Analysis of the expected density of internal equilibria in random evolutionary multi-player multi-strategy games

Abstract

In this paper, we study the distribution and behaviour of internal equilibria in a d-player n-strategy random evolutionary game where the game payoff matrix is generated from normal distributions. The study of this paper reveals and exploits interesting connections between evolutionary game theory and random polynomial theory. The main contributions of the paper are some qualitative and quantitative results on the expected density, f n,d , and the expected number, E(n, d), of (stable) internal equilibria. Firstly, we show that in multi-player two-strategy games, they behave asymptotically as √ d − 1 as d is sufficiently large. Secondly, we prove that they are monotone functions of d. We also make a conjecture for games with more than two strategies. Thirdly, we provide numerical simulations for our analytical results and to support the conjecture. As consequences of our analysis, some qualitative and quantitative results on the distribution of zeros of a random Bernstein polynomial are also obtained.

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

“…For sufficiently large d, when r increases from 0 (uncorrelated) to 1 (identical), E(r , d) reduces from O(d 1/2 ) at r = 0, to O(d 1/4 ) for 0 < r < 1 and to 0 at r = 1. This theorem generalises and improves the main results in [15] for the case r = 0: the asymptotic behaviour,…”

“…Similar to the foundational concept of Nash equilibrium in classical game theory [42], the study of equilibrium points and their stability in EGT has been of significant importance and extensive research [4,10,12,14,15,27,28,36]. They represent population compositions where all the strategies have the same average fitness, thus predicting the coexistence of different strategic behaviours or types in a population.…”

“…In contrast to the equilibrium analysis of concrete games, a recent body of works investigates random games where individual payoffs obtained from the games are randomly assigned [10,14,15,25,27,28,36]. This analysis has proven useful to provide answers to generic questions about a dynamical system such as its overall complexity.…”

“…However, in general, a polynomial of degree five or higher is not analytically solvable [1]. Therefore, the direct approach cannot be generalised to larger d. More recently, in [14,15] the authors introduce a novel method using techniques from random polynomials to calculate E(d) with an arbitrary d, under the assumption that the entries of the payoff matrix are independent normal random variables. More precisely, they derive a computationally implementable formula for E(d) for arbitrary d and prove the following monotonicity and asymptotic behaviour of E(d):…”

Evolution of commitment and level of participation in public goods games

“…3. In the two-player setting (namely, the Prisoner's Dilemma) the behavior of proposing prior commitments has been shown to promote the evolution of cooperation if the cost of arranging commitment is sufficiently small compared to the cost of cooperation [19,23,25]. But when larger groups of actors are involved, decision-making becomes much more complex [11,12,15,26,70]. Instead of a clear, full commitment or, no-commitment, from the co-player as in the two-player game, when moving to the multi-player decision scenario of the PGG, there can be several possible intermediate degrees of participation (as many as the size of the group, i.e.…”