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Evolutionary Game Dynamics and the Moran Model

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Evolutionary Game Dynamics and the Moran Model U.U.D.M. Project Report 2020:39 Evolutionary Game Dynamics and the Moran Model William Norman Examensarbete i matematik, 15 hp Handledare: Ingemar Kaj Examinator: Martin Herschend Juni 2020 Department of Mathematics Uppsala University Evolutionary Game Dynamics and the Moran Model William Norman 1 Contents 1 Introduction 3 2 Evolutionary Game Theory 3 3 The Replicator Dynamics 5 3.1 Two Strategies: Generic Cases . .7 3.2 Rock-Paper-Scissors . .8 4 The Moran Model 9 4.1 Fixation Probabilities and Times . 11 4.1.1 Fixation Probabilities . 11 4.1.2 Fixation Times . 14 4.2 The Moran Process and Weak Selection . 16 4.3 Moran Process with Constant Fitness . 21 5 Evolutionary Graph Theory and the Spatial Moran Model 23 5.1 The Isothermal Theorem . 24 5.2 Suppressing and Amplifying Selection . 26 5.2.1 Suppressors . 27 5.2.2 Amplifiers . 27 6 Conclusion 28 2 1 Introduction Evolutionary game theory is a branch of game theory used for modelling pop- ulations under recurrent strategic interactions. Although originally intended for evolutionary biology the theory has found widespread applications such as solving social dilemmas, contributing to the understanding of evolution of language, financial risk decision-making and creating neural networks. Evolutionary game dynamics, which is the focus of this thesis, is the part of population modelling that describes the manner in which strategies spread among the population. A strategy can be interpreted in many different ways. In biology we may interpret a strategy as a gene, a phenotype or a species; in economics, an investment strategy; in social science, a behaviour. Evolutionary game dynamics act on populations. It is populations that evolve, not strategies. The aim of this thesis is to introduce the reader to the theory of evolution- ary game dynamics by showcasing two commonly used dynamics. In section 2 we introduce basic concepts of evolutionary game theory. In section 3 we consider the deterministic replicator dynamics, a system of ordinary non- linear differential equations. In section 4 we consider stochastic dynamics in the form of birth-death processes, in particular, we consider the Moran process. In chapter 5 we develop a framework for analyzing how population structure influences evolution under the spatial Moran process. 2 Evolutionary Game Theory Game theory is used to model strategic interactions among rational players acting to maximize payoffs by finding optimal strategies in a single game. Further, each player is assumed to have full information of the game rules. In contrast to the strict assumptions of game theory, evolutionary game theory adopts a philosophy based on survival of the fittest and trial and error. Individuals with bounded rationality in a population act to the best of their abilities with limited understanding of the world (using their strategy). Fur- ther, instead of a single game we consider repeated games called evolutionary games. After each game, the proportions of strategies in the population is updated according to payoff comparisons and the choice of game dynamics. This shift allows us to use the underlying framework of game theory and expand upon it to model evolutionary games between, for example, genes which evolve by trial-and-error rather than by a rational approach. 3 Definition 2.1. A payoff matrix U for a game with m strategies is a m × m matrix with real-valued entries called payoffs. Payoff uij represents the reward for an individual with strategy i interacting with an individual with strategy j. Example 2.1. The following is a payoff matrix for a (symmetric) two player game with strategies A and B AB . A a b B c d We interpret a and d as the payoffs of A and B self-interacting, respectively. b is the payoff of A interacting with B and c is the payoff of B interacting with A. In this thesis we consider only symmetric, normal-form games. Normal- form meaning that each game is completely determined by a payoff matrix. Symmetric meaning that payoffs are dependent solely on the employed strat- egy, not on who employs the strategy. Definition 2.2. For a population with m strategies, the population state is a stochastic vector x = (x1; x2; :::; xm) where entry xi is the proportion of strategy i in the population. It is standard practice in traditional game theory to analyze how rational players will behave through static solution concepts such as the Nash equilib- rium. A Nash equilibrium is a game state in which no player has an incentive to switch strategy in the sense that no action results in an incremental benefit assuming all other players stick to their strategy. Definition 2.3. Let U be the payoff matrix of a game. A population state x is a Nash equilibrium (NE) if for all other population states y xT Ux ≥ yT Ux: We call x a strict NE if the inequality is strict for all population states y 6= x. We call x a weak NE if the inequality holds for all states but there is an equality for some population state y 6= x. 4 Definition 2.4. Let U be the payoff matrix of a game. A population state x is a evolutionary stable state (ESS) if for all population states y 6= x either of the following conditions hold 1. xT Ux > yT Ux (strict NE condition), 2. xT Ux = yT Ux and xT Uy > yT Uy (stability against other strategies). Remark. From the definitions we get Strict NE =) ESS =) Weak NE: 3 The Replicator Dynamics Assume that there is a large population with m strategies. The population is well-mixed or structureless, meaning that the probability that an individual using strategy i interacts with j is simply the proportion of strategy j. We define the population space, the set of possible population states, as m m Pm the m-simplex: ∆ = fx 2 [0; 1] : i=1 xi = 1g. Let U be the payoff matrix for an evolutionary game. The expected payoff for an individual with strategy i interacting with a random individual in the population is given by m X (Ux)i = uijxj: (1) j=1 The average payoff for an individual chosen at random interacting with a random individual in the population is given by m m m T X X X x Ux = xi(Ux)i = xiuijxj: (2) i=1 i=1 j=1 Assume that proportions are differentiable with respect to time t and denote the derivative of xi(t) asx _i. This results in continuous evolution (change in proportions). Let evolution be governed by the continuous time replicator equation T x_i = xi[(Ux)i − x Ux] i = 1; 2; :::; m: (3) Under the replicator dynamics, evolution is governed by m − 1 non-linear ordinary differential equations that encapsulates the notion of 'survival of the fittest’ as seen by T x_i > 0 () (Ux)i > x Ux; T x_i = 0 () (Ux)i = x Ux or xi = 0; (4) T x_i < 0 () (Ux)i < x Ux: 5 Further, the simplex is invariant under the replicator dynamics,5 meaning that trajectories starting inside the simplex remain inside the simplex forever. To determine the evolutionary outcome of an evolutionary game governed by the replicator dynamics we borrow concepts from the theory of dynamical systems. Definition 3.1. A point x∗ is an equilibrium point of the differential equation x_ = f(t; x) if f(t; x∗) = 0 for t ≥ 0. Definition 3.2. Let d be the euclidean metric. An equilibrium point x∗ of x_ = f(t; x) with initial condition x(0) = x0 is stable if 8 > 0, 9δ > 0 : ∗ ∗ d(x0; x ) < δ =) d(x(t); x ) < , 8t ≥ 0. An equilibrium point is unstable if it is not stable. Further, an equilibrium point is asymptotically stable if it is stable and the state of the system converges to x∗ as t ! 1. Remark. For a time-invariant, non-linear system of ordinary differential equa- tions, such as the replicator system of equations, we may determine the sta- bility of an equilibrium point by considering the Jacobian matrix J(x) of the system evaluated at the equilibrium point; if J(x∗) has only strictly nega- tive eigenvalues then x∗ is asymptotically stable; if at least one eigenvalue is positive then it is unstable. The following theorem connect equilibrium concepts of game theory and dynamical systems making deterministic games easier to analyze. Theorem 3.1. 1;5 In an evolutionary game, the following is true for a pop- ulation state x∗ under the replicator dynamics. 1. If x∗ is a stable equilibrium point then x∗ is a NE, 2. If x∗ is a NE then x∗ is a equilibrium point, 3. If x∗ is a ESS (or strict NE) then x∗ is asymptotically stable equilibrium point. 6 3.1 Two Strategies: Generic Cases Consider a game of two strategies A and B given by the payoff matrix AB . A a b B c d Denote x = xA = 1 − xB. Evolution in this case is determined by the single replicator equation T x_ = x[(Ux)1 − x Ux] = x[ax + b(1 − x) − x(ax + b(1 − x)) − (1 − x)(cx + d(1 − x))] (5) = x(1 − x)[(a − b − c + d)x + b − d]: The equation has trivial equilibrium points x∗ = 0 and x∗ = 1. If also ∗ d−b [a > c ^ d > b] or [a < c ^ d < b] then x = a−b−c+d is also an equilibrium point in the open interval (0; 1). To determine the stability of the equilibrium points, consider the Jacobian of the system dx_ d J(x) = = x(1 − x)[(a − b − c + d)x + b − d] dx dx (6) = 3x2(−a + b + c − d) + 2x(a − 2b − c + 2d) + b − d: Analyzing the stability of the equilibrium points we deduce the following.
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