Extrapolating weak selection in evolutionary games.

Area of Science:

• Evolutionary Biology
• Theoretical Biology
• Mathematical Biology

Background:

• Previous research (Wu et al., 2013) explored evolutionary games with negligible mutation rates, finding strategy rankings were stable with increasing selection strength for two strategies.
• However, for games with three or more strategies, the 2013 study indicated that strategy rankings could indeed change as selection intensity increases.
• The prior work relied on numerical computations for a fixed population size.

Purpose of the Study:

• To derive analytical formulas for invadability probabilities in evolutionary games with multiple strategies.
• To investigate how strategy rankings change with varying selection strength, particularly for small and large values of selection.
• To provide a theoretical framework for understanding strategy dynamics beyond numerical simulations.

Main Methods:

• Analytical derivation of formulas for invadability probabilities by allowing selection strength to vary while keeping population size fixed.
• Formulas were expressed as integrals on [0, 1], which are analytically intractable but numerically evaluable.
• Numerical evaluation of these integral formulas to determine strategy rankings under different selection strengths.

Main Results:

• Derived simple, closed-form formulas for strategy rankings applicable to small and large selection strengths (c).
• Demonstrated that the ordering of strategies is not always constant and can shift as selection intensity changes.
• Provided a theoretical basis for the observed changes in strategy ranking identified in previous numerical studies.

Conclusions:

• The study provides a theoretical framework for understanding how selection strength influences strategy rankings in multi-strategy evolutionary games.
• The derived formulas offer a more general approach than numerical simulations for predicting evolutionary dynamics.
• These findings are crucial for predicting the long-term evolutionary outcomes in systems with competing strategies.