Question

In a beautiful mind, 4 students find themselves faced with a dilemma while at a bar. There are five available women in the bar for them to approach, four brunettes, and a blond. Each student can approach and win the attention of one of one young women only.

The payoffs of 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.

if two or more young men go for the blonde, she rejects all of them. However, if only one man approaches the blond, she will be more reasonable and accept him. If any of the brunettes are approached second, will reject the men because they don't want to be the second choice, if approached first they will accept. Unfortunately, all four students saw a Marilyn Monroe movie recently and prefer blondes. Each of the means are hoping to be successful with the blonde.

One student named Nash says that their approach is wrong and if they all compete for the blonde they will all prevent each other from succeeding, so their optimal strategy is to ignore the blonde completely and each approach a different brunette to make sure they get dates. In other words, don’t think only about your self-interest, think of the group, too. In his excitement Nash tells the rest “governing dynamics, gentlemen, Adam Smith was wrong”, and then shows his nerdiness by accidentally bumping into the blonde on his way out of the bar and despite getting an inviting smile, just walks out to the woman’s astonishment.

a) First consider a simpler situation where there are only 2 men instead of 4. There are 2 brunettes and 1 blonde in this situation. Show the game in normal form and solve it.

b) Show the three-dimensional payoff table representing this game, where there are now 3 men, 3 brunettes, and 1 blonde. Solve the game and explain.

c) Without the normal form, solve the game for the case where there are 4 men, 4 brunettes, and one blond. Explain.

d) What does the movie get wrong? Explain why the dramatic Nash equilibrium cannot be a Nash equilibrium for this game?

Answer 1