In the film A Beautiful Mind, John Nash and three of his graduate-school colleagues find themselves.

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.

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(a) First consider a simpler situation in which 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 payoff table for the game, and find all of the
pure-strategy Nash equilibria of the game.

(b) Now 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). Again, find all of the Nash equilibria of the game. (c)
Without the use of a table, give all of the Nash equilibria for the case in
which there are four young men (as well as four brunettes and a blonde).

(d) (Optional) Use your results to parts (a), (b), and
(c) to generalize your analysis to the case in which there are n young men. Do
not attempt to write down an n-dimensional payoff table; merely find the payoff
to one player when k of the others choose Blonde and (n 2 k 2 1) choose
Brunette, for k = 0, 1, . . . (n = 1). 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 really be a Nash equilibrium of the game?

 

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