In the film, there is a famous scene where John Nash and four of his friends are sitting in a bar and a beautiful blonde girl walks in, along with four brunette friends. As they all want to go for the blonde girl Nash proposes that they should make a better strategy. If all of them go for the blonde, she would feel overwhelmed and no one wins. Also, the brunettes will feel as a second choice and are more likely to reject the guys. However if all of them go for a brunette each person will succeed, and all of them receive a payoff.

In this scene, the director portrays some fundamental concepts of game theory. It is intuitively reasonable to say that the concept of Nash Equilibrium is seen because each guy has a winning strategy to go after a brunette and get something instead of nothing. Also, the concept of a social welfare maximizer is evident because in this situation every person wins the same amount of payoff. Although this is seen as a big enlightening moment in the movie, this is a simplistic way of approaching the scene. If we properly employ Nash Equilibrium we can notice that Nash’s strategy in this scene is wrong.

In a Nash Equilibrium, no player can improve their strategy, thus they are going to stick with it, and they will not deviate. However, in this case, if they all go for the brunettes, they could all improve their strategy by deviating because they really want the blonde. Thus they can either go for the brunette or the blonde. If most of the guys go for brunettes, a player will have no competition for the blonde, and thus he would talk to the blonde. Therefore that would be the player’s best response.

If we put this situation into a matrix in which Bl is for Blonde and Br is for Brunette we can see that: