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 overcoming this restrictive assumption, here we study equilibrium points in random games with correlated entries. Namely, we analyse the mean value and the distribution of the number of (stable) internal equilibria in multi-player two-strategy evolutionary games where the payoff matrix entries are correlated random variables. Our contributions are as follows. We first obtain a closed formula for the mean number of internal equilibria, characterise its asymptotic behaviour and study the effect of the correlation. We then provide analytical formulas to compute the probability of attaining a certain number of internal equilibria, and derive an approximate formula for the computation of this probability. Last but not least, we reveal some universal estimates that are independent of the distribution of the payoff matrix entries, and provide numerical simulations to support the obtained analytical results.
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 number of internal (stable) equilibria. We then characterise the asymptotic behaviour of this important quantity and study the effect of the correlation. Numerical results are provided to support the obtained analytical results.
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overcoming this restrictive assumption, here we study equilibrium points in random games with correlated entries. Namely, we analyse the mean value and the distribution of the number of (stable) internal equilibria in
this paper, we overcome this restrictive assumption by considering
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Our contributions are as follows. We first obtain
Using techniques from the random polynomial theory we establish
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equilibria, characterise its asymptotic behaviour
(stable) equilibria. We then characterise the asymptotic behaviour of this important quantity
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We then provide analytical formulas to compute the probability of attaining a certain number of internal equilibria, and derive an approximate formula for the computation of this probability. Last but not least, we reveal some universal estimates that are independent of the distribution of the payoff matrix entries, and provide numerical simulations
Numerical results are provided