Nagel's guessing/beauty contest game: a famous experiment in game theory

Much of game theory concerns how ideally rational players should behave when they interact with other ideally rational players, when everyone's rationality and information is common knowledge. Practical market design, of course, has to also concern itself with how games will be played among humans, which is one of the reasons experimental game theory (and experimental economics generally) is such a useful tool, as a complement to game theory.
An important 1995 paper in the AER by Rosemarie Nagel (“Unraveling in Guessing Games: An Experimental Study") reported an experiment in which this tension was made very clear. In one form of the game, a large number of people are asked to pick a real number between 0 and 100, and the winner will be the person who picks the number closest to 2/3 of the median of all the numbers chosen.
This game shows off the tension between playing a game with perfectly rational players and playing the same game with a sample of humans, even if you are perfectly rational yourself. If all the players are perfectly rational, no one will ever choose a number other than 0, and this is the only equilibrium. (You can prove it this way: no matter how close the median number chosen is to 100, 2/3 of 100 will be closer to 2/3 of the median than will any higher number, so no rational player will ever choose a number larger than 66.66. So the median number chosen in the world of all perfectly rational players will never be higher than 66.66, and so 2/3 of the median will never be higher than 2/3 of 66.66, and so no rational player will choose a larger number, and so forth: in the world of all rational players, no one chooses any number larger than 0.
But of course, among humans, many people do choose numbers larger than 0, and so if you cleverly choose zero, you will know that you are smarter than they are, but you won't win the game. The game will be won by someone who chose a positive number nearest to 2/3 of the median. Maybe someone who chose a number near 2/3 of 66, or 2/3 of 2/3 of 66... And indeed that has been the case, with the modal number chosen dropping as players gain experience. Many experiments have followed that original 1995 experiment, some conducted in newspapers among thousands of participants. (Here are the slides I presented about that series of experiments in the first lecture of my Experimental Economics course this past Spring.) A recently reported experiment was among chess players, who turn out to be a lot like humans: Six thousand chess players took part in a beauty contest! By Christoph Bühren and Björn Frank, University of Kassel. (See also Jeff Ely's nice account here: Grandmasters Play the Beauty Contest Game)
One of the things that struck me about the report by Bühren and Frank was that they include A historical note on the Beauty Contest. They trace the game back to a 1981 French puzzle magazine, which, astonishingly, also seems to have run a large scale experiment among their readers:

"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 9th, 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 winning number being the one closest to two third of the average! The average turned out to be 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. 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 result in the pretty high depth of reasoning."


Scientists and scholars spend a lot of effort tracing back "first" discoveries, and this one is a great find. Nagel's original paper already referred to the fact that the great game theorist Hervé Moulin discussed the game in one of his books, and elaborated on the dominance solvable structure underlying the proof I sketched above. (It was this iterative-reasoning structure that Nagel's original experiment was designed to investigate, and the game has found wide use since then for that purpose, see e.g. the 2006 AER paper by the two (now) British economists Miguel Costa Gomes and Vincent Crawford, "Cognition and Behavior in Two-Person Guessing Games: An Experimental Study". The two-person games have a simplicity that lets alternative hypotheses be more easily separated, in this connection see also the 2008 GEB paper by Brit Grosskopf and Rosemarie Nagel, "The two-person beauty contest. )

Of course, it often happens that, once an important contribution is made and understood and disseminated and built upon, prior discoveries are uncovered. It's good to rediscover earlier attempts, that may have been forgotten because they were "before their time", i.e. because they didn't lead to a lasting scientific or cultural conversation when first proposed.

I once wrote about this as follows:
"Columbus is viewed as the discoverer of America, even though every school child knows that the Americas were inhabited when he arrived, and that he was not even the first to have made a round trip, having been preceded by Vikings and perhaps by others. 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..." (Roth and Sotomayor 1990, p. 170):

(In a similar vein, Jack Rosenthal, writing the On Language column in the NY Times Sunday Magazine, speaks of "...Stigler’s Law of Eponymy, which states that any scientific discovery named for someone is not in fact named for its actual discoverer. ...Stephen Stigler, the University of Chicago statistician who proposed the law in 1980, attributed it to the sociologist Robert Merton, who suggested something similar in 1968.")

Some final notes (for those of you who have followed to the end of this long post).

Stigler's Law might equally well be applied to the name of Nagel's experimental game. She initially called it a "guessing game," but also referred to Keynes' famous metaphor about investing in the stock market
""[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."

This isn't a perfect analogy to the guessing game; as Nagel suggested, if the goal were to pick the average number (instead of 2/3 of the average), the game would be as difficult for perfectly rational players as for humans, since there would be a continuum of equilibria. But the "beauty contest" name has stuck. As it happens, there's another important family of experimental games, introduced in a 1990 AER paper by John B. Van Huyck, Raymond C. Battalio, and Richard O. Beil, "Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure" that is much closer to Keynes' beauty contest.

The Nagel paper also has "unraveling" in the title, which refers to the sense in which the "p beauty contest game" with p less than 1 is a reduced form model of the kind of unravelling we see in labor markets and other kinds of markets, e.g. in which employers sometimes try to be earlier than their competitors, with a resulting race to the bottom.

Full (and proud) disclosure: back in the previous millenium, Rosemarie was a postdoctoral fellow of mine, at the University of Pittsburgh, 1994-5. She suggests I mention that her initial inclination to do a careful experiment on this game came from considering her own thought processes when playing the game, and that "one should participate in many experiments when being or wanting to become an experimenter..."