(Optional) Recall Exercise S13 of Chapter 4, which was based on the bar scene from the film A Beautiful Mind. Here we consider the mixed-strategy equilibria of that game when played by n . 2 young...

(Optional) Recall Exercise S13 of Chapter 4, which was based on the bar scene from the film A Beautiful Mind. Here we consider the mixed-strategy equilibria of that game when played by n . 2 young men.

(a) Begin by considering the symmetric case in which each of the n young men independently goes after the solitary blonde with some probability P. This probability is determined by the condition that each young man should be indifferent between the pure strategies Blonde and Brunette, given that everyone else is mixing. What is the condition that guarantees the indifference of each player? What is the equilibrium value of P in this game?

(b) There are also some asymmetric mixed-strategy equilibria in this game. In these equilibria, m , n young men each go for the blonde with probability Q, and the remaining n - m young men go after the brunettes. What is the condition that guarantees that each of the m young men is indifferent, given what everyone else is doing? What condition must hold so that the remaining n₂
m players don’t want to switch from the pure strategy of choosing a brunette? What is the equilibrium value of Q in the asymmetric equilibrium?