John Forbes Nash, Jr., known for the Nash Equilibrium and having schizophrenia, died in a car crash on Sunday along with his wife. Much sads. So I'm going to talk about his Equilibrium. Suppose you have a two players playing a game. It's a specific type of game, wherein each player has a pre-specified, finite set of possible moves, and the payoff from any particular move is known in advance. In particular, if player 1 makes move a and player 2 makes move ii, then both players know how many points or whatever each player will get as a result of those choices. We further assume that both players are completely rational and both want to win and they both know that they both know that blah blah blah etc. Everybody knows everything except for which particular choice the other player is going to make. We call this a game of perfect information. Furthermore, we allow what are called "mixed strategies", where instead of picking a particular move, each player is allowed to assign a probability to picking each move, so maybe player 1 picks move a with probability 1/2, and b with probability 1/3, and c with probability 1/6. The expected payoff is then the total of the payoffs for each choice of moves from the two players, weighted by the probability of them picking those choices. So what can player 1 do? Without knowing what player 2 is going to do, player 1 wants to find a strategy that gets her an expected win no matter what player 2 does. But that doesn't always exist. In lieu of that, player 1 maybe wants a strategy that is expected to be better than any other strategy player 1 can make, regardless of what player 2 does. But that also might not exist. So maybe player 1 wants a strategy that is expected to be better than any other strategy player 1 can make, assuming that player 2 makes the best strategy as far as player 2 is concerned. But what's the best strategy for player 2? That depends on what player 1 does! So this is kind of circular. Player 1's best strategy depends on what player 2's best strategy is, and player 2's best strategy depends on what player 1's best strategy is. So we need a new way of looking at this. Suppose that if player 2 picks strategy IV, then player 1's best strategy is strategy A. And suppose that if player 1 picks strategy A, then player 2's best choice is choice IV. Then if player 1 picks strategy A then player 2 is going to pick strategy IV, and vice-versa. So we have here an equilibrium, a pair of strategy that, if the two players both think that pair is going to be chosen, neither will want to deviate from it. Of course, this kind of situation can arise if you have any number of players, not just 2. As long as everybody knows the payoffs for everybody else, everybody can figure out what the optimal choice is supposing that they know what all the other players are going to do, so they can figure out if there is an equilibrium. Nash's theorem is then that such an equilibrium has to exist. It may not be unique, but it must exist. This might not be the best payoff for all, or even any player! Like a compromise that nobody wants but is the only thing that will make it through the committee, it could be that there are other sets of choices that will net both players more points, but are only possible if everyone cooperates. But if the players can't trust one another, then the equilibrium might only give a small or even a negative payoff for each player, and the only reason it isn't worse is that neither player wants to be the only person to deviate. Think of global thermonuclear war: nobody wants to be the only person not launching nukes, even though the best option would be for nobody to launch nukes. Moreover, if you think that the other person is going to launch a nuke, you want to launch first. So Nash equilibria sometimes lead to what appears to be irrational behavior, even though everyone is acting rationally, taking the best option available according to what they know of everybody else. This theorem has obvious consequences for game theory and economics and population biology and anything that can be modeled as a bunch of agents making choices. It depends on having perfect information and assuming rationality of everyone involved, and also on the threats of deviation being credible. Successive work by Nash and others generalized his original idea to the cases of non-perfect information and non-credible threats; what if you don't know how much value other countries put on not being nuked, and how likely they are to actually launch nukes once they start threatening to? There is also work describing when we should expect there to be a unique Nash equilibria, and when that unique equilibrium is actually the best outcome for everyone. Nash also made several large contributions to other branches of mathematics. For instance, the Nash-Moser theorem talks about existence of solutions of certain types of partial differential equations useful in celestial mechanics, and the Nash Embedding theorem allows us to move from descriptions of geometric objects as patched-together chunks of Rm to descriptions as subsets of Rn. He was actually on his way home from receiving the Abel Prize, the analogue of the Nobel Prize for mathematics, for his work on partial differential equations when he died.