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 fictional newspaper context, in which entrants are asked to choose a set of six faces from photographs of women that were the "most beautiful". Those who picked the most popular face are then eligible for a prize.

A naïve 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.

Other, more explicit scenarios help to convey the notion of the beauty contest as a convergence to Nash Equilibrium. For instance in the "p"-beauty contest game (Moulin 1986), all participants are asked to simultaneously pick a number between 0 and 100. The winner of the contest is the person(s) whose number is closest to p times the average of all numbers submitted, where p is some fraction, typically 2/3 or 1/2. If p<1 the only Nash equilibrium solution is for all to guess 0. By contrast, in Keynes's formulation, p=1 and there are many possible Nash equilibria.

In play of the p-beauty contest game (where p differs from 1), players exhibit distinct, boundedly rational levels of reasoning as first documented in an experimental test by Nagel (1995). The lowest, `Level 0' players, choose numbers randomly from the interval [0,100] . The next higher, `Level 1' players believe that all other players are Level 0. These Level 1 players therefore reason that the average of all numbers submitted should be around 50. If p=2/3, for instance, these Level 1 players choose, as their number, 2/3 of 50, or 33. Similarly, the next higher `Level 2' players in the 2/3-the average game believe that all other players are Level 1 players. These Level 2 players therefore reason that the average of all numbers submitted should be around 33, and so they choose, as their number, 2/3 of 33 or 22. Similarly, the next higher `Level 3' players play a best response to the play of Level 2 players and so on. The Nash equilibrium of this game, where all players choose the number 0, is thus associated with an infinite level of reasoning. Empirically, in a single play of the game, the typical finding is that most participants can be classified from their choice of numbers as members of the lowest Level types 0, 1, 2 or 3, in line with Keynes' observation.

External links

* [http://etext.library.adelaide.edu.au/k/keynes/john_maynard/k44g/chapter12.html The State of Long-Term Expectation] , Ch 12. General Theory of Employment Interest and Money
* [http://www.beautyexpectations.com Beauty Expectations] , a simple beauty contest

References

Keynes, John Maynard (1936). "The General Theory of Employment, Interest and Money". (New York: Harcourt Brace and Co.).
Moulin, Herve (1986). "Game Theory for the Social Sciences," 2nd Ed. (New York: NYU Press).
Nagel, Rosemarie (1995). "Unraveling in Guessing Games: An Experimental Study," "American Economic Review" 85, 1313-1326.