In the film A Beautiful Mind, John Nash and three of his graduate school colleagues find themselves faced with a dilemma while at a bar. There are four brunettes and a single blonde available for them to approach. Each young man wants to approach and win the attention of one of the young women. The payoff to each of winning the blonde is 10; the payoff of winning a brunette is 5; the payoff from ending up with no girl is 0. The catch is that, if two or more young men go for the blonde, she rejects all of them and then the brunettes also reject the men because they don't want to be second choice. Thus, each player gets a payoff of 10 only if he is the sole suitor for the blonde.

(a) [8 pts] First consider a simpler situation where there are only two young men, instead of four. (There are two brunettes and one blonde, but these women merely respond in the manner just described and are not active players in the game.) Show the playoff table for the game, and find all of the pure-strategy Nash equilibria of the game.

(b) [10 pts] Show the (three-dimensional) table  for the case in which there are three young men (and three brunettes and one blonde who are not active players).A gain, find all of the Nash equilibria of the game.

(c) [10 pts] Use your results to part (a) to generalize your analysis to the case in which there is some arbitrary number, n, of young men (So there are n brunettes and one blonde.) Hint: first find the payoff to one player when k of the others choose Blonde and (n - k -l) choose Brunette, for k = 0, l, . . . (n - 1). Then find all of the pure-strategy Nash equilibria. Can the outcome specified in the movie as the Nash equilibrium of the game---that all of the young men choose to go for brunettes---ever be a true Nash equilibrium of the game? Please explain.

(d) [10 pts] Here we consider the mixed-strategy equilibria of that game when played by n> 2 young men. Begin by considering the symmetric case in which all n young men go 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?

(e) [12 pts] There are asymmetric mixed-strategy equilibria in this game also. 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 n - 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?