Brain Teasers · Two-Thirds of the Room
Guess 2/3 of the Average - Beauty Contest Puzzle
The Keynesian beauty contest as a story: pick the number closest to two-thirds of the room's average.
What is the Keynesian beauty contest?
The term comes from John Maynard Keynes's 1936 book The General Theory of Employment, Interest and Money, where he compared professional investing to a newspaper contest in which readers pick the face from a set of photographs that they think most other readers will pick as prettiest - not the face they personally find prettiest. Keynes's point was that markets often reward guessing average opinion about average opinion, ahead of fundamental value, and that this chain of anticipation can run to several levels.
The idea was later formalised as a playable game, usually called the p-beauty contest or the guess-2/3-of-the-average game, and studied experimentally by the economist Rosemarie Nagel in 1995. Every player privately picks a number, often between 0 and 100; the winner is whoever's number is closest to some fraction p (commonly 2/3) of the average of everyone's guess.
Solved by iterated elimination of dominated strategies, the game has a unique Nash equilibrium at 0 when p is less than 1: if everyone reasons all the way through, no number above 0 can survive being beaten by a lower one, and the only guess nobody can be talked out of is 0. Reaching that equilibrium requires unlimited common knowledge of rationality - every player reasoning through infinitely many levels of 'what will everyone else guess.'
Real groups never get there. Experiments consistently find winning guesses clustering well above zero, typically in the 20s, because most players stop after one or two levels of reasoning rather than iterating to the limit. This produced the influential idea of 'levels of reasoning' or 'cognitive hierarchy' models: a level-0 player guesses randomly, a level-1 player best-responds to random play, a level-2 player best-responds to level-1 players, and so on - a real crowd is a mix of these levels.
Because the game is literally a model of guessing what other people are collectively going to guess, it is used as a stylised model of speculative markets, where the profitable move is often not 'what is this asset really worth' but 'what will everyone else think everyone else will pay.'
How it works
A hundred people each write a number from 0 to 100. The winner is whoever comes closest to two-thirds of the average. Nobody sees anyone else's card and the rule is common knowledge. The story hands you the blank card and asks what you write. You lock in a guess, then step through the answer and breakdown.
How scoring works
No grading - your guess is displayed back beside the answer. The story teasers share one lifetime unlock for free users.
The key insight
Each level of reasoning multiplies the target by 2/3: random play averages 50, so level-1 writes 33, level-2 writes 22, and iterating forever lands on 0 - the equilibrium if everyone is perfectly rational and knows it. Real rooms stop at one to three levels, so winning numbers cluster around 20-30. The skill is estimating how deep this room goes and stopping one level past it.
The full step-by-step solution lives at /brain-teasers/beauty-contest/solution.
A worked example
No winning guess can exceed 67, since two-thirds of the largest possible average is about 66.7. Delete those and nothing above 44 can win; delete again for 30, then 20, and the ladder only stops at 0. Against a normal crowd, best-responding to level-1 play means writing about 22.
Common mistakes
• Writing 33 and stopping - that best-responds to a random room but ignores that others reason too.
• Writing 0 against real people - correct only if the whole room iterates forever.
• Giving one number without conditioning on who is in the room; the answer is 0 against theorists, low twenties against a crowd.
Why interviews test this
Quant desks ask this because markets are this game: you are pricing what average opinion expects average opinion to be. The written solution at /brain-teasers/beauty-contest/solution covers level-k reasoning.
Play Two-Thirds of the Room on Outcry
Two-Thirds of the Room is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play Two-Thirds of the Room and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.