8.5 The Academy Award-winning movie A Beautiful Mind about the life of John Nash dramatizes Nash's scholarly contribution in a single scene: his equilibrium concept dawns on him while in a bar bantering with his fellow male graduate students. They notice several women, one blond and the rest brunette, and agree that the blond is more desirable than the brunettes. The Nash character views the situation as a game among the male graduate students, along the following lines. Suppose there are #males who simultaneously approach either the blond or one of the brunettes. If male i alone approaches the blond, then he is successful in getting a date with her and earns payoff

a. If one or more other males approach the blond along with i, the competition causes them all to lose her, and i (as well as the others who approached her) earns a payoff of zero. On the other hand, male i earns a payoff of > 0 from approaching a brunette, since there are more brunettes than males, so i is certain to get a date with a brunette. The desirability of the blond implies a >

b. a. Argue that this game does not have a symmetric pure-strategy Nash equilibrium.

b. Solve for the symmetric mixed-strategy equilibrium. That is, letting p be the probability that a male approaches the blond, find p*.

c. Show that the more males there are, the less likely it is in the equilibrium from part

(b) that the blond is approached by at least one of them. Note: This paradoxical result was noted by S. Anderson and M. Engers in "Participation Games: Market Entry, Coordination, and the Beautiful Blond," Journal of Economic Behavior & Organization 63 (2007): 120-37.