The winner is whoever is closest to two thirds of the average of all the numbers named. Here you play not against bots but against the site's other visitors: your number joins a shared game, and after you answer you will see who is winning right now. Name your number, then check it with a game, a simulation and the explanations.
The chain "the others are no fools either" cuts the number by a third each time and drives it to 0 (and, for whole numbers, also to 1). That is the equilibrium for perfectly rational players. But real people take one or two steps of the chain, so in big games the winning numbers were between 13 and 22. The winner is one step ahead of the crowd, not infinitely many.
This game has been played in laboratories, in lectures and in newspapers with thousands of readers. The rules are simple, and every player knows them.
Each player secretly names a whole number from 0 to 100 inclusive. Nobody sees anyone else's number until the game is over.
Once everyone has named a number, the average of all numbers, yours included, is taken, and two thirds of it is the target. For example, three players name 10, 30 and 80: the average is 40, the target 2/3 · 40 ≈ 26.7.
Whoever's number is closest to the target wins. In the example that is 30: it is 3.3 from 26.7, while 10 is 16.7 away. If several are equally close, they share the prize.
The other players are people just like you: they read the same rules and want to win too. Which number do you name?
"The numbers will be random, the average 50, so 33" or "everyone is smart, so 0"
Both answers lose. 33 assumes the others are more naive than you, 0 assumes everyone is infinitely clever. The winner is whoever guesses how many steps the others really take, and takes one more.
Two thirds of the average is always below the average itself, and the largest possible target is 66.7, when everyone names 100.
A whole number from 0 to 100. It joins the site visitors' game: whoever is closest to 2/3 of the average of all answers wins.
One answer only: you can't change it after checking.
A breakdown of your number and a game against bots will appear here.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
Start simple: if all the others name numbers at random, roughly what will the average be? And two thirds of it?
But the others can take that step too. What happens to the target if everyone reasons the way you just did? And if everyone takes the next step as well?
You win by being closer to the target than the others, not by being the most logical. How many steps of this chain do you think an ordinary person seeing the game for the first time will take?
The bots think to different depths. A level-0 bot names a random number. Level 1 assumes everyone else is random and takes 2/3 of 50. Level 2 assumes the others are level 1, and so on. Some bots go straight to 0 or 1. After each round the bots see the average and adjust. By default the crowd is a mystery: work it out from the first rounds.
First name your number in round 1. The game against bots shows which numbers win, so it opens after your answer.
To round 1| Round | Target | You | Place |
|---|
Now 1,000 bots play. Each takes a random number of thinking steps: some none, some one, some three. The first slider sets how many steps they take on average. The second sets the share of bots who go straight to 0. The lines are the exact values, the dots the simulation. The numbers here are fractional so they match the theory exactly.
The simulation shows where the target lands, so it opens after your answer in round 1.
To round 1The first layer has no formulas, the second has a table, a chart and formulas, the third covers Keynes, the game's history and the newspaper contests.
“In plain words” is hidden on all puzzles.
If everyone picks at random, the average will be about 50, so you should say 33. But the others will work that out too and say 33. Then the average is 33, and you should say 22. The others will foresee that as well… Each such step multiplies the number by 2/3: 33, 22, 15, 10, 7 and so on all the way to 0.
That is the answer for perfectly rational players. This card explains where 0 comes from, but not why 0 loses. That is the next card.
The target depends on what the others name. If most stop at 33 and 22, the average comes out around 25–35 and the target around 17–23. Whoever said 0 is off by 20 or more: they thought infinitely many steps ahead, when one more than the crowd was enough.
That is why this game has no single right answer. The best number depends on the people at the table.
If all the others said 1, the average is close to 1 and the target close to 2/3. The number 1 is a third away from it, and 0 two thirds. So when everyone says 1, nobody gains by switching to 0. "Everyone says 0" and "everyone says 1" are two stable situations, provided there are at least four players. Fractional numbers remove the second.
The students in Rosemarie Nagel's study played for the first time, and none of them named 0 in the first round. The newspaper's readers had two weeks to think, and there were noticeably more zeros among them, yet the target still came out at 12.6.
“Rigorous” is hidden on all puzzles.
| Step k | Bound, fractions: 100·(2/3)k | Bound, whole numbers | Level k: 50·(2/3)k |
|---|
The target never exceeds 66.7, so any number above 67 is worse than 67 whatever the others do: it is dominated. If everyone understands that, the target does not exceed 2/3 · 67 ≈ 44.7, and numbers above 45 drop out. And so on. For fractional numbers the bound 100·(2/3)k tends to 0 but never reaches it, so there are infinitely many steps. For whole numbers the chain 100 → 67 → 45 → 30 → 20 → 13 → 9 → 6 → 4 → 3 → 2 → 1 stops after 11 steps at 1: 0 and 1 remain.
The last column is a different chain: the level-k model. Here a player does not discard the impossible but responds to a concrete guess about the others, starting from "the average will be 50". These very numbers, 33 and 22, are the most frequent in real games.
N players name xi ∈ {0, 1, …, 100}. The target and the winner:
After k elimination steps only numbers up to this bound remain. It tends to 0, so the only equilibrium for fractional numbers is everyone naming 0.
Whole numbers: if all others say 1 and one player says 0, then 0 is 2(N−1)/(3N) from the target and 1 is (N+2)/(3N) away. Zero is closer only when N < 4, so "everyone says 1" is also an equilibrium with at least four players (at N = 4 the distances are equal).
Level k responds to the guess "everyone else is level k − 1", starting from 50. If the levels in the crowd follow a Poisson distribution with mean τ (as in round 3), the crowd's average and target are:
If the same crowd plays again and everyone counts from the last average, while a share z says 0 straight away, the target in round t falls geometrically:
“Deeper” is hidden on all puzzles.
In chapter 12 of The General Theory of Employment, Interest and Money, John Maynard Keynes compared professional investment to the newspaper competitions of his day: pick the six prettiest faces from a hundred photographs, and the prize goes to whoever's choice is closest to the choice of all the competitors together. You should pick not the faces you like but those the others will like.
Keynes writes of "the third degree", where we devote our intelligences to anticipating what average opinion expects average opinion to be, and adds that some practise the fourth, fifth and higher degrees. The 2/3-of-the-average game makes these degrees measurable: each degree multiplies the number by 2/3.
The game is usually credited to the French journalist Alain Ledoux: in 1981 it served as a tie-breaker in a readers' contest in the magazine Jeux & Stratégie. In 1986 the game theorist Hervé Moulin used it in a textbook as an example of rationalizable strategies: those that survive when bad ones are discarded step by step.
In 1995 Rosemarie Nagel, in the American Economic Review, was the first to run the game as an experiment: groups of 15–18 students played four rounds in a row, and the winner of a round received 20 marks. She showed that people's choices cluster around 50·(2/3)k for k = 1, 2 and less often 3, and explained this by depth of reasoning.
In 1997 Richard Thaler ran the game in the Financial Times, and Antoni Bosch-Domènech and Rosemarie Nagel in the Spanish business daily Expansión, unaware of each other. Soon after, Nagel and Reinhard Selten repeated it in the German magazine Spektrum der Wissenschaft. The winning numbers: 13, 16.99 and 14.7.
In all three the most popular numbers were 33, 22 and the equilibrium 0 (or 1). So the crowd was made of people who took one or two steps and people who went all the way. The winners were in between.
After Nagel, economists built models in which players take a finite number of steps. In the cognitive hierarchy model of Camerer, Ho and Chong (2004) people's number of steps follows a Poisson distribution, and in many games an average of about 1.5 steps fits the data best. Round 3 uses a simplified version of this idea, with Nagel-style levels.
The models reach the same conclusion as the game: equilibrium describes where a crowd moves when it plays again and again, but not what it does the first time.
First the maths (a different multiplier instead of 2/3), then the logic (thinking about what others think about others), then practice (when to leave a bubble that everyone can see).
For any p < 1 the chain is the same, only the step differs: at p = 1/2 each step halves the number, at p = 0.9 it trims only a tenth. In Nagel's study the game with p = 1/2 had a first-round average of 27.05, with p = 2/3 it was 36.73.
At p = 1 the winner is whoever guesses the average itself: any common number is stable, and the game becomes a coordination game. At p > 1 the chain runs upwards and the equilibrium is 100. Nagel tested this too: at p = 4/3, 59% and 68% of players named 100 in the third and fourth rounds.
The average can be replaced by the median: the elimination logic is the same, and so is the equilibrium. For whole numbers, a multiplier closer to 1 gives more stable equilibria: at p = 0.9, "everyone says 0, 1, 2, 3 or 4" all qualify.
| You take steps | Your number | Distance to target |
|---|
If all the others take j steps, they name 50·(2/3)j, and the target is 50·(2/3)j+1. That is exactly the number of whoever takes j + 1 steps. Fewer steps leave you above the target, more leave you below. The best is exactly one step ahead.
Hence the main point: cleverness here is not absolute. A player who thinks ten steps ahead loses to a crowd that thinks one. Game theorists call the ideal case common knowledge of rationality: everyone is rational, everyone knows everyone is rational, everyone knows that everyone knows… It is enough for the chain to break somewhere, and equilibrium stops predicting the game.
A toy model, not advice: the price grows 1% a day, and everyone knows it is a bubble. Each trader plans a selling day from 0 to 100. The bubble bursts on the day equal to 2/3 of the average planned day: by then there are too many sellers. Whoever is late loses half.
Keynes invented the beauty contest to explain the stock market: a price depends not on what a company is worth but on what others think others will think. In a bubble everyone knows the price is too high, and everyone hopes to sell one step before the crowd. Whoever left "rationally" at the very start misses the rise. Whoever waited with everyone else loses.
A check: if the others plan day 45, the bubble bursts on day 30. Selling on day 29 earns 1.0129 − 1 ≈ 33%. But this is a toy with a known formula. In a real market nobody knows the day of the crash, the "one step early" rule has no guarantee, and this model is not investment advice.
The idea is older than the game. Keynes described a newspaper contest where you must guess not the prettiest faces but those the others will pick, and compared the stock market to it. That is why the 2/3-of-the-average game is often called a beauty contest.
The game is thought to have first appeared in the French magazine Jeux & Stratégie as a contest tie-breaker. Readers had to name a whole number from 1 to a billion, and the one closest to 2/3 of the average won.
In the first round of Nagel's experiment the average was 36.73, and not one student named 0. Over four rounds the numbers fell noticeably, yet 0 never became the best response to what the others did.
Richard Thaler's Financial Times contest allowed whole numbers only: the paper's lawyer thought a game with fractions would become a pure game of chance, which was not allowed in Britain. 1,476 readers produced an average of 18.91, 13 won, and the prize was two plane tickets from London to New York or Chicago.
The Financial Times, Expansión and Spektrum der Wissenschaft ran the game in 1997, with almost 7,900 players in total. The winning numbers 13, 16.99 and 14.7 fell between 12 and 17, although the readers were different and from different countries.
In 2005 the Danish newspaper Politiken collected 19,196 answers. The winning value came out at 21.6, and five people shared the prize. That is higher than in the three 1997 papers: crowds differ.
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