John F. Nash Jr. is widely known as the subject of the Oscar-winning film “A Beautiful Mind,” but his contributions to the advancement of human knowledge are far greater. Nash paved the way for game theory to spread from a collection of toy cases in mathematics to a generalizable theory applicable to virtually anything — board games, economics, politics, international relations — to the point where now it’s practically a mode of critical thinking in its own right.
On Saturday, the 86-year-old mathematician and his wife, Alicia, were killed in a car crash in New Jersey. There have been many excellent obituaries about Nash’s life and accomplishments, so I wanted to briefly discuss a few details about what he did that was so important, and why his work is still so relevant today.
Nash’s 1951 article “Non-Cooperative Games” refined the definition of an “equilibrium” as a situation in which each player is employing a strategy that is optimal given the strategies of all the other players. For example, in the Prisoner’s Dilemma — a game formalized by Nash’s thesis adviser Albert W. Tucker — the two suspects betraying each other is an equilibrium, despite the best overall outcome being for both of them to remain silent. Nash’s definition would (appropriately) become known as the “Nash Equilibrium” — a term familiar to students in a wide range of academic disciplines.
Given this new definition, Nash was able to prove that a “mixed-strategy” equilibrium exists for virtually any finite game. A “mixed” strategy is one where, instead of choosing a single action, a player chooses a mix of actions with a certain probability for each — for example, the strategy of choosing rock, paper or scissors one-third of the time each in Rock Paper Scissors. It doesn’t matter if your opponent knows what strategy you’re playing, they can’t do anything about it.
One area in which Nash’s legacy continues to be especially relevant and fruitful is sports analysis. One of the cleanest examples is the mini-game of penalty kicks in soccer. If a player always kicks the same direction, the goalkeeper can profitably adapt by always diving in that direction. Thus “always kick right” or “always kick left” can’t be equilibrium strategies. Similarly, if the goalkeeper always dived in one direction, the player would be able to profitably deviate by kicking in the other. Thus the equilibrium strategy for the kicking player is to “mix it up” (use a mixed strategy) by kicking one way some of the time and the other way some of the time. Presuming the player selects randomly and doesn’t telegraph his moves, the goalkeeper can do no better than guessing. Therefore, the goalkeeper’s equilibrium response is to also mix it up, diving in each direction often enough to keep the kicker from exploiting his tendencies. This is why you often see wildly inaccurate dives: It’s not necessarily because they were faked out, it’s just that they picked scissors when the striker picked rock.
And, of course, the same principles apply in much more complicated scenarios, like run/pass balance in football. If teams pass too often, defenses will exploit it by keying against the pass. If defenses key against the pass too much, offenses will exploit it by running more. Indeed, a key insight of game theory is that how you balance the different options in an “optimal” (meaning equilibrium) strategy isn’t just a matter of how good each option seems in a vacuum; it matters how your opponent will adapt to your strategy overall. For example, it doesn’t matter if you’re theoretically “better” at passing than running: If your opponent is defending optimally, you should be indifferent between the two. Thus your “optimal” balance between the two should actually be a matter of ensuring that the defense has nothing to exploit. (Of course, if the defense isn’t defending optimally, you should do more of whichever gives you the best results.)
Let’s use a concrete example: Should the Seattle Seahawks have run or passed at the end of their ill-fated Super Bowl drive? The game theory-savvy answer is basically “neither”: They should use whatever strategy gives the Patriots the maximum headache on defense — likely a mix of passes and runs.
Another situation relevant to sports headlines today is the recent discussions and controversies over how valuable 3-point shots are in basketball and the viability of strategies like Houston Rockets GM Daryl Morey’s (in a nutshell: abandon midrange shooting). While the math suggests 3-point shots are underutilized, if teams shoot more and more from beyond the arc, defenses should adjust to defend those shots. In equilibrium, teams should be indifferent (on average) between 3-point and midrange shooting, and the idea that 3-point shots are intrinsically more valuable than long jumpers can only be true if defenses are literally incapable of diverting any more resources to their defense. From a game-theoretical perspective, underutilized and underdefended are basically the same thing.
Similar questions and scenarios come up in baseball, hockey, tennis and virtually every other sport. Indeed, once you get in this mode of thinking, you start seeing it everywhere (much like with Bayesian inference, or Tetris).
In 1994, for his contributions to the field of game theory, Nash received the Nobel Prize in economics. But his greatest accomplishment may be the role he played in the emergence of a whole new and important way of thinking about the world and the things that happen in it.
Rest in peace.