Homophily, payoff distributions, and truncation selection in replicator dynamics by Bryce Morsky A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Doctor of Philosophy in Applied Mathematics Guelph, Ontario, Canada ©Bryce Morsky, April, 2016 ABSTRACT HOMOPHILY, PAYOFF DISTRIBUTIONS, AND TRUNCATION SELECTION IN REPLICATOR DYNAMICS Bryce Morsky Advisors: University of Guelph, 2016 Chris T. Bauch & Daniel Ashlock This dissertation explores the field of replicator dynamics by examining extensions to and relaxations of the classical replicator equation and complimentary agent-based mod- els. We extend the replicator equation by the incorporation of homophilic imitation, a form of tag-based selection. We show that though the equilibria are not affected by this mod- ification, the population’s diversity may increase or decrease depending on two invasion scenarios we detail, and there is significant impact on the rates of convergence to equi- libria. Two important assumptions of the replicator equation that we relaxed are: mean payoffs, where all replicators earn the mean payoff of the underlying game; and propor- tional selection, where the probabilities for survival and reproduction are proportional to the difference between the fitness of a replicator and the mean fitness of the population. Our models thus comprise payoff distributions and two types of truncation selection: in- dependent, where replicator above a threshold, φ, survive; and dependent, where the top τ of replicators survive. The reproduction rates are equal for all survivors. We show that the classical replicator equation is a special case of our independent truncation equation. Further, for any boundary fixed point, we may choose a φ such that that point is stable (or unstable). We observed complex and transient dynamics in both truncation methods. We applied this framework to evolutionary graphs that included diffusion, and show where cooperation is facilitated by these models in comparison to spatial and non-spatial propor- tional selection. Alfred Russel Wallace reasoned that the relatively unfit could coexist with the fit, and it has been argued that this would result in a genotypically diverse population resistant to extinction. This is because natural selection, rather than Spencer’s “survival of the fittest,” may be better encapsulated by the phrases: “survival of the fit,” or “non- survival of the non-fit.” We argue that truncation selection, here explored, can model this phenomenon, and thus is an important addition to the theoretical biology literature. iv ACKNOWLEDGEMENTS I would like to acknowledge the contributions and support of the Department of Mathe- matics & Statistics, my advisory committee, and my examining committee. I also wish to express my deep gratitude to my advisor, Chris Bauch, for his mentorship. v Table of Contents List of Tables viii List of Figures ix 1 Introduction 1 1.1 The replicator equation . .2 1.1.1 ESS . .3 1.1.2 Limitations . .3 1.1.3 Games . .4 1.2 Extensions to the replicator equation . .6 1.2.1 Imitation dynamics . .6 1.2.2 Truncation selection . .7 1.3 Outline . .8 2 Homophilic replicator equations 11 B. MORSKY, R. CRESSMAN, & C. T. BAUCH 2.1 Abstract . 11 2.2 Introduction . 12 2.2.1 Replicator equations . 14 2.2.2 Games . 16 2.3 Methods . 17 2.3.1 The Model . 17 2.3.2 Measures of diversity . 19 2.3.3 Simulations . 20 2.3.3.1 Recurrent mutations . 21 2.3.3.2 Further simulations . 21 2.4 Results . 22 2.4.1 Fixed points and stability . 22 2.4.2 The two-tag Snowdrift game with recurrent mutations . 25 2.4.3 Coat-tailing and diversity . 25 vi 2.4.3.1 Invasion scenario 1 .................... 25 2.4.3.2 Invasion scenario 2 .................... 28 2.4.4 Rates of convergence . 31 2.5 Discussion . 36 2.6 Appendix: Games . 38 3 Truncation selection and payoff distributions applied to the replicator equa- tion 40 B. MORSKY & C. T. BAUCH 3.1 Abstract . 40 3.2 Introduction . 41 3.3 Methods . 44 3.3.1 Fitness distributions . 44 3.3.2 The replicator equation . 47 3.3.3 Proportional selection . 49 3.3.4 Truncation selection . 50 3.3.5 Agent-based models . 52 3.4 Results . 52 3.4.1 Evolutionary stability . 52 3.4.2 Agent-based simulations vs the 2-strategy model . 55 3.4.2.1 Independent truncation . 55 3.4.2.2 Dependent truncation . 58 3.5 Discussion . 60 3.6 Conclusions . 62 3.7 Appendix: mean payoffs and the proportional selection . 63 3.8 Fixed points and stability proofs . 65 4 Truncation selection facilitates cooperation on random spatially structured populations of replicators 68 B. MORSKY & C. T. BAUCH 4.1 Abstract . 68 4.2 Introduction . 70 4.3 Methods . 75 4.4 Results . 78 4.4.1 Proportional selection . 78 4.4.2 Independent truncation . 80 4.4.3 Dependent truncation . 84 4.5 Discussion . 86 5 Discussion 90 5.1 Summary . 90 5.2 Directions for future work . 92 vii A Appendix of Java code 94 A.1 Code for chapter 3 .............................. 94 A.1.1 DepFogel. java . 94 A.1.2 IndepFogel. java . 97 A.1.3 Paper2. java . 102 A.1.4 Player. java . 108 A.2 Code for chapter 4 .............................. 109 A.2.1 CA. java . 109 A.2.2 CAdep. java . 116 A.2.3 CAindep. java . 119 A.2.4 Paper3. java . 124 A.2.5 Player. java . 135 References 136 viii List of Tables 3.1 Parameter values for the Hawk-Dove game. 46 ix List of Figures 2.1 Relative entropy vs. rjj for invasion scenario 1................ 27 2.2 Relative entropies vs. κ for invasion scenario 1................ 29 2.3 Relative entropy vs. κ for invasion scenario 2................. 32 2.4 Relative entropy vs. κ for invasion scenario 2................. 34 2.5 HRE rates of convergence. 35 3.1 Equilibria for the independent truncation equation. 55 3.2 Independent truncation agent-based simulation averages and extinction rates. 56 3.3 Independent truncation agent-based simulation results. 57 3.4 Equilibria for the dependent truncation equation vs. simulation results. 58 3.5 Dependent truncation agent-based simulation results. 59 3.6 Time series of the agent-based dependent truncation model. 60 4.1 Heatmap of cooperation in parameter space. 71 4.2 Heatmaps of cooperation in parameter space for proportional selection. 79 4.3 Heatmaps of cooperation in parameter space for independent truncation. 81 4.4 Cooperator density, ρc, of each game vs. φ.................. 83 4.5 Heatmaps of cooperation in parameter space for dependent truncation. 85 4.6 Cooperator density, ρc, of each game vs. τ................... 87 1 Chapter 1 Introduction In this dissertation, we explore evolutionary dynamics- the mathematical scheme by which we study evolution. Our focus is upon replicators, evolutionary agents that interact with each other and evolve over time. Within the framework of evolutionary game the- ory (Maynard Smith and Price (1973); Maynard Smith (1974)), replicators earn payoffs from their interactions with each other. The aggregation of these payoffs determines fit- ness, which in turn determines survival and replication. Replicator dynamics have been used to study a variety of fields- biological evolution, animal behaviour, genetics, ecology, chemistry, sociology, evolutionary economics, and even cryptography (Dosi and Nelson (1994); Dugatkin and Reeve (1998); Hines (1987); Hammerstein et al. (1994); Hofbauer and Sigmund (2003); Nowak and Sigmund (2004); Schuster and Sigmund (1983)). Here, we study different mathematical models of replicator dynamics with the purpose of better understanding social group formation, biological evolution, and cooperation. 2 1.1 The replicator equation Introduced in Taylor and Jonker (1978), the replicator equation is the classical im- plementation of evolutionary dynamics (Schuster and Sigmund (1983)). It models the change in time of frequencies of replicator phenotypes, which are represented as strate- gies of a game. Mathematically, we say that si ∈ S is the strategy of xi, the frequency of replicators playing si. When replicators interact, they play a game, and the strategies of the players determines the payoffs each receives. Since the replicator equation is a mean field model, the fitness of xi, fi(x), is a function of the strategy profile of the population, T x = [x1, x2, . , xn] , which is a vector of the frequencies of each phenotype/strategy. We thus have: n X fi(x) = πijxj. (1.1) j=1 Where πij is the payoff to replicators playing si vs. sj, and is an element of the payoff matrix, Π. We then apply a Darwinian process by letting the change in frequencies of replicators be proportional to the difference between their fitness and the average fitness ¯ Pn of the population, f(x). Since i=1 xi = 1 is an invariant attracting hyperplane for the ¯ Pn system, we have f = i=1 xifi(x). The replicator equation is thus: n ¯ X x˙ i = xi(fi(x) − f(x)) = xi fi(x) − xjfj(x) . (1.2) j=1 Of note, this equation has an important relation to the Lotka-Volterra equation. The replicator equation can be imbedded into the Lotka-Volterra equation; further, the Lotka- 3 Volterra equation with equivalent growth rates for all species can be reduced to the replica- tor equation (Bomze (1983, 1995); Hofbauer and Sigmund (1998)). 1.1.1 ESS The stability concept for replicator equation is the evolutionary stable state (ESS), which is equivalent to an asymptotically stable rest point (Hofbauer and Sigmund, 1998; Weibull, 1997). Let x¯ be a fixed point of the replicator equations. Then, it is an ESS if: ∃ > 0 such that ∀x ∈ B(x¯, ), xT Πx < x¯T Πx. (1.3) A Nash Equilibrium is a strategy profile in which no unilateral deviation in a player’s strategy can be profitable to that player.