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

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 multi-player 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.Comment: Revision from the previous version; 27 page

Covariance of the number of real zeros of a random trigonometric polynomial

For random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ)=∑j=0n{ajcos⁡jθ+bjsin⁡jθ}. The expected number of real zeros of Tn(θ) in the interval (0,2π) can be easily obtained. In this note we show that this number is in fact n/3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals (0,π) and (π,2π). It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined as Pn(θ)=∑j=0naj(ω)cos⁡jθ, is also discussed. Tn(θ) has the advantage on Pn(θ) of being stationary, with respect to θ, for which, therefore, a more advanced method developed could be used to yield the results