Replicator-mutator dynamics of the rock-paper-scissors game: Learning through mistakes.

Area of Science:

• Evolutionary Game Theory
• Mathematical Biology
• Complex Systems Dynamics

Background:

• Generalizations of Bush-Mosteller, Roth-Erev, and social learning models are essential for understanding strategy evolution.
• Incorporating mistakes (mutation) into learning models is crucial for realistic simulations.
• The rock-paper-scissors game serves as a fundamental model for studying cyclical dynamics and strategy interactions.

Purpose of the Study:

• To generalize existing learning models to incorporate mistakes, leading to the replicator-mutator equation.
• To analyze the dynamics of the rock-paper-scissors game under various mutation patterns.
• To investigate the impact of mistakes on learning, strategy convergence, and chaotic behavior.

Main Methods:

• Generalization of Bush-Mosteller, Roth-Erev, and social learning models to include additive or multiplicative mutation.
• Asymptotic analysis to derive the nonlinear replicator-mutator equation.
• Exhaustive investigation of the rock-paper-scissors game with analytically tractable mutation patterns.
• Analysis of game dynamics, including limit cycles and chaotic orbits.

Main Results:

• The replicator-mutator equation with additive or multiplicative mutation emerges from generalized learning models.
• Rich dynamics, including limit cycles and chaotic orbits, are observed in the rock-paper-scissors game.
• Mistakes can facilitate learning and guide players towards rational Nash-equilibrium outcomes in both symmetric and asymmetric games.
• A novel Hamiltonian structure for the replicator-mutator equation is identified.

Conclusions:

• Mistakes, often viewed as detrimental, can play a constructive role in evolutionary learning and strategy optimization.
• The inclusion of mutation in learning models can stabilize complex systems and lead to predictable, rational outcomes.
• The discovered Hamiltonian structure offers new avenues for theoretical analysis of evolutionary dynamics.