Six thousand chess players took part in a beauty contest!

By Christoph Bühren and Björn Frank, University of Kassel

During the past few weeks, over 6000 chess players took part in our online experiment announced on the ChessBase newspages. In the first round the participants guessed a number between 0 and 100. In order to win, this number had to be nearest to 2/3 of the average of all guesses.

Why is this game called Beauty Contest?

The name of this game and its basic idea go back to John Maynard Keynes (picture right, 1883-1946), one of the most influential economists who ever lived. Keynes is currently en vogue again, as he stands for an active, stabilizing role of the state. In his famous book "The General Theory of Employment, Interest, and Money" (1936), he explains why investors' and speculators' behaviour is hard to predict. The reason is that these people do not just face the task of picking the most promising projects or shares. Often their success depends on the number of other people thinking that a particular project will be successful, and those people think that the project will be successful if they believe many others to believe that... You see the point? In case you are not sure, Keynes offers the following metaphor:

"[P]rofessional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one's judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees."

This game has no rational solution, a fact that reflects Keynes' intentions well.

The modern "Beauty Contest"

However, in 1993, the German economist Rosemarie Nagel (picture left) based her experiments on a nice variant of the game that Keynes' newspaper readers play; it has a rational solution, yet still it offers the chance to investigate levels – or degrees – of thinking. This is the game we had invited chess players to play.

We want you to give us a number between 0 and 100 inclusively. The number need not be an integer. You have won if your number is closest to 2/3 of the average of the numbers given by all participants.

There are two ways to see that a unique solution exists. First, one can easily see that no one should submit a number higher than 66, because whatever the others do, a guess higher than 66 cannot be better than 66. However, if no one guesses more than 66, then all numbers between 44 (that is, two third of 66) and 66 are inferior to 44. Hence no one would guess more than 44. That, however, should eliminate any number higher than 2/3 of 44, etc., until only 0 is remaining as a reasonable guess. (Note that a second solution besides 0 is 1 if only integers are allowed).

A second method would be to just try out: Presume you guess a certain number x, and anyone else guesses the same number x, would you still wish to stick to your initial guess x? If so, you have found what game theorists call a Nash equilibrium (named after John Nash – yes, the one played by Russell Crowe in Beautiful Mind). Now the only x to which you would want to stick is 0.

However, as Reinhard Selten (who received the Nobel Prize together with Nash) has said: "game theory is for proving theorems, not for playing games". Guessing 0 only leads to success if you think that everyone is rational and fully understands the solution of the game. Empirically, this is not the case.

Previous results

The first round of Rosemarie Nagel's first Beauty contest experiment resulted in a target number (i.e., 2/3 of the average guess between 0-100) of 24.49. Playing the Beauty Contest over the Internet produces very similar results (24.13 in Rubinstein, Economic Journal 2007), while playing the game as a newspaper or magazine contest gives participants more time, and they think one step further ahead. (The target number of a magazine experiment by Selten and Nagel, Spektrum der Wissenschaft, February 1998, was 14.72; Richard Thaler in the Financial Times, June 16th, 1997 got 12.61).

Are chess players different from other humans?

Why would one think that good chess players are any better in the Beauty Contest than others? Psychological research has found that excellent chess players neither calculate markedly deeper than amateurs, nor do they calculate a larger number of moves. However, they make the more relevant calculations. They have stored a large number of patterns or "chunks" that allow them to focus on the moves that are really relevant. This ability is chess-specific, and one would not expect it to help much in the Beauty Contest. Yet Palacios-Huerta and Volij (2007) tested the rationality of chess players in an experiment that shares one particular feature of ours: it has a game-theoretic equilibrium that is usually never reached by “normal” people. In their experiment all grandmasters chose this equilibrium!

Our results

The target number of the ChessBase experiment turned out to be 21.4769 – two third of 32.21539. As pointed out above, the online environment might have played a role. Yet, a first impression is that chess players' guesses are within the range of other humans. The winning guesses came from Nick Burns, UK (21.473), Jarred Jason Neubronner, Singapore (21.463) and Tanner McNamara, USA (21.46). The latter two argued that they anticipated most of the participants to choose a number near 2/3 of the average of 0 and 100, that is 2/3*50=33.33. Therefore, they choose a number near 2/3*33.33. Nick Burns has asked various groups of people (e.g. mathematics classes) before answering our survey.

But what is the role of chess expertise? Indeed, guesses of better chess players are significantly lower. However, this relation is rather minuscule with regard to its extent: On average, chess players guess one integer lower if they have a rating that is 200 points higher according to our very preliminary results.

And what about the grandmasters, of whom we have 28 in our sample? While the average guess in our complete sample was 32.21539, the grandmasters' average was slightly higher: 32.96482!

One of us is a chess player (Elo 2220) while the other is not, hence we do not quite agree whether chess players are particularly clever. Did our experiment decide the issue? Only seemingly so. It might be that the strong chess players in our sample "saw" the theoretical solution, but that they presumed that the average participant makes a less sophisticated guess. Strong chess players might be particularly good in guessing other people's guesses, though this sounds more like poker than like chess. To take an example, if half of the untitled chess players would have guessed 0 and the other half 64, and if the grandmaster all would have guessed 33, then the average deviation from the winning number would be much lower for the grandmasters. However, this turned out not to be the case for our sample. And as far as the Elo rating is concerned, you are one integer nearer to the winning number if you have a rating that is 333 points higher – better chess players are closer to the winning number, but only by a tiny amount (note: this is a very preliminary result again).

Some further results

For round one, we also calculated the target numbers for different groups of chess players separated by their Elo rating. In our second round, we asked our participants to estimate these target numbers of round one (we wrote an email to every first round participant. If it did not reach you, argue with your spam filter):

Our initial hypothesis had been that Elo rating and Beauty Contest guess might be related. Now that we rubbed this hypothesis under the participants' noses, they presume that such a relationship exists, but not very strongly. The winning guesses for this task came from the Netherlands, Denmark, USA and India: Martijn Pauw, Torill Skytte, Jonathan Tayar and Manish Kashyap. In two categories the winners did not react to our emails so far – these days it is no easy task to convince people and their spam filters that they have really won something; a ChessBase voucher in our case.

And finally there was a second round, in which everyone played against players of his own rating group only. Everyone was informed about the target number of round 1 (21.4769). Hence, in the next round one should see a declining average guess, like in all other multiperiod Beauty Contests before. But how large is the decline? Do stronger chess players react stronger? No, quite the opposite is true. Comparing the first and the second round number, here is the size of what might be called a learning effect:

And the winners of this second round are... Reiner Odendahl, Uwe Stein, and Thomas Seelen from Germany, Mark Huizer and Regis Huc from the Netherlands, Matthew Tapp, UK, Michel de Vathaire, France , Mark Ryan, USA, Alexis Murillo Tsijli, Costa Rica and one chess player who has not replied to our emails yet.

Of course this is not the whole story, there is a lot more to do with the data. But now you know at least why we conducted this experiment. We thank all participants for giving us a lot of useful feedback about this experiment and the way they calculated their numbers. Many thanks also to ChessBase for support and cooperation, and to Alain Ledoux who turned out to be the predecessor of this game; click here if you are interested in details (in PDF).

Attention GM tournament organizers in Europe: If you want us to entertain your participants with further experiments, please contact us. We would gladly come to your tournament and pay real money to participating chess players.

Sources

The Selten quotation is from Goeree/Holt, Stochastic game theory, Proc. Natl. Acad. Sci 96 (1999), 10564-10567

The paper by Palacios-Huerta and Volij (2007) is entitled "Field Centipedes", for download here (PDF):

Nagel portrait courtesy of Rosemarie Nagel; Shop window dummies courtesy of Kanzkeu Dr. Palm

Prof. Dr. Björn Frank

University of Kassel, IVWL

Germany

To contact us: frank@uni-kassel.de c.buehren@uni-kassel.de