13.2 (a)
ESS: Fight
(Fight strictly dominates Share)
(b)
Let p be the proportion of Fighters (so 1–p is the proportion of Sharers). Then the expected payoff
(fitness) for each type is:
F(F) = -50p + 200(1 – p) = 200 – 250p
F(S) = 100(1 – p) = 100 – 100p
When p = 2/3, Fighters and Sharers are equally fit. When p > 2/3 (increased number of Fighters),
Sharers are more fit, so p decreases back to 2/3. When p < 2/3 (increased number of Sharers), Fighters
are more fit, so p increases back to 2/3. ESS: Population consists of 2/3 Fighters and 1/3 Sharers.
(c)
In part (a): V = 200 and C = 100, giving a single monomorphic ESS with all fighters (since Fighters
are analogous to Hawks, and Sharers are analogous to Doves, this is the ESS found in section 6.B).
In part (b): V = 200 and C = 300, giving a single polymorphic ESS (same as found in 6.D; V/C = 2/3).
13.9
When the proportion of S-type males is y, the expected payoff (fitness) for each type of female is:
F(S) = y
F(L) = 2(1 – y) = 2 – 2y
When y = 2/3, both types are equally fit, so any initial mix of female types can be sustained. When y >
2/3, type S females are more fit, so x increases to 1. When y < 2/3, type L females are more fit, so x
decreases to 0. By symmetry, the same results hold for Males (replacing y with x, and x with y, in the
analysis above). This gives rise to two ESS (shown in the population dynamics figure below): one in
which all males and females are type S, and another in which all males and females are type L. Both
types are equally fit when x = 2/3 and y = 2/3, but this is not stable, as any deviation from this mixture
(for either gender) will cause the populations to converge to one of the ESS above, which are
monomorphic for each gender.