Nashevo ravnotežje (ključni prispevek brilijantnega uma) za neekonomiste

Širša javnost Johna F. Nasha, nobelovca za ekonomijo, ki je prejšnjo soboto tragično preminil v prometni nesreči v starosti 86 let, večinoma pozna po filmu “A Beautiful mind” z Russellom Croweom. Zelo malo pa se jih zaveda njegovega brilijantnega prispevka k teoretični ekonomiji oziroma natančneje k teoriji iger. Večini je njegov prispevek zaradi visoke matematike tudi preveč abstrakten, da bi ga lahko razumela. Toda Nashev prispevek je široko uporaben na zelo različnih področjih, kjer imamo opravka s kompleksnimi situacijami. Nash je pokazal, da je tudi v kompleksnih situacijah, kjer imamo opravka s številnimi subjekti in njihovimi individualnimi strategijami, mogoče najti točke (“ravnotežja”), ki predstavljajo najboljši izid za vse udeležence v danem trenutku (t.i. Nashevo ravnotežje). V Nashevem ravnotežju ste, če s spremembo vaše strategije in ob upoštevanju strategije vašega nasprotnika ali partnerja kot fiksne v danem trenutku, ne morete izboljšati svojega položaja in če enako velja za nasprotno stran.

No, slaba novica pri tem je, da to ravnotežje ni samo eno, da jih je možnih neskončno mnogo glede na dane okoliščine in izbrane strategije in da nam teorija ne more povedati, katero je pravo. Dobra novica pa je, da nam teorija iger omogoča vsaj izključiti številne možnosti, kjer skupinskega ravnotežja sploh ni mogoče doseči.

Spodaj je izsek iz zelo poljudnega (za neekonomiste) in zelo dobrega opisa Nashevega prispevka izpod peresa Johna Cassidyja v New Yorkerju. (Mimogrede, ob tem se vedno znova spomnim, kako odlični novinarji obstajajo, ki znajo tako kompleksne zadeve tako poljudno in razumljivo popisati. Kapo dol!)

On one level, Nash’s contribution to game theory was highly mathematical, and, ultimately, somewhat trivial. That is how his intellectual rival at Princeton, John von Neumann, reputedly described it back in 1949, anyway, and he had a point. In co-authoring the 1944 magnum opus “Theory of Games and Economic Behavior,” von Neumann had virtually invented a new subject, complete with its own language. Nash, in diverting from his studies in pure mathematics to this nascent field, showed that in a certain class of games a certain set of outcomes exists: those outcomes are now called “Nash equilibria.”

Enter von Neumann and his co-author, Oskar Morgenstern, who provided an intellectual framework for analyzing such situations: game theory. But despite the monumental nature of their achievement, von Neumann and Morgenstern succeeded in showing that definitive solutions, or “equilibria,” existed for only a fairly narrow category of interactions: so-called zero-sum games, in which one person’s gain is another person’s loss. (Poker is a zero-sum game; so is coin tossing.) Often in real-world situations, though, such as how to divide a market among a few competitors, there is a positive economic surplus to be divided: the question is who gets what, and that depends on which actions (or “strategies”) are adopted.

This is where Nash came in. He started out by defining a particular solution to games—one marked by the fact that each player is making out the best he or she (or it) possibly can, given the strategies being employed by all of the other players. Then, applying a deep-mathematical theory that had been developed earlier by the Dutch mathematician L. E. J. Brouwer, Nash demonstrated that such an equilibrium exists in any game with a finite number of players and a finite number of moves to choose from.

The reason is its broad applicability, which extends well beyond economics. Take, for example, the problem of deciding which side of the road to drive on—a question that clearly involves trying to figure out what everybody else will do. If you are living in the United States, where custom and law dictate using the right lane, sticking to that lane is a Nash equilibrium: it gives you the best chance of getting to your destination in one piece. And since the same logic applies to everybody else, the “stay on the right” solution is pretty stable.

Like the intersection of a supply curve and a demand curve, the concept of a Nash equilibrium appeared to pick out a distinct point where things would inevitably end up. Indeed, once you grasped the idea, it was hard to see how an outcome that wasn’t a Nash equilibrium could be sustained for very long in the absence of coercion or misinformation. If there were a better response available, such as driving down the median or zigzagging from left to right, at least some of the players would eventually adopt it. And that would mean that the original solution wasn’t a stable solution at all.

In the nineteen-sixties, seventies, and eighties, economic theorists worked on extending Nash’s approach. At the same time, however, it became clear that his concept of equilibrium has some serious drawbacks that limit its usefulness. To start with, there is often more than one best-response equilibrium, and, in some cases, there is a very large (or even infinite) number of them. For a methodology that is designed to pick out particular solutions, this non-uniqueness property is a serious problem, especially since there is usually no obvious way of deciding which Nash equilibrium will end up being selected.

To return to the driving example, a moment’s reflection should persuade you that driving on the left can also be a best-response equilibrium. In the United Kingdom and many other countries, it’s the one that has been adopted and enshrined into law. But why do Americans drive on the right and Brits on the left? And if we were starting out from scratch, which convention would be adopted: left or right? Nash, and the many economists who have followed in his footsteps over the past sixty-five years, can’t necessarily provide an answer.

The long and the short of it is that if the purpose of economic theories is to predict which of many possible outcomes will occur, Nash’s methodology often isn’t much help—a point acknowledged by David Kreps, an economic theorist at Stanford, back in 1990. But asking any theory in the social sciences to correctly predict the future is a very demanding requirement. And asking that it accomplish this task across a wide range of areas, such as the ones to which Nash’s approach has been applied, is surely too much.

That’s partly because Nash-influenced game theory isn’t actually a testable scientific theory at all. It is an intellectual tool—a way of organizing our thoughts systematically, applying them in a consistent manner, and ruling out errors. […] But while appealing to the Nash criteria doesn’t necessarily give the correct answer, it often rules out a lot of implausible ones, and it usually helps pin down the logic of the situation.

Vir: John Cassidy, The New Yorker

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