please answer all parts of the question and show your work :)

2. Consider a population consisting of two types, A and B, and assume that the change in the composition of the population is related in the standard way to the average payoffs received by the two types. A-types always play strategy A; B-types always play strategy B. The players are matched randomly and receive payoffs described by the following payoff matrix: Player 2 A B 0.0 3,1 Player 1 B 1,3 2,2 (a) Derive the expected payoff to each of the two types of players as a function of the proportion (1) of players of type A. A (b) Analyze the evolution of the share of type A in the population. Con- sider first the case in which the initial share of A-types is 0.1; consider then the case in which the initial share is 0.9. Describe the evolution- arily stable outcome. (c) Now change the payoffs for when an A player meets an A player in such a way that the A strategy becomes evolutionarily stable. 2. Consider a population consisting of two types, A and B, and assume that the change in the composition of the population is related in the standard way to the average payoffs received by the two types. A-types always play strategy A; B-types always play strategy B. The players are matched randomly and receive payoffs described by the following payoff matrix: Player 2 A B 0.0 3,1 Player 1 B 1,3 2,2 (a) Derive the expected payoff to each of the two types of players as a function of the proportion (1) of players of type A. A (b) Analyze the evolution of the share of type A in the population. Con- sider first the case in which the initial share of A-types is 0.1; consider then the case in which the initial share is 0.9. Describe the evolution- arily stable outcome. (c) Now change the payoffs for when an A player meets an A player in such a way that the A strategy becomes evolutionarily stable