# American Institute of Mathematical Sciences

2015, 12(3): 503-523. doi: 10.3934/mbe.2015.12.503

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

 1 Department of Mathematics, Iowa State University, Ames, IA 50011, United States, United States, United States, United States 2 Department of Statistics, Iowa State University, Ames, IA 50011, United States

Received  September 2014 Revised  November 2014 Published  January 2015

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.
Citation: Yuanyuan Huang, Yiping Hao, Min Wang, Wen Zhou, Zhijun Wu. Optimality and stability of symmetric evolutionary games with applications in genetic selection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 503-523. doi: 10.3934/mbe.2015.12.503
##### References:
 [1] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979). Google Scholar [2] I. Bomze, Regularity vs. degeneracy in dynamics, games, and optimization: A unified approach to different aspects,, SIAM Review, 44 (2002), 394. doi: 10.1137/S00361445003756. Google Scholar [3] J. M. Borwein, Necessary and sufficient conditions for quadratic minimality,, Numer. Funct. Anal. Optim., 5 (1982), 127. doi: 10.1080/01630568208816135. Google Scholar [4] S. Boyd and L. Vandeberghe, Convex Optimization,, Cambridge University Press, (2004). doi: 10.1017/CBO9780511804441. Google Scholar [5] N. A. Campbell, J. B. Reece, L. A. Urry, M. L. Cain, S. A. Wasserman, P. V. Minorsky and R. B. Jackson, Biology,, $8^{th}$ edition, (2008). Google Scholar [6] W. J. Ewens, Mathematical Population Genetics,, Springer-Verlag, (2004). Google Scholar [7] R. A. Fisher, The Genetic Theory of Natural Selection,, Clarendon Press, (1999). Google Scholar [8] G. H. Hardy, Mendelian proportions in a mixed population,, Zeitschrift für Induktive Abstammungs- und Vererbungslehre, 1 (1908). doi: 10.1007/BF01990610. Google Scholar [9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, Cambridge University Press, (1998). doi: 10.1017/CBO9781139173179. Google Scholar [10] J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15. doi: 10.1038/246015a0. Google Scholar [11] T. Motzkin and E. Straus, Maxima for graphs and a new proof of a theorem of Turán,, Canadian J. Math., 17 (1965), 533. doi: 10.4153/CJM-1965-053-6. Google Scholar [12] K. G. Murty and S. N. Kabadi, Some NP-complete problems in quadratic and linear programming,, Math. Programming, 39 (1987), 117. doi: 10.1007/BF02592948. Google Scholar [13] J. Nash, Equilibrium points in $n$-person games,, Proceedings of the National Academy of Sciences, 36 (1950), 48. doi: 10.1073/pnas.36.1.48. Google Scholar [14] J. Nocedal and S. J. Wright, Numerical Optimization,, Springer-Verlag, (2006). Google Scholar [15] P. Pardalos, Y. Ye and C. Han, Algorithms for the solution of quadratic knapsack problems,, Linear Algebra and Its Applications, 152 (1991), 69. doi: 10.1016/0024-3795(91)90267-Z. Google Scholar [16] W. H. Sandholm, Population Games and Evolutionary Dynamics,, The MIT Press, (2010). Google Scholar [17] A. R. Templeton, Population Genetics and Microevolutionary Theory,, John Wiley & Sons Inc., (2006). doi: 10.1002/0470047356. Google Scholar [18] L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, SIAM, (1997). Google Scholar [19] J. W. Weibull, Evolutionary Game Theory,, The MIT Press, (1995). Google Scholar [20] W. Weinberg, Aber den nachweis der vererbung beim menschen,, Jahreshefte des Vereins fur vaterlandische Naturkunde in Wurttemberg, 64 (1908), 368. Google Scholar

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##### References:
 [1] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979). Google Scholar [2] I. Bomze, Regularity vs. degeneracy in dynamics, games, and optimization: A unified approach to different aspects,, SIAM Review, 44 (2002), 394. doi: 10.1137/S00361445003756. Google Scholar [3] J. M. Borwein, Necessary and sufficient conditions for quadratic minimality,, Numer. Funct. Anal. Optim., 5 (1982), 127. doi: 10.1080/01630568208816135. Google Scholar [4] S. Boyd and L. Vandeberghe, Convex Optimization,, Cambridge University Press, (2004). doi: 10.1017/CBO9780511804441. Google Scholar [5] N. A. Campbell, J. B. Reece, L. A. Urry, M. L. Cain, S. A. Wasserman, P. V. Minorsky and R. B. Jackson, Biology,, $8^{th}$ edition, (2008). Google Scholar [6] W. J. Ewens, Mathematical Population Genetics,, Springer-Verlag, (2004). Google Scholar [7] R. A. Fisher, The Genetic Theory of Natural Selection,, Clarendon Press, (1999). Google Scholar [8] G. H. Hardy, Mendelian proportions in a mixed population,, Zeitschrift für Induktive Abstammungs- und Vererbungslehre, 1 (1908). doi: 10.1007/BF01990610. Google Scholar [9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, Cambridge University Press, (1998). doi: 10.1017/CBO9781139173179. Google Scholar [10] J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15. doi: 10.1038/246015a0. Google Scholar [11] T. Motzkin and E. Straus, Maxima for graphs and a new proof of a theorem of Turán,, Canadian J. Math., 17 (1965), 533. doi: 10.4153/CJM-1965-053-6. Google Scholar [12] K. G. Murty and S. N. Kabadi, Some NP-complete problems in quadratic and linear programming,, Math. Programming, 39 (1987), 117. doi: 10.1007/BF02592948. Google Scholar [13] J. Nash, Equilibrium points in $n$-person games,, Proceedings of the National Academy of Sciences, 36 (1950), 48. doi: 10.1073/pnas.36.1.48. Google Scholar [14] J. Nocedal and S. J. Wright, Numerical Optimization,, Springer-Verlag, (2006). Google Scholar [15] P. Pardalos, Y. Ye and C. Han, Algorithms for the solution of quadratic knapsack problems,, Linear Algebra and Its Applications, 152 (1991), 69. doi: 10.1016/0024-3795(91)90267-Z. Google Scholar [16] W. H. Sandholm, Population Games and Evolutionary Dynamics,, The MIT Press, (2010). Google Scholar [17] A. R. Templeton, Population Genetics and Microevolutionary Theory,, John Wiley & Sons Inc., (2006). doi: 10.1002/0470047356. Google Scholar [18] L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, SIAM, (1997). Google Scholar [19] J. W. Weibull, Evolutionary Game Theory,, The MIT Press, (1995). Google Scholar [20] W. Weinberg, Aber den nachweis der vererbung beim menschen,, Jahreshefte des Vereins fur vaterlandische Naturkunde in Wurttemberg, 64 (1908), 368. Google Scholar
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