An Introduction to Evolutionary Game Theory: Lecture 2 - School of ...

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Evolutionary Games in Finite Populations (I)Finite population of 2 species: i individuals of species A and N − iindividuals of species B interact according to the payoff matrix:vs A BA a bB c dProbability to draw a A and B is i/N and (N − i)/N, respectively ⇒Probability that a given individual A interacts with another A is(i − 1)/(N − 1)Probability that a given individual A interacts with a B is(N − i)/(N − 1)Probability that a given individual B interacts with another B is(N − i − 1)/(N − 1)Probability that a given individual B interacts with a A is i/(N − 1)The states i = 0 (All-A) and i = N (All-B) are absorbingExpected payoff for A and B, respectively:Ei A a(i − 1) + b(N − i)=N − 1andEi B ci + d(N − i − 1)=N − 1Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2
Evolutionary Games in Finite Populations (II)() −1F A = 1 + ∑ N−1j=1 ∏j k=1 γ k and FB = F A ∏ N−1k=1 γ k, with γ i = β i /α iE Ai= a(i−1)+b(N−i)N−1and EiB = ci+d(N−i−1)N−1Expected payoffs E A,Biare usually interpreted as fitness.Recent idea (Nowak et al.): Introduce a parameter w accounting forbackground random drift contribution to fitness f Aif Ai= 1 − w + wEi A and fiB = 1 − w + wEiBfor A and f Bifor BAverage fitness: ¯f = (i/N)fiA + (1 − (i/N))fiBParameter w measures the intensity of selection: w = 0 ⇒ noselection (only random drift), w = 1 ⇒ only selection, w ≪ 1 ⇒ “weakselection”Consider a Moran process with frequency-dependent hopping rates:α i = f iA ( )( )i N − iand β i = f B ( )( )i i N − i⇒ γ i = f iB¯f N N¯f N NfiA()Thus, F A = 1/ 1 + ∑ N−1j=1 ∏j k=1 (f k B/f k A) and F B = F A ∏ N−1k=1 (f k B/f k A)Mauro Mobilia Evolutionary Game Theory: An Introduction, Lecture 2
Page 1 and 2: An Introduction to Evolutionary Gam
Page 3 and 4: Replicator DynamicsPopulation of Q
Page 5 and 6: Some Properties of the Replicator D
Page 7 and 8: Replicator Dynamics for 2 × 2 Game
Page 9 and 10: Some Remarks on Replicator Dynamics
Page 11 and 12: Stochastic Dynamics & Moran Process
Page 13: Fixation in the Neutral & Constant
Page 17 and 18: Influence of Fluctuations & Finite-
Page 19 and 20: Influence of Fluctuations & Finite-
Page 21 and 22: OutlookIn this set of lectures dedi

An Introduction to Evolutionary Game Theory: Lecture 2 - School of ...

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