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
