The Triumph and Failure of John Nash’s Game Theory
newyorker.com
newyorker.com
Hollywood schmollywood.
From The Dream Machine, Waldrop's account of the history of computing:
"Von Neumann! Goldstine was awestruck [. . .] he already knew the legends. At age forty, John Von Neumann held a place in mathematics that could be compared only to that of Albert Einstein in physics. In the single year of 1927, for example, while still a mere instructor at the University of Berlin, von Neumann had put the newly emerging theory of quantum mechanics on a rigorous mathematical footing; established new links between formal logical systems and the foundations of mathematics; and created a whole new branch of mathematics known as game theory"
[edit: still liked the article]
Oh and how whilst at the RAND organisation he advocated a pre-emptive nuclear strike on the USSR. Stick that in your Nash equilibrium!
Yeah, Von Neumann is not exactly unappreciated, "Von Neumann machine," etc.
Pretty good article, he says Nash equilibrium is hard to calculate, not really true, it's one of those things we do automatically, soccer players do it in picking how to shoot penalties, pigs have been found to do it, it's not like you have to think to drive on the right side of the road.
If it's a game you haven't seen before you might not get it right the first time, but if you look at what other players do and can figure out best response, people figure it out and converge to a Nash equilibrium pretty quickly.
It's one of those things that seems pretty simple and it takes a scientist to formalize it.
What's harder is living your life so you create positive-sum games, but that's another story.
In the late 1970s, I decided to move from mathematics to business and/or public policy. So I wrote a thesis in game theory, extending von Neumann's MinMax Theorem, about a year after learning what the MinMax Theorem even was. So it's safe to say the mathematics of game theory was a bit thin back in those days, and I haven't heard of great progress in the intervening decades either.
As for applicability -- once again, the author got it right.
I'm very happy my parents were subscribers and belive the avoricious devouring of each weekly issue helped me develop a rounded and intelligent world view.
Now I'm a happy subscriber myself!
> This is just one of the drawbacks of Nash’s approach. Another problem is that game theory is mentally taxing.
I find those depictions a bit skewed. Nash equilibrium is kind of like Thermodynamic Equilibrium, and that's fine. Why does it have to be a one-stop-solve-all?
Mentally tasking: so what? That's just what assuming that people are superrational means. You must expect assumptions to be made when people behaviour is being studied. I would think "Behavioural Economics" must do some complexity theory pushed down there to see how decisions vs. optimality decision making would interact and finally crack the whole cognitive-economy.
They do give lip service to the fact that an economic theory could not possibly predict the future exactly, but that's not even the point. Economic theory contributions must be valued in more than one way, and it is my impression that Nash equilibrium is specifically about stable-points in a field of possible transitions.
I am not sure there is an equivalent Thermodynamic problem.
In Economics, it is typical to search for stricter conditions for equilibrium than Nash equilibrium, which avoid the issues of non-uniqueness and other issues. For example, when designing an auction, one might look for an auction with a "dominant strategy equilibrium". In such an equilibrium, there is a best strategy for a given player no matter what the other players do. E.g. in a second price auction, you should always bet your true valuation of the object, no matter what the strategies of the other players are.
a good remark of Myerson: non cooperative game theory can roughly be define as a practical calculus of incentives, which came into existence all because of Nash.
http://www.uib.cat/depart/deeweb/pdi/hdeelbm0/arxius_decisio...
In noncooperative game theory, a player is supposed to be rational and intelligent. Being rational at least means that player doesn't use a dominated strategy (like in one-shot PD, cooperation is dominated by defection). Being intelligent means that player can bet in the rationality of opponents (which means one knows that the other will not play a dominated strategy ~ cooperation).
-> Just these two conditions can solve the Prisoner's Dilemma. However, in other games, after iterating away all the dominated strategies, player needs a stronger notion of rationality: player maximizes expected utility given accurate prediction about opponent's strategy. As before, with the intelligent assumption standing, both players' keen consistency in their action will lead to Nash equilibrium (NE).
NE is a broad and fundamental concept. It's broad because it bases only on belief, not necessarily on what's real. Given the convergence of action and belief in equilibrium, what wasn't justified before will become justified. However, too broad is also its inherent issue because game theory loses its predictive power in many games by dictating too many possible Nash equilibria. Speaking of that, this concept is still so fundamental that it becomes the unit test for any kind of stricter equilibrium concept that one would like to build.
-> Many giants have tried to make major refinements built on top of NE. But along this literature, the further they go, the heavier the burden of rationality assumptions on modelling decision makers becomes. Player has to bear an exhaustive rationality in analysing the game in all the nodes that may never happen, or in face of background noise, or in thinking from future back to present.. Even making mistake needs to be justified. That requires ability to do complex optimization and infinite recursive thinking of common knowledge. So they say, what is the point of a theory starting with assumptions about players that are not even close to the statistically real truth? And if the collective reality doesn't matter, what matters?
-> Until now the narrative has substantially shifted from these ideal players to less demanding subjects. Just as applied economics of behavioral, experimental, neuro-economics trying to figure out a reasonable approach to modelling decision makers by adding cognitive and psychological knowledge, theorists bring evolution into game theory. Starting from an amusing observation that nature doesn't assume any rationality of its creatures but its solution many times becomes our definition of perfect, evolutionary game theory (EGT) offers a less pervasive way of thinking into games by replacing strict assumptions by adaptive learning and letting the itera- tive selection process do its job. Surprisingly, sometimes it gives us the perfect equilibrium just like in the classical regime.