Week 9

• Evolutionary symmetric games


  Games in finite and well-mixed populations of size N, modelled as birth-death processes
• eC


  (1, 0)T
• eD


  (0, 1)T
• x


  Strategy profile, state of the population described by x = 0.4eC + 0.6eD = (0.4, 0.6)T
• Payoff matrix
• Replicator dynamics


  ẋ = x(1 - x)[x(a - c) + (1 - x)(b - d)]
• When N → ∞, we recover replicator dynamics
• When N < ∞, x(t) fluctuates, requiring stochastic modelling
• Coexistence equilibrium


  x* = (d - b) / (a - c + d - b)
• Coexistence equilibrium is metastable when b > d, c > a in a finite population
• Birth and death rates are functions of the fitness of C and D
• Moran process


  λi = (i/N)(N-i/N)(fC(i)/f̄(i))

  μi = (i/N)(N-i/N)(fD(i)/f̄(i))

  where f̄(i) = (i/N)fC(i) + (1 - i/N)fD(i)
• Fermi process


  λi = (i/N)(N-i/N)2 / (1 + e(fD(i)-fC(i)))

  μi = (i/N)(N-i/N)2 / (1 + e-(fD(i)-fC(i)))
• Dynamics with Moran and Fermi processes are the same in the weak selection limit (s << 1)
• Fixation probability of C


  φC

  i
• Fixation probability of D


  φD

  i
• Fixation probabilities satisfy a 2nd-order linear map
• Fixation probability of a single D

  φD = 1 - φC
• Neutral dynamics (s = 0)


  • φC

  i = i/N

  φC = 1/N
• Constant fitness (fC(i) = r, fD(i) = 1)


  • φC

  i = (rN-i - 1) / (rN - 1)

  φC = (rN-1 - 1) / (rN - 1)
• Goal: Fixation probability in 2-player evolutionary games with 2 pure strategies in finite populations
• Pure strategies


  Strategies where the player always chooses the same action
• Infinite populations


  Populations where the number of individuals is very large
• Weak selection


  Selection intensity is small (0<s<<1)
• Fixation probability of C, ϕC, has an exact formula
• Fixation probability of C, ϕC, has examples
• Fixation probability of C, ϕC, has a weak selection limit
• Birth-death process


  Stochastic process where individuals are born and die, leading to changes in the population composition
• Absorbing boundaries


  States where the process cannot leave once entered (fixation of C or D)
• In 2-player games with 2 pure strategies, the outcome is fixation of either C or D
• Fitness-dependent Moran process


  Birth-death process where the birth and death rates depend on the fitness of the individuals
• Fitness-dependent Fermi process


  Birth-death process where the birth and death rates depend on the fitness of the individuals via a Fermi function
• Under weak selection, the fixation probability of a single C has a simple expression
• Selection favours fixation of C

  When the fixation probability of C is greater than 1/N
• Selection opposes fixation of C


  When the fixation probability of C is less than 1/N
• Under weak selection and large N, selection favours fixation of C if a+2b > c+2d
• Under weak selection and large N, selection opposes fixation of C if a+2b < c+2d
• Coordination game


  Game where a > c and d > b
• In coordination games under weak selection and large N, there is an unstable coexistence equilibrium at a frequency x* < 1/3 of C
• In coordination games under weak selection and large N, selection favours fixation of C if the basin of attraction of x* is greater than 2/3