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

Manh Hong Duong,The Anh Han,Hoang Minh Tran

doi:10.1007/s13235-018-0276-4

Manh Hong Duong, The Anh Han + Show 1 more

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https://doi.org/10.1007/s13235-018-0276-4

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Abstract

The analysis of equilibrium points in random games has been of great interest in evolutionary game theory, with important implications for understanding of complexity in a dynamical system, such as its behavioural, cultural or biological diversity. The analysis so far has focused on random games of independent payoff entries. In this paper, we overcome this restrictive assumption by considering multiplayer two-strategy evolutionary games where the payoff matrix entries are correlated random variables. Using techniques from the random polynomial theory, we establish a closed formula for the mean numbers of internal (stable) equilibria. We then characterise the asymptotic behaviour of this important quantity for large group sizes and study the effect of the correlation. Our results show that decreasing the correlation among payoffs (namely, of a strategist for different group compositions) leads to larger mean numbers of (stable) equilibrium points, suggesting that the system or population behavioural diversity can be promoted by increasing independence of the payoff entries. Numerical results are provided to support the obtained analytical results.

Highlights

1.1 MotivationEvolutionary Game Theory (EGT) was originally introduced in 1973 by Maynard Smith and Price [41] as an application of classical game theory to biological contexts, providingDynamic Games and Applications (2019) 9:458–485 explanations for odd animal behaviours in conflict situations
We have studied the mean value, E(r, d), of the number of internal equilibria in d-player two-strategy random evolutionary games where the entries of the payoff matrix are correlated random variables (r is the correlation)
We have provided analytical formulas for E(r, d) and proved that it is decreasing as a function of r

Summary

Introduction

1.1 MotivationEvolutionary Game Theory (EGT) was originally introduced in 1973 by Maynard Smith and Price [41] as an application of classical game theory to biological contexts, providingDynamic Games and Applications (2019) 9:458–485 explanations for odd animal behaviours in conflict situations. 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, predicting the coexistence of different strategic behaviours or types in a population. The maximal number of equilibria, the stability and attainability of certain equilibrium points in concrete games have been well established; see for example [4,11,50,56,62]

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