In this thesis, I started with looking at the optimality conditions for evolutionary games. Consider an evolutionary game defined by a symmetric fitness matrix A ∈ Rn×n. Let Sn =
{x ∈ Rn: x ≥ 0, P
ixi = 1} be the set of all mixed strategies. Solving a direct evolutionary
game is to find an optimal strategy x∗ ∈ Sncalled a Nash equilibrium strategy such that
x∗TAx∗ ≥ xTAx∗, for all x ∈ Sn. (9.1)
This problem is a symmetric evolutionary game. It has important applications in population genetics, where it can be used to model and study distributions of genes in given populations when they are under certain selection pressures. We explored the necessary and sufficient conditions for the equilibrium states in detail. These conditions were applied to solving direct evolutionary games.
In order to obtain the fitness matrix for solving direct games, I investigated the inverse games. An inverse game targets on recovering the payoff matrix A for the evolutionary game model based on the data and replicator equations, whose parameters are the components of the payoff matrix. To obtain the estimation and inference on A, we first unified the different types of data sets. Then we used non-parametric spline methods to estimate the derivatives. With the smoothed data, we applied the least squares method to obtain the estimation of the payoff matrices, and we used the parametric bootstrap sampling method to obtain the inferences of the payoff matrices.
I also discussed computational schemes for solving direct games, including a specialized Snow-Shapley algorithm, a specialized Lemke-Howson algorithm, and a searching algorithm based on the solution of a complementarity problem on a simplex. The Snow-Shapley proce- dure is a classical algorithm for finding all extreme optimal strategies via exhausting all sub-
systems and is a purely combinatorial algorithm. The Lemke-Howson algorithm is a classical simplex type algorithm developed to search for Nash equilibria of two-player, finite-strategy games. For the evolutionary game, we make some assumptions to obtain the specialized Lemke-Howson method. Using our theoretical results, the necessary and sufficient conditions of Nash equilibrium provide complementarity conditions that lead to the complementarity method. It solves the direct evolutionary game by solving a reformulated quadratic program- ming problem.
After addressing the direct and inverse game, I introduced the evolutionary game dy- namics. The dynamics of the replicator equation system were investigated in detail. The relationships among Nash equilibria, KKT points, and ESS were discussed. It provided an- other way for approaching to the Nash equilibrium by utilizing the dynamics of the replicator equation system.
Another topic I focused on is the sparsest and densest Nash equilibria in direct evolutionary games. Based on the necessary and sufficient conditions of Nash equilibrium, we derived a new algorithm, called dual method. It has the same completeness as the specialized Snow-Shapley method on Nash equilibria searching, while it does better on the computational performance. Two numerical examples of direct evolutionary games, including a gene mutation for malaria resistance and a plant succession problem, were presented. The results about solving the direct games were applied to these examples. A solver for evolutionary games, ‘the toolbox for evolutionary dynamics analysis (TEDA)’, was shown with functions and modules, applications in inverse and direct game, stability analysis for equilibria, and dynamics analysis.
BIBLIOGRAPHY
[1] Axelrod, R., and Hamilton, W.D., 1981. The evolution of cooperation. Science, 211: 1390-1396.
[2] Bremermann, H.J., and Pickering, J., 1983. A game theoretical model of parasite viru- lence. J. Theore. Biol., 100: 411-426.
[3] Campbell, N. A. and Reece, J. B., et al., Biology, Pearson Education, Inc., 2009
[4] Cao, J., Fussmann, G.F. and Ramsay, J.O., 2008. Estimating a predator-prey dynamical model with the parameter cascades method. Biometrics, 64(3): 959-967
[5] Clarke, B., Fokoue, E. and Zhang H.H., 2009. Principles and theory for data mining and machine learning. New York: Springer-Verlag.
[6] R. Cressman: Strong stability and density dependent evolutionarily stable strategies. J. Theor. Biology 145(1990), 319-30. MR 91j:92010
[7] Davison, A.C., and Hinkley, D.V., 1997. Bootstrap Methods and their Application. Cam- bridge: Cambridge University Press.
[8] Dieckmann, U., Law, R., and Metz, J.A.J., 2000. The geometry of ecological interactions: simplifying spatial complexity. Cambridge: Cambridge University Press
[9] Dieckmann, U., Marrow, P., and Law, R., 1995. Evolutionary cycling in predator-prey interactions: population dynamics and the red queen. J. Theor. Biol., 176: 91-102. [10] Doebeli, M., Hauert, C., and Killingback, T., 2004. The evolutionary origin of cooperators
[11] Fisher, R.A., 1930. The distributio of gene ratios for rare mutations. P. Roy. Soc. Edinb. B., 50: 205-220
[12] Gautschi, W., 1997. Numerical analysis: an introduction. Birkhauser
[13] Hamilton, W., 1964. The genetical evolution of social behaviour. I. Journal of Theoretical Biology 7 (1): 116.
[14] Hammerstein, P., 1996. Darwinian adpatation, population genetics, and the streetcar theory of evolution. J. Math. Biol., 34: 511-532.
[15] Hauert, C., De Monte, S., Hofbauer, J., and Sigmund, K., 2002. Volunteering as red queen mechanism for cooperation in public goods games. Science, 296: 1129-1132. [16] Hofbauer, J., and Sigmund, K., 1998. Evolutionary games and population dynamics.
Cambridge: Cambridge University Press.
[17] Huang, Y., Liu, D. and Wu, H.L., 2006. Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system. Biometrics, 62: 413-423
[18] Lemke, C.E., and Howson, J.T.Jr., 1964. Equilibrium Points Of Bimatrix Games. J. SIAM, 12(2): 413-423
[19] Liang, H. and Wu. H.L., 2008. Parameter estimation for differential equation models using a framework of measurement error in regression models. J. Am. Stat. Asso., 103: 1570-1583
[20] Lieberman, E., Hauert, C., and Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature, 433: 312-316
[21] Lipsitch, M.E., Herre, E.A., and Nowak, M.A., 1995. Host population structure and the evolition of virelence: A law of diminishing returns. Evolution, 49: 743-748
[23] May. R.M., and Anderson, R.M., 1983. Epidemiology and genetics in the coevoluion of parasites and hosts. P. Roy. Soc. Lond. B Bio., 219: 281-313
[24] May. R.M., and Oster. G.F., 1976. Bifurcation and dynamics complexity in simple eco- logical models. Am. Nat., 110: 573-599
[25] May. R.M., and Nowak. M.A., 1994. Superinfection, metapopulation dynamics, and the evolution of diversity. J. Theor. Biol., 170: 95-114
[26] Maynard Smith, J., 1979. Hypercycles and the origin of life. Nature, 280: 445-446 [27] Murray, J.D., 2002. Mathematiacl Biology I: An introduction. 3rd ed. New York:
Springer-Verlag.
[28] Nash, J.F., 1950. Equilibrium points in n-person games. P. Natl. Acad. Sci. USA, 36: 48-49
[29] Nocedal, J. and Wright, S. J., Numerical Optimization, Springer, 2006
[30] Nowak, M.A., 2006. Evolutionary dynamics: Exploring the equations of life. Cambridge: Harvard University Press.
[31] Nowak, M.A., Bonhoeffer, S., and May, R.M., 1994. Spatial games and the maintenance of cooperation. P. Natl. Acad. Sci. USA, 91: 4877-4881
[32] Nowak., M.A., and May, R.M., 1992. Evolutionary games and spatial chaos. Nature, 359: 826-829.
[33] Nowak, M.A., Page, K.M., and Sigmund, K., 2000. Fairness versus reason in the ultima- tum gene. Science, 289: 1773-1775
[34] Nowak, M.A., and Sigmund, K., 2004. Evolutionay dynamcis of biological games. Science, 303: 793-799.
[35] Pardalos, P., Ye, Y., and Han, C., 1991. Algorithms for the solution of quadratic knapsack problems, Linear Algebra and Its Applications 152: 69-91
[36] Ramsay, J.O., Hooker, G., Campbell, D. and Cao, J., 2007. Parameter estimation for differential equations: a generalized smoothing approach. J. R. Statist. Soc. B, 69: 741- 796
[37] Roughgarden, T.: Computing equilibria: a computational complexity perspective. Econ Theory 42 (2010)
[38] Schreiber, S., 2001. Urn models, replicator processes, and random genetic drit. SIAM J. Appl. Math., 61: 2148-2167
[39] Shahshahani, S., 1979. A new mathematical framework for the study of linkage and selection. Mem. Am. Math. Soc. 211. Providence, RI: American Mathematical Society [40] Shapley, L.S., and Snow, R.N., 1953. A Value for n-person Games, In Contributions to
the Theory of Games volume II, H.W. Kuhn and A.W. Tucker (eds.): 27-37.
[41] Smith, J.M., and Price, G.R., 1973. The logic of animal conflict. Nature, 246: 1518. [42] Sousa, W.P., 1979. Experimental investigations of disturbance and ecological succession
in a rocky intertidal algal community. Ecol. Monogr. 49: 227-254
[43] Steward, F.M., and Levin, B.R., 1984. The population biology of bacterial viruses: Why be temperate. Theor. Popul. Biol., 26: 93-117
[44] Templeton, A. R., Population Genetics and Microevolutionary Theory, John Weley & Sons, Inc., 2006
[45] Trefethen, L. N. and Bau, III, D., Numerical Linear Algebra, SIAM, 1997
[46] Turner, P.E., and Chao, L., 1999. Prisoner’s dilemma in an RNA virus. Nature, 398: 441-443.
[47] van Hulst, R, 1987. Invasion Models of Vegetation Dynamics. Vegetatio, 69: 123-132 [48] van Domselaar, B., and Hemker, P.W., 1977. Nonlinear parameter estimation in initial
[49] Volterra, V., 1926. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. Acad. Lincei., 2: 31-113 (Translation in an appendix to R.N. Chapman, Animal ecology, New York, 1931)
[50] Webb, J. N., Game Theory: Decisions, Interaction and Evolution, Springer, 2006 [51] Weibull, J.W., 1995. Evolutionary game theory. Cambridge: MIT Press
[52] Wright, S., 1931. Evolution in Mendelina populations. Genetics, 16: 97-159
[53] Williams, G.S., 1966. Adpatation and natural selection. Princeton, NJ: Princeton Uni- versity Press
[54] Wakeley,J. 2004. Metapopulation models for historical inference. Mol. Ecol., 13: 865-875 [55] Xue, H.Q., Miao, H.Y. and Wu., H.L., 2010. Sieve estimation of constant and time varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error. Annals of Statistics, 38: 2351-2387