# On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game

Manh Hong Duong, The Anh Han

Research output: Contribution to journalArticle

3 Citations (Scopus)

### Abstract

In this paper, we analyze the mean number E(n, d) of internal equilibria in a general d-player n-strategy evolutionary game where the agents’ payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next, we characterize the asymptotic behavior of E(2 , d) , estimating its lower and upper bounds as d increases. Then we provide a closed formula for E(n, 2). Two important consequences are obtained from this analysis. On the one hand, we show that in both cases, the probability of seeing the maximal possible number of equilibria tends to zero when d or n, respectively, goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger n and d, numerical results are provided and discussed.

Original language English 324-346 23 Dynamic Games and Applications 6 3 https://doi.org/10.1007/s13235-015-0148-0 Published - 1 Sep 2016

### Fingerprint

Evolutionary Game
Upper and Lower Bounds
Asymptotic Behavior
Infinity
Tend
Internal
Closed
Numerical Results
Interval
Zero
Demonstrate
Strategy

### Cite this

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abstract = "In this paper, we analyze the mean number E(n, d) of internal equilibria in a general d-player n-strategy evolutionary game where the agents’ payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next, we characterize the asymptotic behavior of E(2 , d) , estimating its lower and upper bounds as d increases. Then we provide a closed formula for E(n, 2). Two important consequences are obtained from this analysis. On the one hand, we show that in both cases, the probability of seeing the maximal possible number of equilibria tends to zero when d or n, respectively, goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger n and d, numerical results are provided and discussed.",
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In: Dynamic Games and Applications , Vol. 6, No. 3, 01.09.2016, p. 324-346.

Research output: Contribution to journalArticle

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T1 - On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game

AU - Duong, Manh Hong

AU - Han, The Anh

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N2 - In this paper, we analyze the mean number E(n, d) of internal equilibria in a general d-player n-strategy evolutionary game where the agents’ payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next, we characterize the asymptotic behavior of E(2 , d) , estimating its lower and upper bounds as d increases. Then we provide a closed formula for E(n, 2). Two important consequences are obtained from this analysis. On the one hand, we show that in both cases, the probability of seeing the maximal possible number of equilibria tends to zero when d or n, respectively, goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger n and d, numerical results are provided and discussed.

AB - In this paper, we analyze the mean number E(n, d) of internal equilibria in a general d-player n-strategy evolutionary game where the agents’ payoffs are normally distributed. First, we give a computationally implementable formula for the general case. Next, we characterize the asymptotic behavior of E(2 , d) , estimating its lower and upper bounds as d increases. Then we provide a closed formula for E(n, 2). Two important consequences are obtained from this analysis. On the one hand, we show that in both cases, the probability of seeing the maximal possible number of equilibria tends to zero when d or n, respectively, goes to infinity. On the other hand, we demonstrate that the expected number of stable equilibria is bounded within a certain interval. Finally, for larger n and d, numerical results are provided and discussed.

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