Via Marginal Revolution, here is a report on an experiment wherein top chess players played a textbook example of a game in which “rational” play is never matched in practice.  6000 chess players picked a number between 0 and 100.  The winner was the player whose guess was closest to 2/3 of the average.  The winner earns his guess in cash.

Nash equilibrium, or even iterative elimination of dominated strategies implies that no player will guess more than 1.  (Nobody should guess more than 66, but then nobody should guess more than 44, but then …)However, in experimental trials, the winning guess is usually around 25.

Most experiments involve volunteers at Universities.  Would professional chess players, being generally smarter and also trained to think strategically do “better?”  Well, they didn’t. But let’s look at it more carefully.

Casual discussion of the predictions of game theory usually blur an important distinction:  between playing rationally and knowing that others will play rationally. To be rational and make smart decisions is one thing, and no doubt the chess players are better at this than college students.  But that doesn’t go very far because to make a rational guess just means starting with some hypothesis about how others will guess and then guess 2/3 of the average of that. What really drives a wedge between the theory and the experiments is that experimental subjects have good reason to doubt that the others are rational.

Even a rational player in the beauty contest experiment will not guess anything close to zero if he is not convinced that all of the other players are rational.  For example, guessing 33 is rational if you think that most of the other players are not rational and on average they will guess the midpoint of 50.

And it is not enough just to know that everyone else is rational.  If you know that everyone else is rational but you are not convinced that everyone else knows that everyone is rational then you would reasonably predict that everyone else will guess 33 and so you should guess 22.

As long as there is some doubt that others have some doubt that others have … that everyone is rational, then even a rational player will guess something far from 0.

To see the effect of this in action, suppose that 100 subjects are playing.

• 10 of them are not rational and will guess 100,
• 10 are rational (but don’t know that others are rational) and guess 66,
• everyone else is as sophisticated as you wish

Then the average guess cannot be less than (100 + 66 )/10 = about 17, and so the winning guess will be no smaller than 11. And since the winning guess will be no smaller than 11, the highly sophisticated players will not guess less than 10.  But then this means that the average guess cannot be less than (100 + 66 + 88)/10 = about 25, yielding a winning guess of no more than 16!  The iterated reasoning is going in the opposite direction now!

Ultimately, the 80 highly sophisticated players will guess the value x that solves

(100 + 66 + 8x)/10 = 3x/2

which is about 23. (The winning guess in the experiment involving chess players was 21.5)