$ are the probabilities of playing, respectively, rock, scissors or paper at the next interaction. Each agent is able to recall his last $m$ 