This problem is inspired by an incident in the movie, A Beautiful Mind, about John Nash. Two boys meet two girls in a bar. Both boys like both girls, but both prefer Girl A to Girl B. Both boys assign a cardinal utility of 5 to talking with Girl A and 4 to talking with Girl B, and a utility of 0 to talking with neither. The boys’ attitudes towards gambles about outcomes are given by expected utilities. For example the expected utility of talking to Girl A with probability p and to nobody with probability 1-p is 5p+0(1 p) = 5p. The expected utility of talking to Girl B with probability p and nobody with probability 1-p is 4p+0(1 p) = 4p. The girls like both boys equally. If both boys approach the same girl, she will choose one of them at random. The girl who wasn’t approached is offended and won’t talk to the rejected boy. If one boy approaches A and the other approaches B, then each of the girls will talk to the boy who approached her.
A) The decision about who to approach can be modeled as a game between the two boys, where each boy has two possible strategies Approach A and Ap- proach B. Show this game in strategic form.
B) Find a symmetric Nash equilibrium for this game.
C) In symmetric Nash equilibrium, what is the probability that at least one boy approaches Girl A? Girl B? What is the probability that both girls are approached by one of the boys?