Alain Ledoux, who was one of over 6,000 chess players taking part in Bühren and Frank´s (2012) online Beauty Contest experiment, turned out to be the forgotten inventor of that game. We reconstruct the birth of the Beauty Contest. In section 1 of our note, its first two authors outline the history of the game that metamorphosed into the famous guessing game experiment which was first run in the lab by Rosemarie Nagel. In section 2, Rosemarie Nagel adds further remarks and thoughts about the development of the experimental Beauty Contest.

Björn Frank

Christoph Bühren

Rosemarie Nagel

English

Research Papers in Economics

 
 
Joint Discussion Paper 
Series in Economics 
by the Universities of 
Aachen · Gießen · Göttingen 
 Kassel · Marburg · Siegen 
ISSN 1867-3678 
 
 
 
No. 11-2012 
 
 
Christoph Büren, Björn Frank and Rosemarie Nagel 
 
 
 
 
A Historical Note on the Beauty Contest 
 
 
 
 
 
 
 
 
 
 
This paper can be downloaded from 
https://www.uni-marburg.de/fb02/makro/forschung/magkspapers 
 
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A Historical Note on the Beauty Contest 
by Christoph Bühren
, Björn Frank
♦
and Rosemarie Nagel
 (University of Kassel) 
∗
Updated version: February 20th, 2012 
 (ICREA, BGSE, Universitat Pompeu Fabra) 
 
 
 
Abstract 
Alain Ledoux, who was one of over 6,000 chess players taking part in Bühren and Frank´s 
(2012) online Beauty Contest experiment, turned out to be the forgotten inventor of that 
game. We reconstruct the birth of the Beauty Contest. In section 1 of our note, its first two 
authors outline the history of the game that metamorphosed into the famous guessing game 
experiment which was first run in the lab by Rosemarie Nagel. In section 2, Rosemarie Nagel 
adds further remarks and thoughts about the development of the experimental Beauty Contest. 
 
                                                           
 c.buehren@uni-kassel.de, University of Kassel, Institute of Economics, Nora-Platiel-Straße 4, 34127 Kassel, 
Germany, Tel.: +49-561-804-7267, Fax: +49-561-804-3083 
♦
frank@uni-kassel.de  
∗ rosemarie.nagel@upf.edu  
2 
 
1. The History of the Beauty Contest Experiment 
Rosemarie Nagel (1993, 1995) rightfully became known for the Beauty Contest experiment 
that she initially called guessing game. Ho, Camerer and Weigelt (1996, 1998) were the first 
to call this guessing game “p-Beauty Contest”, inspired by Nagel´s reference in 1995 to 
Keynes'  (1936, p. 156) famous comparison of stock market investments and newspaper 
beauty contests. The Beauty Contest is an import tool as it provides researchers with a clear 
and feasible concept of "depth of reasoning". It is Nagel's achievement to have discovered this 
potential of the game, she does not claim to be its inventor - in the initial footnote (Nagel, 
1995, p. 1313) she writes: "I learned about the guessing game in a game-theory class given by 
Roger Guesnerie, who used the game as a demonstration experiment." Probably Guesnerie's 
source was Hervé Moulin (1986) who is the first who published this game in a social science 
context. In his textbook, the game was called "Guess the average", with each player picking 
an integer between 1 and 999. The game served as the introductory example to the chapter on 
successive elimination of dominated strategies. Moulin on his part was inspired by a source 
that had slipped his memory when later, in the nineties, consulted by Rosemarie Nagel. Our 
recent call for participation in an online Beauty Contest experiment 
 (Bühren and Frank 2012) 
finally unearthed Moulin's source.  
As a reminder: Participants in Nagel's first experiments (as in Guesnerie’s class) were asked 
to guess a (real) number between 0 and 100, the winning number being the one that comes 
closest to a share p of the average. In Nagel (1995), p takes the values 1/2, 2/3 and 4/3, with 
p = 2/3 being most popular in later replications and variants of the experiment.  
In 1981, the French magazine "Jeux & Stratégie", a popular magazine devoted mainly to 
strategic board games, but also covering card games and mathematical games, arranged a big 
readers' competition consisting of mathematical puzzles but also problems from games such 
as chess, bridge and go. Ledoux (1981) reports on almost 15,000 participants, 4,078 of them 
being ex aequo, hence the winner had to be decided in a playoff. All first round winners 
received a letter with new puzzles, and to avoid another round with multiple winners, chief 
editor Alain Ledoux invented in the last question of this letter what is today known as the 
Beauty Contest (the name given to it by Ledoux, according to an email to us from July 9
th, 
2009,  was “psycho-statistique”, although this does not appear to have appeared in print). 
Readers were asked to state an integer between 1 and 1,000,000,000, the wining number 
being the one closest to two third of the average! The average turned out to be  
3 
 
134,822,738.26, two third of this being 89,881,825.51. This is 8.99 percent of the maximum 
number, markedly less than what is typically found in first rounds of Beauty Contest 
experiments (Bosch-Domènech, Montalvo, Nagel and Satorra, 2002). However, as explained 
above, the participants had been pre-selected, having solved a series of puzzles in the first 
round of the contest, and they knew that everyone else was pre-selected. Both facts should 
have resulted in the pretty high depth of reasoning. The large interval (with 1,000,000,000 
instead of 100 as the upper bound) could also have played a role - an untested hypothesis 
suggested to us by Rosemarie Nagel. 
In 1983, Jeux & Stratégie held another readers' contest. There was a reference to the first 
experiment but not to its result. However, some readers might have remembered or looked up 
the result of the previous round. Again, the Beauty Contest was used as a tie break between 
equal players on the previous questions - 2898 participants if or reading of Ledoux (1983) is 
correct. The target number in 1983 (two third of the average guess) was 67,329,453, or 6.73 
percent of the maximum number. 
Our preliminary conclusion is that Alain Ledoux should be given some credit for starting this 
fascinating line of research, though no one is to blame for the fact that researchers have 
previously overlooked him. He did not even sign the articles in Jeux & Stratégie with his 
name, and not a single public or scientific library in Germany holds this journal. We would be 
glad to receive hints on similar cases already known; the best one that occurred to us so far is 
the  balanced budget multiplier,  typically credited to Haavelmo, but see Gelting  (1941), 
written in a language alien to almost any economist.
1
 
 
                                                           
1 See also Al Roth in his blog on “Nagel's guessing/beauty contest game: a famous experiment in game 
theory” where he quotes from Roth and Sotomayor (1990, p.170): "What is important about Columbus' 
discovery of America is not that it was the first, but that it was the last. After Columbus, America was never lost 
again."   
4 
 
2. Some Further Remarks by Rosemarie Nagel
2
1.  In November 1990 in the LSE Tore Ellingsen presented a one shot guessing game in 
his master student IO class. I chose 22, according to 50*2/3*2/3. I did not further think 
about this game.  
  
2.  Shortly thereafter I saw the game again in Gueneries’ Phd game theory class, like 
some other students who were also in Tore’s class. I chose a number a bit lower than 
in the other class and won. Then Guenerie asked us to play again. And again I won. I 
also saw the choices and saw some numbers near and at 33 and 22. I asked Guenerie 
for the data set and analyzed it, and I also did a pilot in LSE. …. I find this important 
to add because it shows that sometimes you need to have been your own subject to see 
the beauty of some idea. Remember, the game was around for some years, but nobody 
thought it was interesting to do an experiment. First of all at that time grade students 
were not interested/educated in doing experimental economics. And the theorists only 
saw that behaviour is not in equilibrium, which is of course trivial. I needed two inputs 
of classes to see its beauty. At the time I was already a researcher-phd student in 
experimental economics and searching for a topic far away from ultimatum games 
which I had done in my first paper.  Guenerie used this game as a demonstration 
experiment to show that rationalizability doesn’t work in this game while I used this 
game to test the 50*p
n model, which I had actually used in Tore Ellingsen's class. 
However, I needed this second class by Guenerie to appreciate the game. 
3.  One Phd student, Klaus Kultti, like me in LSE in 1991, now Helsinki University, 
pointed out that the game (maybe also my reasoning process) is due to Keynes beauty 
contest. 
4.  Reinhard  Selten, my supervisor,  was against the name beauty contest  as Keynes´ 
contest has multiple equilibria  and so we named it guessing game. (John Duffy 
insisted to call it Keynes beauty contest in Duffy and Nagel (EJ 1997)), which indeed 
is also correct since it is just a special case with the parameter equal 1 (1*average).  
5.  In Pittsburgh in 1995  Oliver  Schulte  a graduate student in Computer science and 
philosophy in Carnegie Mellon  (now computer science prof.) found the guess the 
average game in Moulin’s book which Oliver used for teaching purposes.  
                                                           
2 See also Coricelli and Nagel (2010) for a personal account on the guessing game history.   
5 
 
6.  I saw in 1995 in a public choice conference (I think) Herve Moulin and told him that I 
had done experiments in his game. He then referred me to Pour la Science (French 
version of Scientific American) as his original source, which now actually turned out 
to be Jeux & Stratégie.  
Now finally the game has found  his founding father, who, however, saw no special 
interest in it besides providing a tie breaker.  
 
  
6 
 
References 
Bosch-Domènech, Antoni, José G. Montalvo, Rosemarie Nagel and Albert Satorra (2002), 
One, Two, (Three), Infinity, ... : Newspaper and Lab Beauty-Contest Experiments, American 
Economic Review 92, 1687-1701 
Bühren, C. and B. Frank (2012), Chess Players’ Performance Beyond 64 Squares: A Case 
Study on the Limitations of Cognitive Abilities Transfer, Talent Development and Excellence, 
forthcoming. 
Coricelli, Giorgio and Rosemarie Nagel (2010), Walking with Reinhard Selten and the 
Guessing Game: From the Origin to the Brain of the Guessing Game, in The Selten School of 
Behavioral Economics A Collection of Essays in Honor of Reinhard Selten Ockenfels, Axel; 
Sadrieh, Abdolkarim (Eds.) 1st Edition, 2010, XV, Springer Verlag 
Duffy, John and Rosemarie Nagel (1997), On the Robustness of Behaviour in Experimental 
‘Beauty Contest’ Games, Economic Journal 107, 1684-1700 
Gelting, Jørgen (1941), Nogle Bemaerkninger om Finansieringen af offentlig Virksomhed, 
Nationaløkonomisk Tidsskrift 79, No. 5, 293-299  
Ho, Teck-Hua, Colin Camerer and Keith Weigelt (1996), Iterated Dominance and Iterated 
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Keynes, John M. (1936), The General Theory of Interest, Employment and Money, London: 
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[Ledoux, Alain] (1981), Concours résultats complets. Les victimes se sont plu à jouer le 14 
d'atout, Jeux & Stratégie 2, No. 10, 10-11 
[Ledoux, Alain] (1983), Concours J&S 83. On a retrouvé la stanta barbara!, Jeux & Stratégie 
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Moulin, Hervé (1986), Game Theory for Social Sciences, New York University Press  
7 
 
Nagel, Rosemarie (1993), Experimental Results on Interactive Competitive Guessing, 
Discussion Paper No. B-236, University of Bonn, April 1993 
Nagel, Rosemarie (1995), Unraveling in Guessing Games: An Experimental Study, American 
Economic Review 85, 1313-1326 
Roth, Alvin E. and Marilda A. Oliveira Sotomayor (1990), Two-Sided Matching. A Study in 
Game-Theoretic Modeling and Analysis, Cambridge University Press 
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Game Theory, http://marketdesigner.blogspot.com/2009/09/nagels-guessingbeauty-contest-
game.html, retrieved February 20
th, 2012 
 
 

Rosemarie Nagel and Albert Satorra

Walking with Reinhard Selten and the Guessing Game: From the Origin to the Brain of the Guessing Game, in The Selten School of Behavioral Economics A Collection of Essays in Honor of Reinhard Selten Ockenfels, Axel; Sadrieh, Abdolkarim (Eds.) 1st Edition,

A Historical Note on the Beauty Contest

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A Historical Note on the Beauty Contest

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