Evolutionary Game Theory

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Evolutionary Game Theory
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Game Theory

• Von Neumann Morgenstern (1953)Studying
  economic behavior
• Maynard Smith Price (1973)Why are animal
  conflicts examples of limited wars?

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Assumptions

• Infinite population size
• Random mating
• Asexual reproduction
• Frequency dependent fitness
• Genotype can be mapped directly onto phenotype -
  haplotypes

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Fundamental Concept

• The Evolutionary Stable Strategy (ESS)A
  strategy such that if all members of the
  population adopt it, then no mutant can invade
  the population under the influence of selection

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The Haploid Hawk Dove Game

• Consider two haplod virus genotypes that breed
  true
• The Hawk genotype encodes a virulent virus
  strain.
• The Dove genotype encodes an avirulent virus
  strain

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Fitness payoffs

• The reproductive value of an infected host to a
  virus is V
• When two virulent viruses (H) coinfect a host
  there is a cost associated with morbidity C
• When a virulent virus (H) coinfects with a
  avirulent virus (D), H derives all the benefits
  V.
• When two avirulent viruses (D) infect a host they
  obtain approximately half of the resource each V/2

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Payoff matrix

• H DH 1/2(V-C) VD 0 V/2

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Building the model

• p frequency of H viruses
• W(H) W(D), denote mean fitness
• E(H,D) fitness payoff to H infecting a body
  already infected with D, similar meaning for
  E(H,H), E(D,H) and E(D,D)
• W0 is the fitness of the virus prior to infection
  of the host

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Virus fitnesses

• Upon infection of a single hostW(H) W0
  pE(H,H) (1-p)E(H,D)W(D) W0 pE(D,H)
  (1-p)E(D,D)

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Determining the ESS conditions

• Consider any two genotypes I JW(I) W0
  pE(I,J) (1-p)E(I,I)W(J) W0 pE(J,J)
  (1-p)E(J,I)
• Assume that I is an ESS and J is a rare mutant
  with frequency p
• If I is an ESS then W(I) gt W(J), assuming that p
  ltlt1, then,E(I,I) gt E(J,I) or (Invasion
  condition)E(I,I) E(J,I) and E(I,J) gt E(J,J)
  (Stability)

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ESS solutions to the H D game

• E(I,I) gt E(J,I) or (Invasion condition)E(I,I)
  E(J,I) and E(I,J) gt E(J,J) (Stability)
• E(D,D) gt E(H,D) Never!
• E(H,H) gt E(D,H) only of 1/2(V-C) gt 0

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Mixed ESS solutions

• What if VltC?
• Does this mean that there is no ESS solution to
  the game?
• An alternative ESS solution can exist if the
  biology permits.
• This requires either a genotype capable of
  switching between H and D or some mix of H D
  coexisting in the population.

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Mixed ESS solution

• Consider strategy I as genotype H with
  probability P and genotype D with probability
  (1-P).
• For a mixed ESS to exist thenE(A,I) E(B,I)
  E(C,I) E(I,I)All pure strategies in support of
  I must have the same payoff.

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Finding the mixed ESS

• If I is a mixed ESS then E(H,I) E(D,I)
• E(H,I) PE(H,H) (1-P)E(H,D)
• E(D,I) PE(D,H) (1-P)E(D,D)
• P(1/2)(V-C) (1-P)V P.0 (1-P)V/2
• Solve for P
• P V/C

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Testing I with the ESS conditions

• E(I,I) gt E(J,I) or (Invasion condition)E(I,I)
  E(J,I) and E(I,J) gt E(J,J) (Stability)
• We need to see if I meets the stability
  conditionE(H,I) E(D,I) E(I,I) (True)
• Therefore we require thatE(I,D) gt E(D,D)
  E(I,H) gt E(H,H)
• Calculate the above and show that I is an ESS

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Evolution of virulence genes

• When V gt C then virulent virus always favoured
• When V lt C then some proportion of the population
  given by V/C will be virulent
• Increasing the cost favours avirulent forms
• Reducing the cost favours the virulent forms

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Game Theory Summary

• Fitness of a gene can depend on frequencies of
  all other genes in a population -- fitness is
  frequency dependent
• Game theory provides a tool for determining the
  equilibrium distribution of genotypes in the
  population when fitness is frequency dependent
• Key Reference John Maynard Smith.Evolution and
  the Theory of Games. CUP. 1982.

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Game theory anisogamy
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Game Theory the sex ratio
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Game theory area of application

• Frequency-dependent selection
• Ignorant about genetic mechanisms
• Parthenogenetic inheritance
• Act as an aid to intuition before building more
  complex models
• When we do know about genetics it is best to add
  selection to our population genetics models