One-third rules with equality: Second-order evolutionary stability conditions in finite populations

Area of Science:

• Evolutionary Dynamics
• Game Theory
• Population Genetics

Background:

• The one-third law of evolutionary dynamics provides a robustness criterion for evolution in finite populations.
• It states that if fitnesses differ at an A-frequency of 1/3, mutant fixation probability deviates from the neutral threshold (1/N).

Purpose of the Study:

• To investigate the fixation probability of a single mutant (A) in a population of B players when their fitnesses are equal at an A-frequency of 1/3.
• To determine the role of cross-payoffs in this specific scenario.

Main Methods:

• Mathematical analysis of evolutionary game dynamics.
• Examination of fixation probabilities under conditions of equal fitness at a critical frequency.
• Application of findings to a language game model.

Main Results:

• When fitnesses are equal at 1/3 A-frequency, cross-payoffs dictate fixation probability.
• A higher payoff for A against B than B against A leads to fixation probability > 1/N.
• In partnership games (balanced cross-payoffs), fixation probability is < 1/N.

Conclusions:

• The one-third law's boundary case (equal fitnesses) is sensitive to the relative magnitudes of cross-payoffs.
• Partnership games represent a specific scenario where mutant fixation is less likely than neutral drift.
• These findings have implications for understanding cooperation and evolution in structured populations, illustrated by a language game example.