The movie is a representation of the life of John Nash, a Nobel Laureate in Economics, for his developments in Game Theory. Most of you have probably heard about the notion of a Nash Equilibrium which is, in summary, a situation in which no player wants to change his strategy, based on what the other players are doing. A Nash Equilibrium does not need to be a good nor a desirable outcome, just a best response of all players. In other words, if all players can stop the time and think: am I doing the best I can considering what everyone else did? and they all find that choosing another strategy will yield them worse results, then that sum of strategies of all players is a Nash Equilibrium.
In the movie, there is a famous scene that tries to illustrate this concept:
The idea is simple: if all players do what is best for themselves ignoring that other players will also do what is best for themselves, then they will get a negative outcome. If, in contrast, they consider what other players may be thinking before making a decision, they may prefer to choose another action.
The example is easy to understand and well explained but it is full of mistakes! It is strange that a biographical movie about one of the best mathematicians of the history makes so many mistakes in a crucial part of the movie.
Nash did not study Sequential Games
The illustration of the movie can be represented as a Sequential Game. The men move first and receive a first response from the approached women. Then, they move again and receive a new response from the women. The idea of Sequential Games is that if the men can anticipate by backward induction that they will be rejected by the brunettes in the second subgame, they would prefer to approach them directly in the first subgame.
Although the concept is part of Game Theory (and one of the most interesting and useful topics), it is not what John Nash studied! In fact, he focused on Strategic Games, in which all players have to move simultaneously. In other words, the situation represented in the movie implies a game with two rounds. Therefore, it should not be part of John Nash biographical movie.
The equilibrium is definitely wrong!
This is what actually drives me crazy. In a Nash Equilibrium, all the players must be playing their best response, taking into account what the rest of the players are doing. Therefore, if there are 4 men and 5 women and we assume that the blond is the prettiest and the 4 brunettes are less desirable, then an outcome in which the 4 men approach the 4 brunettes is not a Nash Equilibrium! in fact, all the men would individually be willing to change their action and approach the blond woman (considering that nobody else has approached her). Actually, this game would have four Nash Equilibria. In each equilibrium, one different man finishes with the blond and the three others finish with the three most beautiful brunettes. It is straightforward to verify that this is actually a Nash Equilibrium, because in this case, nobody would be willing to change his action. If one of the men that approached a brunette decides to approach the blond, he will have a worse outcome because he will be rejected by the blond (who was with other guy) and will finish without any woman. In addition, nobody wants to change their woman for the remaining brunette, because she is not as pretty as the ones that they have already approached.
There is also a Mixed Strategy Nash Equilibrium
This may be a bit too technical, and it seems reasonable that the director did not include it in the movie. There are some games in which there is not a Nash Equilibrium in pure strategies. For example, in Rock Paper Scissors, if A did Rock and B Paper, it is not a Nash Equilibrium because A would prefer to change to Scissors. But this is not a Nash Equilibrium because B would prefer to change to Rock, and it continues ad infinitum. However, the situation in which both players randomize and play each action with probability 1/3 is a Mixed Strategy Nash Equilibrium, because if the other one is playing that strategy, my best response is to do it too. The four Nash Equilibria described in the previous section assume that players do not play symmetric actions (one goes with the blond woman and the other ones with the brunettes). However, we can think about a Mixed Strategy Nash Equilibrium in which the 4 men randomize their actions, and approach the blond woman with probability P and a brunette with probability 1-P. The computation of P is not complicated, but we would need to assume some Bernoulli Values in order to maximize their expected utility.
In conclusion, it is a bit frustrating that the best representation of a fascinating topic like Game Theory has so many mistakes in one of the most important scenes. In the movie, Nash says:
Adam Smith said the best result comes from everyone in the group doing what's best for himself. Right? That's what he said, right? Incomplete. Incomplete, okay? Because the best result will come from everyone in the group doing what's best for himself … and the group.
Real life John Nash would agree that Adam Smith theory was incomplete, but he would argue that the mistake was different. The real John Nash would never claim that the best result come from everyone in the group doing what is best for himself and the group, but from everyone doing what is best for himself taking into consideration what the rest of the group has done.
Paraphrasing Russell Crowe´s script, A Beautiful Mind needs revision...definitely.
