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Are you smarter than a New York Times Reader?

This puzzle from the New York Times challenges readers to enter a number between 0 to 100 such that the number is 2/3 of the number chosen most commonly by the other participants of the challenge. After 59,835 contestants, the winning number was 19.

This challenge was a useful example for illustrating a number of game theory topics. A common thought process was to assume other readers guess randomly, leading one to believe that the most common number is 50 in which case one would guess 33. However, stopping at this point is an example of 1 step thinking (where a player accounts only for the actions of other players while assuming other players do not do the same). Iterating this same thought process indefinitely would lead to the Nash Equilibrium solution of 0. Surprisingly, 0 was not the winning guess, and some 0 step thinkers (people who chose largely randomly) even chose numbers greater than 66 (which could never be a winning number since 2/3 of the largest number was 66).

This non-ideal result shows the weakness of many game theoretic models in assuming rational players. Also, the result has many important applications; in finance, the viability of a company as a good investment is depends heavily on how other investors also view the company. To be of even more use for such applications, this study could have been designed to target a more useful summary value, like a number n such that New York Times readers can be summarized as n step thinkers. Currently, the study only reveals that 10 percent of the readers are infinite-step thinkers (as they choose either 0 or 1, both of which are in Nash Equilibrium due to roundoff errors). Such a summary value could, for instance, be used to gauge how all other investors regard a company. Clearly, infinite step rational thinking cannot be expected of all individuals with respect to every decision.

Source: http://www.nytimes.com/interactive/2015/08/13/upshot/are-you-smarter-than-other-new-york-times-readers.html

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