SCHOOL OF ENGINEERING AND COMPUTER SCIENCE

Consider two alleles, A and B. Denote the density of A by $ \rho $.

We have * a 2-by-2 payoff matrix $V(\mathrm{self} | \mathrm{other}) $ * a 2-by-2 matrix of probabilities describing the chance of interacting with other types: $P(\mathrm{other} | \mathrm{self}) $.

Expected fitness of A's:

\[ W(A) \;\; = \;\; W_0 \;\; + \;\; P(A|A) \, V(A|A) \;\; + \;\; P(B|A) \, V(A|B) \]

Similarly, expected fitness of B individuals is

\[ W(B) \;\; = \;\; W_0 \;\; + \;\; P(A|A) \, V(A|A) \;\; + \;\; P(B|A) \, V(A|B) \]

  • Note that $P(B|A) = 1-P(A|A) $, and similarly $P(A|B) = 1-P(B|B) $, so all we really need to specify are $P(A|A)$ and $P(B|B) $.

case $ P(A \rfloor A) $ $ P(A \rfloor A) $
randomly mixed $ \rho$ $ 1-\rho $
perfectly sorted $ 1 $ $ 1 $
partial assortment
$ 0<r<1 $ 		$ r + (1-r) \rho $ $ r + (1-r) (1-\rho) $

Example: prisoner's dilemma

  • agents can choose to help each other: helping benefits the other guy by $ b $, but costs you (the helper) $ c $. We assume $ b > c $ to make it interesting.
  • Consider two pure strategies: C and D (cooperate and defect).
  • Denote density of C by $ \rho $.

Payoff matrix:

$ b-c $ $ -c $
$ b $ 0

Plugging these in gives Expected fitness of C:

\[ W(C) \;\; = \;\; W_0 \;\; + \;\; P(C|C) \, V(A|A) \;\; + \;\; P(B|A) \, V(A|B) \]

Similarly, expected fitness of D:

\[ W(D) \;\; = \;\; W_0 \;\; + \;\; P(A|A) \, V(A|A) \;\; + \;\; P(B|A) \, V(A|B) \]

William D. Hamilton:
hamilton.jpg

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