## A Number Puzzle in Game Theory and Behavioral Economics

The Upshot newsletter of New York Times published an online number puzzle, open to everyone to participate in. This puzzle asked the participant to guess a number between 1 and 100. The correct number would be two-thirds of the average number chosen by all participants of the game. Therefore, in order to win, the participant needs to successfully guess what number other participants would submit.

The results suggested that participants practiced k-step thinking. K-step thinking is the process of thinking ahead. For example, if the participant were to guess that the average would be 50, they would submit their guess, 33, which is two-thirds of 50. This means this participant has a k-step of 1. Furthermore, a participant with a k-step of 2 would submit 22, which is two-thirds of 33. This pattern continues until the number would finally hit 0. So, participants with more extreme thinking would logically conclude that the correct answer would be 0, which is the Nash equilibrium. To put it into context, the Nash equilibrium here would be the number that if everyone guessed it, no one would want to change their guess.

Out of 56,000 participants, about one out of ten participants guessed 0 or 1. That means about 10% of participants reached the Nash equilibrium conclusion. The most popular guess was 33, which proved that most people thought ahead, but usually only one step or a few. 0 was the second most popular guess, and 1 was the fourth most popular guess.

It is interesting to see what the fusion of game theory and behavioral economics is like. Although the whole population cannot be based on a mere sample size of 56,000 people, it is a striking statistic that only 10% will think the furthest ahead. If you were a participant, it might be seemingly impossible to guess what other participants would guess, not mentioning the average of all other guesses. Your guess would affect the average of all guesses; so the correct answer would be to guess the number that can no longer be affected by all guesses. That is why the Nash equilibrium is 0. Regardless of all other guesses, the participant choosing to submit 0 will still submit 0.