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Nash equilibrium's relevance

Source Jim Devine Date 02/04/11/19:53

In the article below, Varian explains Nash equilibrium. As an expert in game

theory, he points out that it's not a realistic prediction of how people

play most actual real-world games.

If so, why is Nash's equilibrium used for all sorts of things, such as

electricity regulation? (If I remember, the movie mentioned that.) Is it

that Nash equilibrium is basically a normative concept and that it's applied

to improve the efficiency of electricity regulation (or what not) rather

than to be an accurate description of the way the world works?

(BTW, it seems quite a major theme in orthodox economics -- which puts a lot

of value on a strong distinction between "normative" and "positive"

economics -- that normative and positive matters are all mixed up in an

ideologically convenient way. For example, the famous Walrasian

(Arrow-Debreu) model is really a normative model, since it is based on all

sorts of unrealistic assumptions, including the existence of God (the

Auctioneer). But then that model is applied to describe -- and worse,

prescribe -- real-world matters.)

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New York TIMES/April 11, 2002

What, Exactly, Was on John Nash's Beautiful Mind?

By HAL R. VARIAN

So what did John Nash actually do? Viewers of the Oscar-winning film "A

Beautiful Mind" might come away thinking he devised a new strategy to pick

up girls.

Mr. Nash's contribution was far more important than the somewhat contrived

analysis about whether or not to approach the most beautiful girl in the

bar.

What he discovered was a way to predict the outcome of virtually any kind of

strategic interaction. Today, the idea of a "Nash equilibrium" is a central

concept in game theory.

Modern game theory was developed by the great mathematician John von Neumann

in the mid-1940's. His goal was to understand the general logic of strategic

interaction, from military battles to price wars.

Von Neumann, working with the economist Oscar Morgenstern, established a

general way to represent games mathematically and offered a systematic

treatment of games in which the players' interests were diametrically

opposed. Games of this sort - zero-sum games - are common in sporting events

and parlor games.

But most games of interest to economists are non-zero sum. When one person

engages in voluntary trade with another, both are typically made better off.

Although von Neumann and Morgenstern tried to analyze games of this sort,

their analysis was not as satisfactory as that of the zero-sum games.

Furthermore, the tools they used to analyze these two classes of games were

completely different.

Mr. Nash came up with a much better way to look at non-zero-sum games. His

method also had the advantage that it was equivalent to the von

Neumann-Morgenstern analysis if the game happened to be zero sum.

What Mr. Nash recognized was that in any sort of strategic interaction, the

best choice for any single player depends critically on his beliefs about

what the other players might do. Mr. Nash proposed that we look for outcomes

where each player is making an optimal choice, given the choices the other

players are making. This is what is now known as a Nash equilibrium.

At a Nash equilibrium, it is reasonable for each player to believe that all

other players are playing optimally - since these beliefs are actually

confirmed by the choices each player makes.

It's a nice theory. But is it true? Does it describe actual behavior in

actual games?

Well, no. Game theory is an idealization: it analyzes how "fully rational"

players should play if they all know they are playing against other fully

rational players.

That assumption of "full rationality" is the problem with game theory. In

real life, most people - even economists - are not fully rational.

Consider a simple example: several players are each asked to pick a number

ranging from zero to 100. The player who comes closest to the number that is

half the average of what everyone else says wins a prize. Before you read

further, think about what number you would choose.

Now consider the game theorist's analysis. If everyone is equally rational,

everyone should pick the same number. But there is only one number that is

equal to half of itself - zero.

This analysis is logical, but it isn't a good description of how real people

behave when they play this game: almost no one chooses zero.

But it's not as if the Nash equilibrium never works. Sometimes it works

quite well. Two economists, Jacob Goeree and Charles Holt, recently

published a clever article, "Ten Little Treasures of Game Theory and Ten

Intuitive Contradictions," that exhibits a number of games in which the Nash

theory works well, and then show that what should be an inconsequential

change to the payoffs can result in a large change in behavior.

In their simplest example, two players, whom we will call Jacob and Charles,

independently and simultaneously choose an amount from 180 cents to 300

cents. Both players are paid the lower of the two amounts, and some amount R

(greater than 1) is transferred from the player who chooses the larger

amount to the player who chooses the smaller one. If they both pick the same

number, they both are paid that amount, but no transfer is made. So if Jacob

chooses 200 and Charles chooses 220, the payoff to Jacob is 200+R and the

payoff to Charles is 200-R.

If Jacob thinks Charles will say 200, then Jacob will want to announce 199.

But if Charles thinks Jacob will announce 199, then Charles should say 198.

And so on. The only consistent pair of beliefs is when each thinks the other

will say 180.

When Mr. Goeree and Mr. Holt performed this experiment with R=180, nearly 80

percent of the subjects picked 180, which is the Nash prediction. When they

set R=5, and reran the experiment (with different subjects), however, the

outcomes were completely reversed, with nearly 80 percent choosing 300.

Findings of this sort have stimulated the development of "behavioral game

theory," which tries to formulate a theory of how to understand games

involving real people, rather than those mythical "fully rational" people.

Consider, for example, the "guess half the average" game described earlier.

Oscar, a simpleminded player, might think that any number between zero and

100 is equally likely, so he would guess 50. Emmy, who is more

sophisticated, might figure that if lots of people were like Oscar and say

50, then she should say 25. Tony, who is yet more sophisticated, figures

that if lots of people think like Emmy, then he should say 12 or 13. And so

on.

An economist named Rosmarie Nagel ran a game like this a few years ago and

found that the choices do tend to cluster around 50, 25 and 12.

In fact, the winning choice turned out to be close to 13, a number chosen by

about 30 percent of the players. In this game the best strategy wasn't the

Nash equilibrium, but it wasn't so far away from it either.

Back to picking up girls. In the movie, the fictional John Nash described a

strategy for his male drinking buddies, but didn't look at the game from the

woman's perspective, a mistake no game theorist would ever make. A female

economist I know once told me that when men tried to pick her up, the first

question she asked was: "Are you a turkey?" She usually got one of three

answers: "Yes," "No," and "Gobble-gobble." She said the last group was the

most interesting by far. Go figure.

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