(Optional) In Section 5.B, we demonstrated for the assurance game that changing Sally’s payoffs does not change her equilibrium mixing proportions—only Harry’s payoffs determine her equilibrium...

(Optional) In Section 5.B, we demonstrated for the assurance game that changing Sally’s payoffs does not change her equilibrium mixing proportions—only Harry’s payoffs determine her equilibrium mixture. In this exercise, you will prove this as a general result for the mixed-strategy equilibria of all two-by-two games. Consider a general two-by-two non-zero-sum game with the payoff table shown below:

(a) Suppose the game has a mixed-strategy equilibrium. As a function of the payoffs in the table, solve for the probability that Rowena plays Up in equilibrium.

(b) Solve for the probability that Colin plays Left in equilibrium.

(c) Explain how your results show that each player’s equilibrium mixtures depend only on the other player’s payoffs.

(d) What conditions must be satisfied by the payoffs in order to guarantee that the game does indeed have a mixed-strategy equilibrium?