The experience
Pick a number from 0 to 100. Everyone else does the same, and the winner is whoever lands closest to two-thirds of the average of all guesses. That is the whole game — one number, one fully disclosed rule. In the station above you play against a recorded crowd: the first-round guesses from the original published classroom experiment, so your number is scored against what real first-time players actually did. One thing is worth knowing before you commit: hardly anyone plays the theoretically perfect move on first encounter. In Nagel's classic 1995 experiment the average first-round guess in the two-thirds game was 36.73 and the median 33, not a single player chose 0, and almost half of all choices fell in the neighbourhoods of 33 and 22 — exactly where one and two rounds of thinking about everyone else would land you.Convergent / replicated Whatever number is forming in your head right now, somebody in that classroom almost certainly wrote it down.
What is happening
Every guess in this game is a theory about other people. Guessing 50 treats the crowd as noise. Guessing 33 assumes a crowd of 50-guessers. Guessing 22 assumes a crowd that has already thought of 33 — and the ladder keeps going, because no guess above about 67 can ever be a best reply: even if every other player picked 100, two-thirds of the average would sit near 67. A player who discards those numbers, and trusts everyone else to discard them too, must next discard everything above two-thirds of 67 — and the argument repeats until the only guess left standing is 0, the game's unique equilibrium, in which every player announces zero.Convergent / replicated The mathematics has a clean bottom, yet standing on it is a mistake of its own kind: two-thirds of what a real crowd actually does is nowhere near zero. The winning move is not the equilibrium — it is one step ahead of the crowd you are actually in. You are not solving for an answer; you are estimating how far other people will climb the ladder before they stop.
What this shows
The name comes from chapter 12 of Keynes's General Theory (1936), which likens professional investment to newspaper competitions in which entrants pick the six prettiest faces from a hundred photographs, the prize going to the entrant whose picks best match the average preference. Keynes's competitors reach "the third degree", devoting their intelligences "to anticipating what average opinion expects the average opinion to be" — and some, he believed, practise "the fourth, fifth and higher degrees".Historical / descriptive The number game distils that passage into something measurable — and what it measures is robust: the pattern survives outside the laboratory at enormous scale. When a business daily in the UK, a business daily in Spain and a German popular-science magazine ran the same game as prize contests in the late 1990s, 7,900 readers entered, the most popular numbers everywhere were about 33, about 22 and the equilibrium choices 0 and 1, the three averages ran from 18.91 to 25.47, and the winning numbers all fell between 13 and 17.Convergent / replicated Different countries, different languages, different readerships — the same few rungs of the same ladder.
What this does not show
A tempting reading is that high guessers are fools and the people who write 0 have seen through the game. The evidence points the other way: seeing the equilibrium is not the same as playing it. Among magazine readers whose written comments correctly described the full unravelling to zero, 58% of a sampled fifty still chose a number above the equilibrium — deliberately, because they expected everyone else to stop iterating early — and they were right: the winning numbers in all three newspaper contests sat far above zero.Supported, with limits Nor is one guess a measurement of you. A single number cannot separate luck, habit, ladder-climbing and second-guessing, and your session summary describes only what you did against one recorded crowd — the site draws no conclusions about your reasoning or intelligence.
Research and limitations
How deep do people actually think? much of this behaviour can be captured by assuming people simply differ in how many steps they think. In Camerer, Ho and Chong's cognitive hierarchy model, where each player best-responds to the mix of players reasoning fewer steps than themselves, the median estimate across 24 beauty-contest data sets is about 1.6 thinking steps, and an average of 1.5 steps fits data from many other games — which is why group averages stall around 30 instead of unravelling to zero.Supported, with limits That is a model fit, not a mind-reading: rival accounts describe the same data, and calling any single guess "Level 1" is an inference about a distribution, never a fact about one person's thoughts.
Repetition changes the picture, slowly. played repeatedly with feedback, the crowd does creep toward zero: in Nagel's four-round sessions most players lowered their guesses from round to round and the median fell every period, yet even in the final round of the two-thirds sessions only one player out of 67 chose a number below 1.Supported, with limits Two honesty notes about our version: the crowd you face is an approximate reconstruction of the published first-round distribution, rebuilt from the paper's figures rather than its raw data, and the original samples were 1990s classroom volunteers paid in Deutschmarks and self-selected newspaper readers — a crowd, not the crowd.