Persistence probability of a random polynomial arising from evolutionary game theory

Abstract

In this paper, we obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.

On the expected number of real roots of random polynomials arising from evolutionary game theory

“…It remains elusive to us whether the techniques in [33,24,6] can be applied to the class of random polynomials in this paper. Furthermore, studying other statistical properties such as central limit theorem and the distribution of the number of equilibria also demands future investigations, see for instance [5] for a characterization of the probability that a multiplayer random evolutionary random game has no internal equilibria.…”

Government regulation strategy, leading firms’ innovation strategy, and following firms imitation strategy: An analysis based on evolutionary game theory

“…The basic principles in evolutionary game theory have been widely applied to different fields such as species diversity [ 38 ], climate negotiations [ 39 , 40 ], public health [ 41 , 42 ], and traffic flow [ 43 ]. Duong and Pham [ 44 ] used evolutionary game theory to obtain asymptotic formulas for determining the continuous probability of a positive holomorphic line with a random polynomial, whose key ingredient was a degree index order bounded close to the random polynomial by an appropriate central smooth Gaussian process. Sekiguchi and Ohtsuki [ 45 ] described the stochastic evolution of an infinite population of 2 × 2 bimatrix game kinetics, obtaining a fixed probability of convergence of the evolutionary dynamics from a given primitive state to a specific absorbing state; they proved that evolutionary dynamics is biased toward fairness.…”

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

“…We also provide several universal upper and lower bound estimates, which are independent of the underlying payoff distribution, for the probability of obtaining a certain number of internal equilibria. In addition, the asymptotic behaviour of the probability of having no internal equilibria is then obtained (Can et al, 2018). The distribution of equilibria provides more elaborate information about the level of complexity or the number of different states of biodiversity that will occur in a dynamical system, compared to what obtained with the expected number of internal equilibria.…”