On the Expected Number of Internal Equilibria in Random Evolutionary Games with Correlated Payoff Matrix

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Colleges, School and Institutes


The analysis of equilibrium points in random games has been of great interest in evolutionary game theory, with important implications for understanding of complexity in a dynamical system, such as its behavioural, cultural or biological diversity. The analysis so far has focused on random games of independent payoff entries. In this paper, we overcome this restrictive assumption by considering multiplayer two-strategy evolutionary games where the payoff matrix entries are correlated random variables. Using techniques from the random polynomial theory, we establish a closed formula for the mean numbers of internal (stable) equilibria. We then characterise the asymptotic behaviour of this important quantity for large group sizes and study the effect of the correlation. Our results show that decreasing the correlation among payoffs (namely, of a strategist for different group compositions) leads to larger mean numbers of (stable) equilibrium points, suggesting that the system or population behavioural diversity can be promoted by increasing independence of the payoff entries. Numerical results are provided to support the obtained analytical results.


Original languageEnglish
JournalDynamic Games and Applications
Publication statusPublished - 28 Jul 2018


  • Evolutionary game theory, Multiplayer games, Replicator dynamics, Random polynomials, Number of equilibria, Random games