Could you do first-year economics?

Once again, I’m teaching EC102, a mathematics-oriented introduction to economics for first-year undergraduate students at the London School of Economics.  The following is a question from one of the students’ weekly quizzes:

A population consists of two types, “friendlies” (Fs) and “aggressives” (As). Each individual interacts with a randomly chosen member of the population. When two Fs interact, they each earn 3 units. When two As interact, they each earn 0. When an F and an A interact the former gets 1 unit and the latter 5 units. The growth rate of each type is proportional to their average payoff. What will be the equilibrium population share of Fs?

There are some hints over the fold …
First hint: In general, an equilibrium is a state of the system that is stable over time.  In this question, the state of the system is the proportions of (F)riendly and (A)ggressive types in the population.

Second hint: Let ‘f’ be the fraction of the population that is ‘F’ (friendly). Try to put everything as a function of ‘f’.

Third hint: What is the expected (i.e. average) payoff for each type of agent?

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