5. A Beautiful Mind, a movie about John Nash, fails to properly demonstrate a Nash equilibrium. It attempts to do so in a bar scene where men at a bar (Nash and his friends) plan to ask women to dance. There is one beautiful woman that the men consider the most attractive, as well as several other women. Nash assumes the “less attractive” women will only accept an offer to dance if the man extending the offer has not first been rejected by the beautiful woman. In the movie, Nash proposes that all the men agree to not ask the beautiful woman in the first place.

a. Nash’s proposal may lead to a good outcome for each man, but it is not a Nash equilibrium. Why not?

b. The movie initially shows all the men asking the beautiful woman to dance. To be fair, this is also not a Nash equilibrium. Why not? Why is it a Nash equilibrium if exactly one man asks the beautiful woman to dance?