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

Abstract

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.

On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games

“…We recall that finding an equilibrium point of the replicator-mutator dynamics for d-player two-strategy games is equivalent to finding a positive root of the polynomial (17) with coef-ficients given in (18). In this section, by employing techniques from random polynomial theory, we provide explicit formulas for the computation of the expected number of internal equilibrium points of the replicator-mutator dynamics where the entries of the payoff matrix are random variables, thus extending our previous results for the replicator dynamics [4][5][6][7]. We will apply the following result on the expected number of positive roots of a general random polynomial.…”

“…Equilibrium properties of the replicator dynamics, particularly the probability of observing the maximal number of equilibrium points, the attainability and stability of the patterns of evolutionarily stable strategies have been studied intensively in the literature [2,3,12,13,17]. More recently, we have provided explicit formulas for the computation of the expected number and the distribution of internal equilibria for the replicator dynamics with multi-player games by employing techniques from both classical and random polynomial theory [4][5][6][7]. For the replicator dynamics, that is when there is no mutation, the first condition in (2) means that all the strategies have the same fitness which is also the average fitness of the whole population.…”

“…In this paper, we explore further connections between classical/random polynomial theory and evolutionary game theory developed in [4][5][6][7] to study equilibrium properties of the replicator-mutator dynamics. For deterministic games, by using Descartes' rule of signs and its recent developments, we are able to fully characterize the equilibrium properties for social dilemmas.…”

Persistence probability of a random polynomial arising from evolutionary game theory

“…In [GT10, HTG12, GT14], the authors provide analytical and simulation results for random games with a small number of players (n ≤ 4) focusing on the probability of attaining the maximal number of equilibrium points. In [DH15,DH16,DTH17b], the authors derive a closed formula for the expected number of internal equilibria, characterize its asymptotic behaviour and study the effect of correlations. Related work on the expected number of equilibrium points of random large complex systems arising from physics and ecology are presented in [Fyo04,FN12,FK16], see also references therein.…”

On the Expected Number and Distribution of Equilibria in Multi-player Evolutionary Games

“…In (Duong et al, 2018b) we generalize our analysis for random evolutionary games where the payoff matrix entries are correlated random variables. In social and biological contexts, correlations may arise in various scenarios particularly when there are environmental randomness and interaction uncertainty such as in games of cyclic dominance, coevolutionary multi-games or when individual contributions are correlated to the surrounding contexts (e.g.…”