The Guess-Two-Thirds-of-the-Average Game

A room of players each secretly picks a number between 0 and 100. Whoever's guess is closest to two-thirds of the average of all the guesses wins. It's a favorite interview and classroom game precisely because the "correct" theoretical answer and what real, smart people actually submit are famously different — and understanding the gap is the whole lesson.

Working out the game-theoretic answer

Suppose everyone believes the average guess will be uniformly distributed and picks near 50, the midpoint. Two-thirds of 50 is about 33, so a smart player guesses 33 instead of 50. But if everyone reasons this far, the average shifts down to around 33, and two-thirds of that is about 22 — so a player who anticipates everyone else reasoning one step further guesses 22. This logic doesn't stop at any particular level: if every player is fully rational and knows every other player is fully rational, and that's common knowledge, the only guess that survives this endless one-step-ahead reasoning is 0 — because at any positive number $x$, two-thirds of $x$ is strictly less than $x$, so $x$ can never be a stable guess once everyone applies the same logic to it. This mirrors iterated elimination of dominated strategies: any guess above roughly 67 is dominated immediately (it can never be two-thirds of anything below 100), and once those are eliminated, the effective maximum keeps shrinking, chasing the same logic down to zero, round after round.

Worked example

Suppose four players submit guesses of 40, 30, 20, and 10. The average is $\frac{40+30+20+10}{4} = 25$, and two-thirds of that is $\dfrac{2}{3} \times 25 \approx 16.7$. Whichever guess is closest to 16.7 wins — here, 20 (distance 3.3) beats 10 (distance 6.7). Notice the winner wasn't the lowest guess, just the closest to two-thirds of the actual average, which is why blindly guessing 0 doesn't win either, unless everyone else has also converged near 0. This is the real-world twist: the game-theoretic equilibrium (0, if everyone is perfectly rational and expects everyone else to be too) is rarely what wins in practice, because real groups only iterate the reasoning a few steps before stopping — a room of first-timers tends to average somewhere in the 20s to 30s, not 0, and the winning guess reflects that real, finite depth of reasoning rather than the theoretical limit.

What this means in practice

This game is a compact demonstration of the gap between full common-knowledge rationality (the mathematical equilibrium) and how real, smart-but-finite people actually reason — a gap that matters enormously in markets, where prices reflect the actual, bounded depth of reasoning traders apply, not the infinite-regress equilibrium of game theory. Interviewers use it to see whether a candidate can articulate both the formal equilibrium and why real behavior deviates from it, since knowing only one half misses the point of the exercise.