Keynesian Beauty Contest - Subsequent Theory

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 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 . 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, 6435, 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.

(This contest has been held in Spektrum der Wissenschaft, November 1997, with results as PDF. Both links in German; Google translation.)