Mathematician John Forbes Nash Jr and his wife, Alicia (née Lopez-Harrison de Lardé), died Saturday, May 23 in a car crash in New Jersey. Mr Nash shared a Nobel Prize in 1994 for his theories, known as the “Nash equilibria” on noncooperative games, first published in 1951. Those theories have become pervasive in the work of other scientists in fields as diverse as economics and evolutionary biology. He was 86 and she 82.
The taxi in which Mr and Mrs Nash were traveling lost control and hit a guard rail and another vehicle, ejecting both of them. They were pronounced dead at the scene. The taxi driver and the driver of the other vehicle were treated for non-life threatening injuries. No criminal charges have been filed in connection with the crash.
Dr Nash, certainly one of the great mathematicians of the 20th century, was known for “the originality of his thinking and for his fearlessness in wrestling down problems so difficult few others dared tackle them,” the New York Times noted. The paper quoted a one-sentence letter written in support of his application to Princeton’s doctoral program in math, which said simply, “This man is a genius.”
The simple idea at the heart of his theory is this: You can’t predict the result of the choices many different decision makers will make if all you do is consider each of those decisions in isolation. What you have to do is ask what each decision maker would do, taking into account the decisions made by others. In the movie A Beautiful Mind, it was explained like this, as four college boys, including Mr Nash, plotted a scheme to pick up a woman at a bar:
If we all go for the blonde and block each other, not a single one of us is going to get her. So then we go for her friends, but they’ll all give us the cold shoulder because no one likes to be second choice. But what if none of us goes for the blonde? We won’t get in each other’s way and we won’t insult the other girls. It’s the only way to win. It’s the only way we all get laid.
Perhaps a more famous, albeit less Hollywood, example of the Nash equilibrium is the situation in which two hunters have to choose between hunting a stag or hunting a rabbit. According to the rules of the game, each hunter can kill a rabbit by himself, but a rabbit has much less meat and so gets that hunter less nutrition than he would receive if he were to kill a stag. Unfortunately, the rules of the game say he’ll need the other hunter’s help to kill a stag.
Each hunter must decide, independently but by considering the choice he thinks the other hunter will make, whether to go for a rabbit, worth one point, or a stag, worth four. If he decides to go for the stag and the other hunter decides to go for the rabbit, he’ll lose and end up with nothing. But Dr Nash noticed in the real world, there were few situations that had only winners and losers like this, known as a zero-sum game. So if the hunter is confident the other one will help him hunt the stag, netting four points to be split between the two of them, he will come out ahead of where he would be if he had decided to hunt the rabbit, as will the other hunter.
Nash equilibria even have applications in areas closer to our own lives. The state championship series in high school football, for example, reaches a Nash equilibrium at the end of the season. The eventual state champion’s trophy is a little bigger than the one given to the runner-up or third-place finisher, but not by much.
And besides, assuming that all participating teams are making decisions they believe will lead to victories and that they take into account the strategies they think their opponents will bring, all participants gain something from playing the game, from working out and practicing, and from hearing the instructions and encouragement from coaches, parents, peers, and school officials. The pep rallies for a second-place finisher in the state don’t look very different from the ones given to the state champions.
Likewise, marching band competitions, like those conducted by Bands of America or even by individual high schools in Illinois, aren’t zero-sum games either. Sure, a grand champion will be named eventually, but members of all participating bands gain something by performing, by practicing for the performance, by hearing the applause, by listening to judges’ comments, by talking with peers, parents, and other proponents of music, and by celebrating excellent performances in several ways throughout the season.
The idea of mutual gain by a game’s competitors is at the center of Mr Nash’s theories, which directly support the idea that there’s a lot to be gained by young people through participation in noncooperative competition in the fine arts and in athletics. This in no way refutes the idea that a lot can be gained through cooperation as well, but noncooperative competition, as long as schools emphasize the gains made by students, rather than the absolute winners and losers assumed by zero-sum theory, stands on solid footing in our schools.
It is fitting, albeit tragic, that Mr and Mrs Nash died together. In real life, they were divorced for a while but got back together after he won his Nobel Prize. In the movie, he spoke the following as he accepted his Nobel Prize:
What truly is logic? Who decides reason? My quest has taken me to the physical, the metaphysical, the delusional, and back. I have made the most important discovery of my career—the most important discovery of my life. It is only in the mysterious equations of love that any logic or reasons can be found. I am only here tonight because of you. [looking at and speaking to Alicia] You are the only reason I am. You are all my reasons.
Stunned…my heart goes out to John & Alicia & family. An amazing partnership. Beautiful minds, beautiful hearts. https://t.co/XF4V9MBwU4
— Russell Crowe (@russellcrowe) May 24, 2015
— Ron Howard (@RealRonHoward) May 24, 2015