Overview
Keynes described the action of rational agents in a market using an analogy based on a fictional newspaper contest, in which entrants are asked to choose a set of six faces from photographs of women that are 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 the 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 their knowledge of public perceptions. This can be carried one step further to take into account the fact that other entrants would each have their own opinion of what public perceptions are. 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] that, to the best of one’s judgment, are really the prettiest, nor even those that 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 practice 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 think their fundamental value is, but rather on what they think everyone else thinks their value is, or what everybody else would predict the average assessment of value is.
Subsequent theory
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 p="1">
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.
In another variation of reasoning towards the beauty contest, the players may begin to judge contestants based on the most distinguishable unique property found scarcely clustered in the group. As an analogy, imagine the beauty contest where the player is instructed to choose the most beautiful six faces out of a set of hundred faces. Under special circumstances, the player may ignore all judgment-based instructions in a search for the six most unique faces (interchanging concepts of high demand and low supply). Ironic to the situation, if the player finds it much easier to find a consensus solution for judging the six ugliest contestants, he may apply this property instead of beauty to in choosing six faces. In this line of reasoning, the player is looking for other players overlooking the instructions (which can often be based on random selection) to a transformed set of instructions only elite players would solicit, giving them an advantage. As an example, imagine a contest where contestants are asked to pick the two best numbers in the list: {1, 2, 3, 4, 5, 6, 7, 8, 2345, 6345, 9, 10, 11, 12, 13}. All judgment based instructions can likely be ignored since by consensus two of the numbers do not belong in the set.
Via: wikipedia
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