Guess 2/3 of the Average: Puzzle and Step-by-Step Solution
Game Theory
Medium
Reported at Jane Street, Optiver, SIG, Citadel
The puzzle
A hundred people each write down a whole number from 0 to 100. The winner is whoever comes closest to two-thirds of the average of all the numbers written. What number do you write?
Short answer: Pure logic drives the answer to 0, but against a real crowd the winning number is usually in the low twenties, so the right play depends on who is in the room.
Step 1: Eliminate the dominated guesses
The average can never exceed 100, so two-thirds of the average can never exceed 66.7. Any guess above 67 is beaten by 67 whatever anyone else writes. A rational player never writes it.
If nobody writes above 67, the average cannot exceed 67, so the target cannot exceed 44.4. Then 29.6, then 19.8, then 13.2. Each round of 'nobody sensible would write that' multiplies the ceiling by two-thirds again.
Step 2: The equilibrium is zero
Repeat the elimination forever and the ceiling falls to 0. If everyone is perfectly rational and everyone knows everyone else is too, every player writes 0, and 0 is two-thirds of the average of 0. It is the unique Nash equilibrium.
With whole numbers there is a small wrinkle: if everyone writes 1, the target is 0.67, which rounds closer to 1, so 1 can also survive. Mentioning that detail shows you checked the edge case rather than reciting the textbook answer.
Step 3: Level-k: the game inside the game
Real rooms are not infinitely rational. A level-0 player picks at random, averaging 50. A level-1 player best-responds to level-0 players with 33. A level-2 player best-responds to a room of level-1 players with 22. Most people do one to three steps of this reasoning.
Published runs of this experiment, including ones with newspaper readers and with finance professionals, typically land the winning number between about 13 and 25. Groups who play repeatedly converge toward zero, round by round.
Step 4: The answer to give
The correct answer is conditional: 0 against a room of game theorists, roughly 20 against an ordinary crowd, and somewhere in between against a room of quant candidates who all read this page. Saying exactly that, and why, is the winning move.
If you are allowed to see the guesses arrive over several rounds, update: estimate the room's average from the last round and write two-thirds of it. That adaptive rule is how a trader treats a market that is itself a beauty contest, and it is the follow-up interviewers most often ask.
Why interviewers ask it
This is Keynes's beauty contest, and firms ask it because markets are this game: prices track what average opinion expects average opinion to be. Price the room, not the textbook.
Knowing the proof is half of it. Work Guess 2/3 of the Average interactively, then take it under a clock with the rest of the Outcry simulators.