Statistics of the number of internal equilibria: Evolutionary Game Theory meets Random Polynomial Theory

Evolutionary Game Theory has become one of the most diverse and far reaching theories in biology finding applications in a plethora of disciplines such as ecology, population genetics, social sciences, economics and computer sciences.  Similar to the foundational concept of Nash equilibrium in classical game theory, the analysis of equilibrium points is of great importance in evolutionary game theory providing essential implications for understanding of complexity in a dynamical system, such as its behavioural, cultural or biological diversity.  In this talk, we study the statistics of the number of internal equilibria of a multiplayer two-strategy random evolutionary game. We show intriguing connections to random polynomial theory and using these connections we compute the mean value and the distribution of the number of internal equilibira and study their asymptotic behavior as the number of players goes to infinity.