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The analysis of equilibrium points in random games has been of great interestin evolutionary game theory, with important implications for understanding ofcomplexity in a dynamical system, such as its behavioural, cultural orbiological diversity. The analysis so far has focused on random games ofindependent payoff entries. In this paper, we overcome this restrictiveassumption by considering multi-player two-strategy evolutionary games wherethe payoff matrix entries are correlated random variables. Using techniquesfrom the random polynomial theory we establish a closed formula for the meannumbers of internal (stable) equilibria. We then characterise the asymptoticbehaviour of this important quantity for large group sizes and study the effectof the correlation. Our results show that decreasing the correlation amongpayoffs (namely, of a strategist for different group compositions) leads tolarger mean numbers of (stable) equilibrium points, suggesting that the systemor population behavioural diversity can be promoted by increasing independenceof the payoff entries. Numerical results are provided to support the obtainedanalytical results.

On the expected number of internal equilibria in random evolutionary games with correlated payoff matrix