Download Evolutionary Game Theory

Evolutionary Game Theory
Game Theory
• Von Neumann & Morgenstern (1953)
Studying economic behavior
• Maynard Smith & Price (1973)
Why are animal conflicts examples of
‘limited wars’?
Assumptions
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Infinite population size
Random mating
Asexual reproduction
Frequency dependent fitness
Genotype can be mapped directly onto
phenotype - haplotypes
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”
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
Fitness payoffs
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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
Payoff matrix
H
H
D
1/2(V-C)
0
D
V
V/2
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
Virus fitnesses
• Upon infection of a single host:
W(H) = W0+ pE(H,H) + (1-p)E(H,D)
W(D) = W0 + pE(D,H) + (1-p)E(D,D)
Determining the ESS
conditions
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Consider any two genotypes I & J:
W(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) > W(J), assuming that p <<1,
then,
E(I,I) > E(J,I) or (Invasion condition)
E(I,I) = E(J,I) and E(I,J) > E(J,J) (Stability)
ESS solutions to the H & D
game
• E(I,I) > E(J,I) or (Invasion condition)
E(I,I) = E(J,I) and E(I,J) > E(J,J)
(Stability)
• E(D,D) > E(H,D) Never!
• E(H,H) > E(D,H) only of 1/2(V-C) > 0
Mixed ESS solutions
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What if V<C?
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.
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 then:
E(A,I) = E(B,I) = E(C,I)…= E(I,I)
All pure strategies in support of I must
have the same payoff.
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
Testing I with the ESS
conditions
• E(I,I) > E(J,I) or (Invasion condition)
E(I,I) = E(J,I) and E(I,J) > E(J,J) (Stability)
• We need to see if I meets the stability
condition:
E(H,I) = E(D,I) = E(I,I) (True)
• Therefore we require that:
E(I,D) > E(D,D) & E(I,H) > E(H,H)
• Calculate the above and show that I is an
ESS
Evolution of virulence genes
• When V > C then virulent virus always
favoured
• When V < 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
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.
Game theory: anisogamy
Game Theory: the sex ratio
Game theory: area of application
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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