A Keynesian Beauty-Contest in the Classroom

by Rosemarie Nagel, Universitat Pompeu Fabra, Barcelona, Januar, 2000

Most models of economic behavior are based on the assumption of rationality of economic agents and common knowledge of rationality. This means that an agent selects a strategy that maximizes his utility believing that all others do the same (are equally rational) and that all agents believe that all others believe that all agents are rational etc.

The p-beauty contest game is an appropriate game to test the assumption of this kind of reasoning. 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. Think of a seller in the stock market. He wants to sell his shares just before the average person is selling, thus 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 is thinking like him, the selling time is unraveling. Unraveling can be seen in many real world markets, such as entry-level medical labor markets or clinical psychology internship as documented in Roth and Xing (1994). The name of the game is due to Keynes (1936, 156) who compared a clever investor to a participant in a newspaper beauty-contest where the aim was to guess the average preferred face among 100 photographs.

The experiment can be introduced in many different courses and at all levels of teaching. For example in game theory in order to discuss the issue of iterated elimination of (weakly) dominated strategies and common knowledge of rationality; in macroeconomics to discuss rational expectation; in microeconomics to discuss strategic interaction between players.

II. The rules of the basic beauty-contest game:

The rules of the game are straightforward. Elements of the rules (indicated by bold/italic) can in many ways (below we give some suggestions about different treatments):

Each person of N-players is asked to choose a (real or integer) number from the interval 0 to 100. The winner is the person whose choice is closest to p times the mean of the choices of all players (were p is for example 2/3). The winner gets a fixed prize of \$20. In case of a tie the prize is split amongst those who tie.

The same game may be repeated several periods. Subjects are informed of the mean, 2/3 mean and all choices in after each period.

The students should write down a brief comment how they came to their choice.

Time to think: about 5 minutes or as a take home task.

III. The game theoretic solution and the contrast to a bounded rationality model using the basic game

In equilibrium all players have to choose zero. Figure 1a. describes the process of iterated elimination of weakly dominated strategies for p=2/3. A rational player does not simply choose a random number or his favorite number, nor does he choose number above 100p, since it is dominated by 100p. Moreover, if he believes that the others are rational as well, he will not pick a number above 100p2, and if he believes that the others are rational and that they also believe that all are rational, he will not pick a number above 100p3 and so on, until all numbers but zero are eliminated. If the number of players is 2, 0 is a dominant strategy. There are no other equilibria in the (in) finitely repeated game. If p>1 then the upper bound of the interval is also an equilibrium. For p=1 any number chosen by all players can be an equilibrium.

Equilibrium ¬ ¾ ¾ ¾ ¾ ITERATION ¬ ¾ ¾ ¾ ¾

 ... ... E(4) E(3) E(2) E(1) E(0)

0 13.17 19.75 29.63 44.44 66.66 100

Figure 1a.) Infinite process of iterated elimination of dominated strategies for p=2/3-mean. E(0) is the area of dominated choices, E(1) is the area of one iteration of elimination and so on. Adapted with permission from Ho et al. (1998).

In contrast to the iterated elimination of dominated strategies, figure 1b shows another process, the process of iterated best reply, which explains better actual behavior. The elimination process does not start at 100 but instead at 50, because a player will, for insufficient reasoning, think that any number is equally likely; therefore the mean should be 50. Thus best reply is 2/3*50=33.33. If everybody thinks that way best reply should be 22.22 and so on.

Equilibrium ¬ ¾ ¾ ¾ ¾ ITERATION

 E(3) E(2) E(1)

0 14.9 22.22 33.33 50 100

Figure 1 b.) Infinite process of iterated elimination of best replies, starting at 50.

IV. Why is a study of human behavior with this game interesting?

1. There is a clear distinction between bounded rationality and game theoretic solution in the basic game. Game theory predicts a unique game theoretic solution that all players choose 0. Most actual players do not behave according to this solution. There are two explanations: Either because a subject chooses a strategy at random or just reasons 1 level using the hints of the mean and the parameter 2/3. Or he thinks that the others are not as clever himself and therefore does not reason very far himself.
2. This kind of boundedness does not arise as an outcome of motivational factors as fairness or cooperation, which typically explain the deviations from theory in other games. We are facing here a pure strategic game (constant some game) where there is no room for cooperation. The behavior can be interpreted as "pure bounded rationality".
3. The behavior can be categorized into different levels of reasoning via iterated best reply or iterated elimination of dominated strategies.

Each single aspect can be also found in other games but the combination of all three are not easily met at once in other games. Furthermore, it is very easy to change the rule of the game in such a way that the iteration process leads with a few or even infinitely many eliminations to equilibrium.

V. Results of actual behavior

a.)First period:

Figure2 a) lab-data. With permission from Nagel (1995)

Figure 2 b) game theorist data

Figures 2a-c. show the behavior in the first period of a.) lab-experiments with Bonn undergrad students; b.) Game theorists participating in that game in a conference; and c.) of experiments run in different newspapers. Note that the modal frequency in all games are around or near 22 and 33 in all treatments. Game theorists and newspaper readers also exhibit a large proportion of behavior at or near the equilibrium zero. Dominated choices are rarely chosen. One can show that a combination of theorist choices and lab-experiments produce the newspaper distribution. The following comment by a high school class (submitted to one of the newspaper studies, the Spektrum der Wissenschaft) summarizes the most important thought processes:

I would like to submit the proposal of a class grade 8e of the Felix-Klein- Gymnasium Goettingen for your game: 0.0228623. How did this value come up? Johanna …asked in the math-class whether we should not participate in this contest. The idea was accepted with great enthusiasm and lots of suggestions were made immediately. About half of the class wanted to submit their favorite numbers. To send one number for all, maybe one could take the average of all these numbers.

A first concern came from Ulfert, who stated that numbers greater than 66 2/3 had no chance to win. Sonja suggested to take 2/3 of the average. At that point it got too complicated to some students and the finding of the decision was postponed. In the next class Helena proposed to multiply 33 1/3 with 2/3 and again with 2/3. However, Ulfert disagreed, because starting like that one could multiply it again with 2/3. Others agreed with him that this process then could be continued. They tried and realized that the numbers became smaller and smaller. A lot of students gave up at that point, thinking that this way a solution could not be found. Other believed to have found the path of the solution: one just has to submit a very small number.

However, one could not agree how many of the people who participate realized this process. Johanna supposed that the people who read this newspaper are quite sophisticated. At the end of the class 7 to 8 students heatedly continued to discuss this problem. The next day the math teacher received the following message: .. We think it best to submit number 0.0228623.

b.) Over time:

Figure 3 shows the behavior over time in some selected treatments. In general behavior converges to equilibrium. The speed of convergence depends on the parameter used. Typically the reasoning process from period to period is anchored to the mean of the previous period. Level 1-reasoning is then 2/3 times the mean of the previous period, level 2-reasoning is 4/9 times the mean of previous period and so on. The average level of reasoning typically applied by subjects does not increase over time.

Figure 3: Mean behavior over time for different variations.

1. Further readings and treatment suggestions

Nagel (1995) discusses the basic game with parameters p=2/3, ½ and 4/3 played for 4 periods with about 15 students. Ho et al. (1998) discuss the game with group size 3 and 7, and intervals from [0,100] or [100,200] and several parameters. Thaler (1998) discusses his newspaper results in connection with financial economics. Nagel et al (1999) compare the behavior in experiments run in three newspapers, run with game theorists and lab experiments. Nagel (1998) presents a survey of beauty-contest experiments and analyses the case where the prize to the winner is his choice paid in dollars. Rubinstein (1999) lets his students play the basic game among other games on his webpage.

References:

Nagel, R. A. Bosch-Domènech, A. Satorra, and J. García-Montalvo (1999). "One, Two, (Three), Infinity: Newspaper and Lab Beauty-Contest Experiments" Univerisitat Pompeu Fabra working paper 438 (download http://www.econ.upf.es/cgi-bin/onepaper?438)

Ho, T., Camerer, C. and Weigelt, K. (1998). Iterated Dominance and Iterated Best-Response in Experimental ‘P-Beauty Contests’, American Economic Review 88, 4, 947-969.

Keynes, J.M. (1936). The General Theory of Interest, Employment and Money. London: Macmillan.

Nagel, R. (1995). "Unraveling in Guessing Games: An Experimental Study, American Economic Review," 85 (5), 1313-1326

Nagel, R. (1998). "A Survey on Experimental "Beauty-Contest Games:" Bounded Rationality and Learning," in Games and Human Behavior, Essays in Honor of Amnon Rapoport. Eds. D. Budescu, I. Erev, and R.Zwick. Publisher: Lawrence Erlbaum Associates, Inc., New Jersey (1998), 105-142.

Roth, A. and X. Xing (1994). Jumping the gun: Imperfections and Institutions Related to the Timing of Market Transactions. American Economic Review, 84,992-1044.

Rubinstein, A. (1999). "Experiments from a Course in Game Theory: Pre and Post-class Problem Sets as a Didactic Device." Games and Economic Behavior, 28 (1999), 155-170. (See also http://www.princeton.edu/~ariel/99/gt100.html)

Thaler, R. (1997). Giving Markets a Human Dimension. Financial Times, section Mastering Finance 6, June 16, 1997.