A Beautiful Mind Blu Ray
Elvina Cannizzaro
LinkedIn and 3rd parties use essential and non-essential cookies to provide, secure, analyze and improve our Services, and to show you relevant ads (including professional and job ads) on and off LinkedIn. Learn more in our Cookie Policy.
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 aNash 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 is a Nash Equilibrium.
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.
The illustration of the movie can be represented as a Sequential Game. All 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 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 that implies a game with two rounds, should not be part of John Nash biographical movie.
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 a blond, he will have a worse outcome because he will be rejected by the blond (that 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.
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 infinitely. 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.
Real life John Nash would agree that Adam Smith theory was incomplete, but 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.
While finishing my dissertation at Princeton, I had the distinct pleasure of taking a seminar with Nobel Laureate, John Nash. If you've read the book or seen the movie, "A Beautiful Mind," you'll know that Nash suffered from schizophrenia and this seminar was one of his first appearances since having brought that mental ailment under control. You'll also know that Nash, a genius, essentially "invented" non-cooperative game theory.
Not knowing what to expect, most of us graduate students signed up just to spend time with Nash. He was a legend. And though we'd seen him often lurking around Firestone library with stacks of paper scribblings, none of us ever spoke to him. Yet, we were curious about what a true genius might actually have to say and how one might think.
Strangely enough, Nash's seminar didn't make everyone on the faculty happy. I remember one game theorist telling me attending it would be a 'distinct waste of time'. Others even chuckled at the thought of a person with a mental disability giving a coherent seminar at Princeton. All this seemed ironic and cruel, given Nash's importance in the field and the fact that, with his work, most of their work would lack any importance at all.
Non-cooperative game theory is the study of how individuals or institutions might interact strategically if they don't communicate and Nash won the 1994 Nobel Prize for presenting the first, stable solution to such a situation. A player can do no better in taking an action given the actions of other players, given the inability to cooperate.
If you loved the films, "The Usual Suspects" or "LA Confidential," for example, those plots demonstrated Nash Equilibrium in its great, Hollywood form. In a nutshell, the principle characters are backed into situations where they make the best decisions they can given what they expect others to do. Everything becomes interesting when circumstance gives the good guys the upper hand.
Surprisingly, however, Nash didn't speak about non-coop game theory in our seminar. Instead he presented his work on cooperative game theory. Coop-game theory is about how groups of individuals enforce behaviour to achieve certain outcomes. Just about every spy movie with a dastardly syndicate influencing its members involves coop theory.
If non-cooperative game theory demonstrates powerlessness in the face of others actions, coop-theory does the opposite. It explains how power can be amassed, wielded and maintained--with due punishment as an enforcement technique. At the end of the day, however, what made Nash's presentation amazing was not so much the insights he presented but the way he presented them. He looked at things differently and that perspective influenced how he presented his ideas.
Nash painted the picture of an idea in an audience member's mind, then slowly brought that picture to life in the audience member's head. In watching and listening to him, it was truly a glimpse of how genius can be communicated.
After the presentation, our group went to a small room with Nash for cookies and discussion. Of course, while Nash wanted to talk about his research results and possible extensions, we wanted to know about his life and about how became "Nash".
I asked him, for example, who had influenced his choice of dissertation topic and he said, no one. Another person wondered whether Nash's work was accepted readily by people around the mathematics department at the time and Nash said, no. Finally, however, we hit the jackpot when we asked Nash how his current research model could explain the politics of the world we currently see.
All of a sudden, this 70-something year old man who'd seemingly awakened from 50 years of dormancy stood up and spoke to us with a 20-year old's energy. He started illustrating the interesting point that powerful, cooperative relationships actually stem from the lack of power, not from the possession of it. Enforcing the rules of a coalition was really about playing on the fears and insecurities of other members, not those of the less powerful, as one might think.
All of that made a sort of counterintuitive sense to us. It made me think, for example, about Hitler's rise to power and his ability to create alliances with the leadership of 'seemingly nice countries'. In school, we were taught that most of these countries fell under Hitler's sway for fear of Nazi invasion. Nash's model, however, suggested insecurity was what might have motivated the Axis powers to cooperate.
But while we pondered Nash's ideas, what further amazed us was his humility and hunger for intellectual interaction, even with us lowly grad students. He spoke to us as colleagues, not as pupils. He invited us to challenge, not merely to validate his thinking. He also encouraged us when he believed we were on the right track, and gently corrected us when we got things wrong.
4a15465005