Experiment

Guess Two-Thirds of the Average

One number, one rule, and everyone else who has played on this site. It takes about a minute, most of it thinking.

Pick a whole number from 0 to 100. Every reader who plays picks one too. The winning number is two-thirds of the average of all the guesses, and whoever is closest to it wins.

So if the guesses averaged 60, the winning number would be 40. What do you pick?

Only the number is saved, anonymously, so later readers have something to play against. Nothing else is recorded.

How deep did you go?

When others played

In Rosemarie Nagel's 1995 experiments with German students, first-round guesses averaged about 36, and they bunched near 33 and 22. In 1997 the Financial Times ran the game for its readers at the economist Richard Thaler's suggestion. About 1,500 entered, the average was 18.9, and the winning number was 13. When the Danish newspaper Politiken ran it for roughly 20,000 readers, the winning number was 21.6, with visible spikes of guesses at 33 and 22.

Here is the reasoning most people go through. If everyone picked at random, the average would be about 50, so you should pick 33. But if everyone thinks that, the average will be 33, so you should pick 22. But if everyone thinks that, you should pick 15, then 10, and so on. Game theory's answer follows the chain all the way down: the only number that survives every step is 0. If every player reasons perfectly and knows that every other player does too, everyone picks 0 and everyone wins.

Real players never do that, and the people who pick 0 almost never win. The winning number depends on where everyone else stopped, so the real question is not "what is the logical answer?" but "how many steps will the others take?" That has no fixed answer. Each extra step is a guess about how many steps the others are guessing you will take. John Maynard Keynes described the same problem in 1936, comparing the stock market to a newspaper beauty contest where you win by picking the face other people will pick: "We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be."

The essay The Empty Fort Problem follows the same regress through a famous scene from Romance of the Three Kingdoms, in which Sima Yi talks himself out of attacking an undefended city by reasoning about what Zhuge Liang expected him to reason. Its point is the one this game makes with numbers: reading another mind that is reading yours has no natural floor. You stop somewhere, and you win only if you stopped one step past everyone else.

The winning number on this page keeps moving as more readers play. Come back later and see where it has settled.

Discussion

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