On the distribution of the number of internal equilibria in random evolutionary games

Abstract

The analysis of equilibrium points is of great importance in evolutionary game theory with numerous practical ramifications in ecology, population genetics, social sciences, economics and computer science. In contrast to previous analytical approaches which primarily focus on computing the expected number of internal equilibria, in this paper we study the distribution of the number of internal equilibria in a multi-player two-strategy random evolutionary game. We derive for the first time a closed formula for the probability that the game has a certain number of internal equilibria, for both normal and uniform distributions of the game payoff entries. In addition, using Descartes' rule of signs and combinatorial methods, we provide several universal upper and lower bound estimates for this probability, which are independent of the underlying payoff distribution. We also compare our analytical results with those obtained from extensive numerical simulations. Many results of this paper are applicable to a wider class of random polynomials that are not necessarily from evolutionary games.

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

“…We will tackle this more intricate question for arbitrary d in a separate paper [17]. We expect that our work in this paper as well as in [17] will open up a new exciting avenue of research in the study of equilibrium properties of random evolutionary games. We discuss below some directions for future research.…”

“…This question has been addressed for games with a small number of players [27,36]. We will tackle this more intricate question for arbitrary d in a separate paper [17]. We expect that our work in this paper as well as in [17] will open up a new exciting avenue of research in the study of equilibrium properties of random evolutionary games.…”

“…where we have used (14) to obtain the last equality. The assertion (15) is then followed from (16) and (17).…”

Mobility restores the mechanism which supports cooperation in the voluntary prisoner’s dilemma game

“…In particular, abstention has attracted attention both for acting as a mechanism to support cooperation and for promoting cyclic behaviour [13][14][15][16][17][18]. The cyclic dominance behaviour is often studied within the bounds of the rock-paper-scissors game, which, different to the VPD game, imposes the cyclic dominance in the payoff matrix [19][20][21][22][23][24].…”

Persistence probability of a random polynomial arising from evolutionary game theory

“…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. More recently, [DTH17a] offers, among other things, an analytical formula for the probability that a multi-player two-strategy game has a certain number of internal equilibria. Although the analytical formula is theoretically interesting, it involves complicated multiple integrals and is computationally intractable when the number of player becomes large.…”