The 2/3 of the average problem posed on Friday is a well known puzzle in game theory, and it illustrates some fundamental game theoretic concepts. To

The Incidental Economist

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2021-06-17 14:00:03

The 2/3 of the average problem posed on Friday is a well known puzzle in game theory, and it illustrates some fundamental game theoretic concepts. To recap, here’s the problem statement:

Suppose everyone in your town selects a real number between 0 and 100, inclusive (i.e. 0 and 100 are both possible choices, as is any other number between). The winner is the individual (or individuals) who selects the number closest to 2/3 of the average of numbers chosen. What number do you choose? Why?

A nice way to view the problem is to start by identifying the set of numbers that no rational player should select. Intuitively, most sense that the winning answer will not be 100. Why? Because the average of numbers no greater than 100 cannot be greater than 100. Thus, 2/3 of the average cannot be 100. One can make a better choice. In fact, 2/3 of the average of numbers no greater than 100 cannot be greater than 66.666… Therefore, it is irrational to select a number higher than 66.666… All numbers above 66.666… are weakly dominated strategies (game theory jargon) meaning that one cannot do worse and may do better by selecting a number outside this range.

If you are rational you will not select a number greater than 66.666… this suggests the next stage of analysis. Can you assume everyone else is rational? If so then all your opponents also eliminate numbers above 66.666… from consideration. This reduces the puzzle to selecting a number between 1 and 66.666… trying to get closest to 2/3 of the average.

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