On the distribution of the number of internal equilibria in random evolutionary games

TL;DR

This paper analyzes the probability distribution of internal equilibria in multi-player two-strategy random evolutionary games using random polynomial theory, combinatorics, and sampling comparisons.

Contribution

It provides a closed-form formula for the probability of having a specific number of internal equilibria in such games, advancing understanding of their equilibrium structure.

Findings

Derived a closed-form probability formula for internal equilibria

Provided estimates using Descartes' rule of signs and combinatorics

Validated analytical results with sampling data

Abstract

In this paper, we study the distribution of the number of internal equilibria of a multi-player two-strategy random evolutionary game. Using techniques from the random polynomial theory, we obtain a closed formula for the probability that the game has a certain number of internal equilibria. In addition, by employing Descartes' rule of signs and combinatorial methods, we provide useful estimates for this probability. Finally, we also compare our analytical results with those obtained from samplings.