Replicator-mutator dynamics of Rock-Paper-Scissors game: Learning through mistakes

Abstract

We generalize the Bush--Mosteller learning, the Roth--Erev learning, and the social learning to include mistakes such that the nonlinear replicator-mutator equation with either additive or multiplicative mutation is generated in an asymptotic limit. Subsequently, we exhaustively investigate the ubiquitous Rock-Paper-Scissors game for some analytically tractable motifs of mutation pattern. We consider both symmetric and asymmetric game interactions, and reveal that mistakes can some-times help the players learn. While the replicator-mutator flow exhibits rich dynamics that include limit cycles and chaotic orbits, it can control chaos as well to lead to rational Nash equilibrium outcome. Moreover, we also report an instance of hitherto unknown Hamiltonian structure of the replicator-mutator equation.