Economic Classroom Experiments/Guessing Game

Students are asked to pick a number between 0 and 100, with the winner of the contest being the student that is closest to 2/3 times the average number picked of all students.

Instructions

 * Guess a number 0 to 100.
 * The guess closest to 2/3 the average number wins a prize.
 * Ties will be broken randomly.
 * Please write your name and your guess on the piece of paper and turn it in to me.
 * Don’t let others see your guess!

After collecting responses, show the results, explain them, and rerun it.


 * Wharton average 40
 * Caltech UG average 30
 * Economics Phds 25
 * CEO average also 40.
 * What would we expect if everyone is rational and thinks everyone else is rational?
 * Now repeat this!

Results
One may want to start by asking the students what guess would be irrational (above 67). Then ask the students what they guessed and why. For instance, pick a student who guessed 33 and ask them why. It is then fairly easy to explain the equilibrium of everyone guessing 0 along the lines of Nagel (1995), who first ran an experiment on this game. Nagel found that people based their guesses on levels of rationality and found lumps of guesses on: level 0 rationality (guessing 50); level 1 rationality, best response to 50 (guessing 33); level 2 rationality, best response to 33 (guessing 22); etc. One can then say that equilibrium theory doesn't necessarily do well.

After explaining the equilibrium and showing the initial results (or results from last year), one can now rerun it. Results will come now much much closer to the equilibrium.

Discussion
Moral of the Story
 * Sometimes markets aren’t always rational or in “equilibrium”. This Guessing game is like a market bubble.
 * Sometimes theory based on rationality doesn’t predict perfectly.
 * You shouldn’t ignore theory (even when it is wrong). Otherwise, it may catch you expectedly.
 * Star Trek: Kirk had several advisors with different views of the world. Spock used logic. Bones emotions. Scotty practicality. When analyzing a problem, it is worthwhile, like Captain Kirk, to take into account several views including the fully rational one.

Keynesian beauty contest
A Keynesian beauty contest is a concept developed by John Maynard Keynes and introduced in Chapter 12 of his masterwork, General Theory of Employment Interest and Money (1936), to explain price fluctuations in equity markets. Keynes described the action of rational agents in a market using an analogy based on a contest that was run by a London newspaper where entrants were asked to choose a set of six faces from 100 photographs of women that were the "most beautiful". Everyone who picked the most popular face was entered into a raffle for a prize.

A naive strategy would be to choose the six faces that, in the opinion of the entrant, are the most beautiful. A more sophisticated contest entrant, wishing to maximize his chances of winning a prize, would think about what the majority perception of beauty is, and then make a selection based on some inference from his knowledge of public perceptions. This can be carried one step further to take into account the fact that other entrants would also be making their decision based on knowledge of public perceptions. Thus the strategy can be extended to the next order, and the next, and so on, at each level attempting to predict the eventual outcome of the process based on the reasoning of other rational agents.


 * “It is not a case of choosing those [faces] 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.” (Keynes, General Theory of Employment Interest and Money, 1936).

Keynes believed that similar behavior was at work within the stock market. This would have people pricing shares not based on what they thought their fundamental value was, but rather based on what they think everyone else thinks their value was, or what everybody else would predict the average assessment of value was.

Guessing Game comparison to a Beauty Contest
What Keynes explicitly describes is only when $$p=1$$. However, his notion of degrees of rationality does describe the behavior in experimental results (with many choosing a peak around 33 and 22 for $$p=2/3$$). Other, more explicit scenarios help to convey the notion of the beauty contest as a convergence to Nash Equilibrium when the agents in the game behave perfectly rationally. The most famous such example is a contest where entrants are asked to pick a number between 0 and 100, with the winner of the contest being the person that is closest to 2/3 the average number picked for all contestants.