Experimental Game 1
The p-Beauty Contest

Many results of game theory assume both the rationality of all players, and the players' common knowledge of this rationality.

The p-beauty contest game tests the assumption of common knowledge of rationality. In this game a player has to guess what the average choice is going to be and the player will win if his choice is closest to some fraction of the average choice.

The reason it is called a beauty contest game is because of the strong resemblance it bears to the newspaper beauty contests discussed by John Maynard Keynes (a famous English economist) some 60 years ago - a collection of women's faces would be printed in the newspaper, and the object was to pick the ones that would be rated "prettiest." So one should not vote for the faces he or she truly thought were prettiest, or even the ones he or she thought most people would think were prettiest, but the ones that one thought most people would think that most people would think were the prettiest, or even at higher orders.

In equilibrium all players have to choose zero. A rational player does not simply choose a random number or his favorite number, nor does he choose a number above 75=(¾)100, since the winning number can never be higher than 75. Moreover, if he believes that the others are rational as well, he will realize that no other player will pick a number above 75. Hence, he will not pick a number above (¾)²100=56.25. If he believes that the others are rational and that they also believe that all are rational, he will not pick a number above (¾)³100=42.1875, and so on, until all numbers but zero are eliminated.

The equilibrium of this game demonstrates a process of unraveling. Similar unraveling may be found in earlier and earlier early-decision deadlines for college applications and earlier search for job market candidates (in the first year of law school, for example).

Most actual players do not behave according to this solution. There are two explanations: First, a person might not be so clever, and only reasons one or two levels deep into the iterated process. The other possibility is that a person fully realizes the equilibrium of the game but does not believe that other players are equally clever. Either of these has the same implication: in reality, people are not fully rational. Instead, people are boundedly rational, which simply means that they, in general, understand the thought process described by game theory, but are unable to carry it out an infinite number of steps.

Because a player did not pick 0, we cannot infer that the player is irrational. It can only be inferred that rationality is not common knowledge. After all, a person who selected 0 would not win. Without common knowledge, any number selected below 75 can be justified. In other words, a rational person can pick a number greater than zero, but only if she believes some in the group to be irrational.

Notice that to win at the p-beauty contest, you have to be just one step more clever than the average person, but not too clever. If you are too clever, you will select a number too low. Think of a seller in the stock market. He wants to sell his shares just before the average person is selling, when the price of the share is at its peak. Therefore, he does not want to sell it too early. As a consequence if everybody thinks like him, the selling time is unraveling. Fortunately, most other investors are boundedly rational, and therefore unraveling does not occur. Therefore, you need to be one step ahead of the average investor, selling just before they do.