Consider an evolutionary version of the game between Baker and Cutler, from Exercise U1 of Chapter..

Consider an evolutionary version of the game between
Baker and Cutler, from Exercise U1 of Chapter 10. This time Baker and Cutler
are not two individuals but two separate species. Each time a Baker meets a
Cutler, they play the following game. The Baker chooses the total prize to be
either $10 or $100. The Cutler chooses how to divide the prize chosen by the
Baker: the Cutler can choose either a 50:50 split or a 90:10 split in the
Cutler’s own favor. The Cutler moves first, and the Baker moves second. There
are two types of Cutlers in the population: type F chooses a fair (50:50)
split, whereas type G chooses a greedy (90:10) split. There are also two types
of Bakers: type S simply chooses the large prize ($100) no matter what the Cutler
has done, whereas type T chooses the large prize ($100) if the Cutler chooses a
50:50 split, but the small prize ($10) if the Cutler chooses a 90:10 split. Let
f be the proportion of type F in the Cutler population, so that (12 f )
represents the proportion of type G. Let s be the proportion of type S in the
Baker population, so that (12 s) represents the proportion of type T.

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(a) Find the fitness of the Cutler types F and G in terms
of s.

(b) Find the fitness of the Baker types S and T in terms
of f.

(c) For what value of s are types F and G equally fit?

(d) For what value of f are types S and T equally fit?

(e) Use the answers above to sketch a graph displaying
the population dynamics. Assign f as the horizontal axis and s as the vertical
axis.

(f) Describe all of the equilibria of this evolutionary
game, and indicate which ones are stable.

 

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