Monday, March 10, 2008

Can you guess x?

Suppose that there are 10 people playing a guessing game. Each of the 10 people choose a number between 0 and 100. The average of these numbers is computed, and this average is multiplied by p = 2/3. We call the resulting number x. The person whose number is closest to x wins a big prize, while all others receive nothing. If everybody selects the same number, then the prize is split equally. What number would you select if your objective is to maximize the chance that you win the prize? Explain your strategy choice and then suggest an economic situation that illustrates the essence of this game.

Congratulations to Josh Busser for submitting a correct answer to this week's question. According to Josh, the dominant strategy in this guessing game is to submit the number 0 (zero). Read the comments to see Josh's explanation. This question comes from Andrew Schotter's Microeconomics (3e) and is a description of a class of games known among economists as "beauty contests."

1 comment:

Joshua Busser said...

In this guess the number game, the optimal number to guess, given a perfectly rational participant, should be zero. In this game, it would be irrational to pick a number above 66.67, as this is the maximum value that X could be, if all choices were 100. A study conducted in 1995 analyzing such a number game as this found different levels of rationality existed among people based on what their choice was. Nagel found that a choice of a number above zero was due to some personal irrationality, belief of irrationality in the other participants, or some inherent belief of irrationality in their "hierarchy of beliefs" (known as zero, first, and second order beliefs, respectively.)
Such a game is a example of game theory in which there are theoretically infinite numbers of participants and there is no incentive for a player to change from their initial choice. A player who makes the decision to change from the equilibrium value ultimately has nothing to gain from making such a decision and is therefore irrational. The selection of zero is not a dominant strategy, however, but simply a pure strategy Nash equilibrium.
Economically speaking, a variation of a silent auction can be an example illustrating this idea. In a silent auction, the winning bidder has the highest bid for the item, but all bids are submitted without any knowledge of the other participants. In this context, the optimal outcome would be for all bidders who were to receive the prize based on some stipulations like those in the number game. Optimally, every person should submit the same bid so as to have the same opportunity to win the item, and provided the item that can be won is able to be divided between each winning bid, the rational outcome would be to bid nothing and get a share of the item along with everyone else.