Random evolutionary games and random polynomials

TL;DR

This paper reveals a deep connection between random asymmetric evolutionary games and Kostlan-Shub-Smale random polynomials, providing analytical insights into the equilibria and diversity in such systems.

Contribution

It establishes an exact link between evolutionary game equilibria and a well-known class of random polynomials, enabling detailed statistical analysis.

Findings

The number of internal equilibria follows a universal distribution.

Symmetry increases the expected number of equilibria.

Analytical characterization of equilibria statistics.

Abstract

In this paper, we discover that the class of random polynomials arising from the equilibrium analysis of random asymmetric evolutionary games is \textit{exactly} the Kostlan-Shub-Smale system of random polynomials, revealing an intriguing connection between evolutionary game theory and the theory of random polynomials. Through this connection, we analytically characterize the statistics of the number of internal equilibria of random asymmetric evolutionary games, namely its mean value, probability distribution, central limit theorem and universality phenomena. Biologically, these quantities enable prediction of the levels of social and biological diversity as well as the overall complexity in a dynamical system. By comparing symmetric and asymmetric random games, we establish that symmetry in group interactions increases the expected number of internal equilibria. Our research establishes new theoretical understanding of asymmetric evolutionary games and highlights the significance of symmetry and asymmetry in group interactions.