Asymptotic expected number of Nash equilibria of two-player normal form games

McLennan, Andrew; Berg, Johannes

doi:10.1016/j.geb.2004.10.008

Cited by 50 publications

(29 citation statements)

References 19 publications

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“…Specifically, Algorithm 1 considers every possible support size profile separately, favoring support sizes that are balanced and small. The motivation behind these choices comes from work such as McLennan and Berg (2002), which analyzes the theoretical properties of the NE of games drawn from a particular distribution. Specifically, for n-player games, the payoffs for an action profile are determined by drawing a point uniformly at random in a unit sphere.…”

Section: Algorithm For Two-player Gamesmentioning

confidence: 99%

Simple search methods for finding a Nash equilibrium

Porter

Nudelman

Shoham

2008

Games and Economic Behavior

170

137

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Section: Algorithm For Two-player Gamesmentioning

confidence: 99%

Simple search methods for finding a Nash equilibrium

Porter

Nudelman

Shoham

2008

Games and Economic Behavior

170

137

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“…As in classical game theory with the dominant concept of Nash equilibrium [37,36], the analysis of equilibrium points in random evolutionary games is of great importance because it allows one to describe various generic properties, such as the overall complexity of interactions and the average behaviours, in a dynamical system. Understanding properties of equilibrium points in a concrete system is important, but what if the system itself is not fixed or undefined?…”

mentioning

confidence: 99%

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

Duong

2016

J. Math. Biol.

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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.

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“…In related work, McLennan and Berg [24] have studied the expected number of Nash equilibria for certain random games. Finally, we remark that extending our work to allow matrix entries to have arbitrary means would give a polynomial-time randomized algorithm for finding approximate Nash equilibria in arbitrary games.…”

Section: Corollarymentioning

confidence: 99%