The stabilization of equilibria in evolutionary game dynamics through mutation: mutation limits in evolutionary games

Area of Science:

• Evolutionary Game Theory
• Dynamic Systems Theory
• Computational Game Theory

Background:

• Multi-population replicator dynamics model coevolving populations and multi-player games.
• Standard replicator dynamics often lack guaranteed convergence and may not yield Nash equilibria.
• Interior equilibria in these dynamics can exhibit cyclic behavior, preventing stable solutions.

Purpose of the Study:

• Introduce a novel equilibrium concept for replicator dynamics: mutation limits.
• Investigate the existence and properties of these mutation limits, particularly attracting mutation limits.
• Provide a robust dynamic solution concept for games where standard dynamics fail.

Main Methods:

• Introducing a mutation mechanism invariant to parameter choice.
• Proving the existence of mutation limits for a broad class of games.
• Analyzing attracting mutation limits in perturbed replicator dynamics.

Main Results:

• Mutation limits are proven to exist for a large class of games.
• Attracting mutation limits are shown to be approximated in mutation-perturbed dynamics.
• Mutation can stabilize systems, making attracting mutation limits attainable dynamic solutions.

Conclusions:

• Mutation limits provide a valuable dynamic solution concept for games, especially when standard replicator dynamics are unstable.
• Attracting mutation limits offer approximate dynamic solutions, bridging the gap between theory and practical game dynamics.
• Further research is needed to characterize games where attracting mutation limits exist.