The playground Rock–Paper–Scissors (RPS) game provides a surprisingly rich source of nonlinear dynamical phenomena. This talk will focus on one aspect of the dynamics: the derivation of the usual mean-field replicator equations for the dynamics from a large and well-mixed, but finite, population of RPS strategists. Including random adoptions of different strategies (i.e. ‘mutations’, occurring with rate \mu) leads to an attracting limit cycle for the mean field dynamics.

 

When the population size N is large but finite, the analysis of the limit cycle divides into three regimes determined by different asymptotic balances between \mu and N. Each regime exhibits a different scaling law for the period T of the limit cycle, and can be analysed through different techniques from dynamical systems and stochastic processes. Randomness introduced by the finite population size (‘demographic noise’) has been well-studied near equilibrium points; here we consider its intriguing effects on a limit cycle. Happily, the analyses in the different regimes match up smoothly