Analysis and Simulations of the Rock-Paper-Scissors Game with Environmental Feedback

The classic Rock-Paper-Scissors (RPS) game has the only internal Nash Equilibrium (NE), which is not an Evolutionary Stable Strategy (ESS). Based on the replicator dynamics of evolutionary game theory, this paper considering the new RPS game that players have mutual feedback with environment, establishes a three-dimensional ordinary differential equation system. Through analysis and numerical simulations of this system, we prove the new RPS game has an ESS when the initial environment is not depleted and find three evolutionary internals including Evolutionary Equilibrium (EE) and periodic solutions.