An Introduction to Evolutionary Game Theory: Lecture 2 Mauro - PowerPoint PPT Presentation

An Introduction to Evolutionary Game Theory: Lecture 2 Mauro Mobilia Lectures delivered at the Graduate School on Nonlinear and Stochastic Systems in Biology held in the Department of Applied Mathematics, School of Mathematics University of Leeds, U.K. 31/03/2009 - 01/04/2009 Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Outline The goal of this lecture is to give some insight into the following topics: Some Properties of the Replicator Dynamics Replicator Equations for 2 × 2 Games Moran Process & Evolutionary Dynamics The Concept of Fixation Probability Evolutionary Game Theory in Finite Population Influence of Fluctuations on Evolutionary Dynamics Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Replicator Dynamics Population of Q different species: e 1 ,..., e Q , with frequencies x 1 ,..., x Q State of the system described by x = ( x 1 ,..., x Q ) ∈ S Q , where S Q = { x ; x i ≥ 0 , ∑ Q i = 1 x i = 1 } To set up the dynamics, we need a functional expression for the fitness f i ( x ) Between various possibilities, a very popular choice is: � � f i ( x ) − ¯ ˙ x i = x i f ( x ) , where, one (out of many) possible choices, for the fitness is the expected payoff : f i ( x ) = ∑ Q i = 1 A ij x j and ¯ f ( x ) is the average fitness : ¯ f ( x ) = ∑ Q i x i f i ( x ) This choice corresponds to the so-called replicator dynamics on which most of evolutionary game theory is centered Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Some Properties of the Replicator Dynamics (I) Replicator equations (REs): ˙ x i = x i [( Ax ) i − x . Ax ] Set of coupled cubic equations (when x . Ax � = 0) Let x ∗ = ( x ∗ 1 ,..., x ∗ Q ) be a fixed point (steady state) of the REs x ∗ can be (Lyapunov-) stable, unstable, attractive (i.e. there is basin of attraction), asymptotically stable=attractor ( =stable + attractive), globally stable (basin of attraction is S Q ) Only possible interior fixed point satisfies (there is either 1 or 0): ( Ax ∗ ) 1 = ( Ax ∗ ) 2 = ... = ( Ax ∗ ) Q = x ∗ . Ax ∗ x 1 + ... + x Q = 1 Same dynamics if one adds a constant c j to the payoff matrix � � ( � Ax ) i − x . � A = ( A ij ) : ˙ x i = x i [( Ax ) i − x . Ax ] = x i Ax , where � A = ( A ij + c j ) Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Some Properties of the Replicator Dynamics (II) Dynamic versus evolutionary stability: connection between dynamic stability (of REs) and NE/evolutionary stability? Notions do not perfectly overlap ⇒ Folks Theorem of EGT: Let x ∗ = ( x ∗ 1 ,..., x ∗ Q ) be a fixed point (steady state) of the REs NEs are rest points (of the REs) Strict NEs are attractors A stable rest point (of the REs) is an NE Interior orbit converges to x ∗ ⇒ x ∗ is an NE ESSs are attractors (asymptotically stable) Interior ESSs are global attractors Converse statements generally do not hold! For 2 × 2 matrix games x ∗ is an ESS iff it is an attractor REs with Q strategies can be mapped onto Lotka-Volterra � � r i + ∑ Q − 1 equations for Q − 1 species: ˙ y i = y i j = 1 b ij y j Replicator dynamics is non-innovative: cannot generate new strategies Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Replicator Dynamics for 2 × 2 Games (I) 2 strategies: say A and B N players: N A are A -players and N B are B -players, N A + N B = N General payoff matrix: vs A B A 1 + p 11 1 + p 12 1 + p 21 1 + p 22 B where selection → p ij and the neutral component → 1 Frequency of A and B strategists is resp. x = N A / N and y = N B / N = 1 − x Fitness (expected payoff) of A and B strategists is resp. f A ( x ) = p 11 x + p 12 ( 1 − x )+ 1 and f B ( x ) = p 21 x + p 22 ( 1 − x )+ 1 Average fitness: ¯ f ( x ) = xf A ( x )+( 1 − x ) f B ( x ) Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Replicator Dynamics for 2 × 2 Games (II) Replicator dynamics: dx x [ f A ( x ) − ¯ = f ( x )] = x ( 1 − x )[ f A ( x ) − f B ( x )] dt = x ( 1 − x )[ x ( p 11 − p 21 )+( 1 − x )( p 12 − p 22 )] xy = x ( 1 − x ) : interpreted as the probability that A and B interact f A ( x ) − f B ( x ) = x ( p 11 − p 12 )+( 1 − x )( p 12 − p 22 ) : says that reproduction (“success”) depends on the difference of fitness Equivalent payoff matrix ( A i 1 → A i 1 − p 11 , A i 2 → A i 2 − p 22 ), with µ A = p 21 − p 11 and µ B = p 12 − p 22 : vs A B 1 + µ A A 1 1 + µ B B 1 dx dt = x ( 1 − x )[ − x µ A +( 1 − x ) µ B ] = x ( 1 − x )[ µ B − ( µ A + µ B ) x ] ⇒ For 2 × 2 games, the dynamics is simple: no limit cycles, no oscillations, no chaotic behaviour Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Replicator Dynamics for 2 × 2 Games (III) dx dt = x ( 1 − x )[ µ B − ( µ A + µ B ) x ] µ A > 0 and µ B > 0: Hawk-Dove game 1 x ∗ = µ B µ A + µ B is stable (attractor, ESS) interior FP µ A > 0 and µ B < 0: Prisoner’s Dilemma 2 B always better off, x ∗ = 0 is ESS µ A < 0 and µ B < 0: Stag-Hunt Game 3 Either A or B can be better off, i.e. x ∗ = 0 and x ∗ = 1 are ESS. x ∗ = µ B µ A + µ B is unstable FP (non-ESS) µ A < 0 and µ B > 0: Pure Dominance Class 4 A always better off, x ∗ = 1 is ESS Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Some Remarks on Replicator Dynamics dx dt = x ( 1 − x )[ µ B − ( µ A + µ B ) x ] For x small: ˙ x = µ B x For x ≈ 1: ˙ y = ( d / dt )( 1 − x ) = µ A ( 1 − x ) Thus, the stability of x ∗ = 0 and x ∗ = 1 simply depends on the sign of µ B and µ A , respectively Another popular dynamics is the so-called “adjusted replicator dynamics”, for which the equations read: � f A ( x ) − f B ( x ) � x f A ( x ) − ¯ dx f ( x ) = = x ( 1 − x ) ¯ ¯ dt f ( x ) f ( x ) These equations equations share the same fixed points with the REs. In general, replicator dynamics and adjusted replicator dynamics give rise to different behaviours. However, for 2 × 2 games: same qualitative behaviour Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Stochastic Dynamics & Moran Process Evolutionary dynamics involves a finite number of discrete individuals ⇒ “Microscopic” stochastic rules given by the Moran process Moran Process is a Markov birth-death process in 4 steps: 2 species, i individuals of species A and N − i of species B An individual A could be chosen for birth and death with 1 probability ( i / N ) 2 . The number of A remains the same An individual B could be chosen for birth and death with 2 probability (( N − i ) / N ) 2 . The number of B remains the same An individual A could be chosen for reproduction and a B 3 individual for death with probability i ( N − i ) / N 2 . For this event: i → i + 1 and N − i → N − 1 − i An individual B could be chosen for reproduction and a A 4 individual for death with probability i ( N − i ) / N 2 . For this event: i → i − 1 and N − i → N + 1 − i Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Stochastic Dynamics & Moran Process Evolutionary dynamics given by the Moran process : Markov birth-death process in 4 steps There are two absorbing states in the Moran process: all-B and all-A What is the probability F i of ending in a state with all A ( i = N ) starting from i individuals A ? For i = 1, F 1 is the “fixation” probability of A Transition from i → i + 1 given by rate α i Transition i → i − 1 given by rate β i F i = β i F i − 1 +( 1 − α i − β i ) F i + α i F i + 1 , for i = 1 ,..., N − 1 F 0 = 0 and F N = 1 Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Moran Process & Fixation Probability What is the fixation probability F 1 of A individuals? = β i F i − 1 +( 1 − α i − β i ) F i + α i F i + 1 , i = 1 ,..., N − 1 F i for = F N = 1 F 0 0 and Introducing g i = F i − F i − 1 ( i = 1 ,..., N − 1), one notes that ∑ N i = 1 g i = 1 and g i + 1 = γ i g i , where γ i = β i / α i ⇒ one recovers a classic results on 1 + ∑ i − 1 j = 1 ∏ j k = 1 γ k Markov chains: F i = j = 1 ∏ j 1 + ∑ N − 1 k = 1 γ k 1 ⇒ Fixation probability of species A is F A = F 1 = 1 + ∑ N − 1 j = 1 ∏ j k = 1 γ k As i = 0 and i = N are absorbing states ⇒ always absorption (all-A or all-B) ⇒ Fixation probability of species B ∏ N − 1 k = 1 γ k is F B = 1 − F N − 1 = j = 1 ∏ j 1 + ∑ N − 1 k = 1 γ k Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2

Fixation in the Neutral & Constant Fitness Cases Fixation Probabilities: 1 k = 1 γ k and F B = F A ∏ N − 1 F A = k = 1 γ k , with γ i = β i / α i j = 1 ∏ j 1 + ∑ N − 1 When α i = β i = γ i = 1, this is the neutral case where there is no selection but only random drift : F A = F B =1 / N This means that the chance that an individual will generate a lineage which will inheritate the entire population is 1 / N Case where A and B have constant but different fitnesses, f A = r for A and f B = 1 for B , α i = ri ( N − i ) i ( N − i ) N ( N +( r − 1 ) i ) and β i = N ( N +( r − 1 ) i ) Thus, F A = 1 − r − 1 1 − r 1 − r − N and F B = 1 − r N If r > 1, F A > N − 1 for N ≫ 1: selection favours the fixation of A If r < 1, F B > N − 1 for N ≫ 1: selection favours the fixation of B Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2