Optimality and stability of symmetric evolutionary games with applications in genetic selection.

Yuanyuan Huang,Zhijun Wu,Min Wang,Wen Zhou,Yiping Hao

doi:10.3934/mbe.2015.12.503

Abstract

Symmetric evolutionary games, i.e., evolutionary games with symmetric fitness matrices, have important applications in population genetics, where they can be used to model for example the selection and evolution of the genotypes of a given population. In this paper, we review the theory for obtaining optimal and stable strategies for symmetric evolutionary games, and provide some new proofs and computational methods. In particular, we review the relationship between the symmetric evolutionary game and the generalized knapsack problem, and discuss the first and second order necessary and sufficient conditions that can be derived from this relationship for testing the optimality and stability of the strategies. Some of the conditions are given in different forms from those in previous work and can be verified more efficiently. We also derive more efficient computational methods for the evaluation of the conditions than conventional approaches. We demonstrate how these conditions can be applied to justifying the strategies and their stabilities for a special class of genetic selection games including some in the study of genetic disorders.

Highlights

We consider an n-strategy evolutionary game defined by a symmetric fitness matrix A ∈ Rn×n
We review the relationship between the symmetric evolutionary game and the generalized knapsack problem, and discuss the first and second order necessary and sufficient conditions that can be derived from this relationship for testing the optimality and stability of the strategies
Some of the conditions are given in different forms from those in previous work and can be verified more efficiently

Summary

Introduction

We consider an n-strategy evolutionary game defined by a symmetric fitness matrix A ∈ Rn×n. Let S = {x ∈ Rn : x ≥ 0, i xi = 1} be the set of all mixed strategies. The problem is to find an optimal strategy x∗ ∈ S such that x∗T Ax∗ ≥ xT Ax∗ for all x ∈ S. We call this problem a symmetric evolutionary game or SEgame for short. The problem has important applications in population genetics, where it can be used to model and study the evolution of genotypes in a given population when their corresponding phenotypes are under selection pressures. Evolutionary biology, population genetics, genetic selection, evolutionary games, generalized knapsack problems, evolutionary stability.

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Conclusion