Pick a number between 0 and 100. Whoever gets closest to two-thirds of the average of everyone's guess wins. Sounds easy — until you realize everyone else is trying to out-think you too!
In real experiments (even among economists!), most people guess between 20 and 35 — around Level 1 or 2 thinking. Very few guess 0, because that requires assuming everyone else reasons infinitely deep too.
Everyone submits a number 0-100 at the same time. The target is 2/3 of the average of all numbers. Whoever's guess is closest to the target wins.
This game, often called the "Beauty Contest Game" (after an analogy economist John Maynard Keynes made about newspaper contests), asks each player to pick a number from 0 to 100. Instead of picking your favorite number or a random one, you should picture what everyone else will guess — because the winner is whoever gets closest to two-thirds of the average of every submission, not closest to some fixed target. This tiny twist turns a simple guessing game into a fascinating window on how people reason about other people's reasoning.
Game theorists describe this using "levels of thinking." A Level-0 thinker just guesses randomly, with no model of other players at all. A Level-1 thinker assumes everyone else is Level-0 (averaging around 50) and so guesses roughly two-thirds of 50, which is about 33. A Level-2 thinker assumes everyone else is Level-1 and guesses about two-thirds of 33, roughly 22. This chain continues — each additional level of reasoning pulls the guess further down. If every single player reasoned perfectly and infinitely deep, expecting everyone else to do the same, the only number that could survive that logic is 0 — because two-thirds of zero is still zero. That's the game's Nash Equilibrium.
But in practice, almost nobody guesses 0! When this experiment has been run with real people — including university students, newspaper readers, and even trading floor professionals — the average guess typically lands somewhere between 20 and 35, suggesting most people reason one or two levels deep but rarely assume everyone else is a perfect logician too. That's exactly why understanding your opponent's likely level of thinking, not just the theoretically "correct" answer, is often the real key to winning these kinds of strategic guessing games.
Gather a group of at least 5 friends or classmates. Have everyone secretly write down a number from 0 to 100 on a slip of paper. Collect them, calculate the average, multiply by two-thirds, and see who got closest. Play a few rounds in a row — you'll likely notice the winning numbers drift lower over time as players learn to anticipate each other's thinking!
What is the 'Guess 2/3 of the Average' game?
Each player picks a number between 0 and 100, and the winner is whoever picks closest to two-thirds of the average of all the numbers chosen.
What is the Nash Equilibrium of this game?
If everyone reasons perfectly, the only stable answer is 0, since two-thirds of zero remains zero no matter how many times you apply the rule.
What are 'levels of thinking'?
Level-0 players guess randomly, Level-1 assumes everyone else is Level-0, Level-2 assumes everyone else is Level-1, and so on, going deeper with each level.
Why doesn't everyone just guess 0?
Because that requires assuming everyone else reasons infinitely deep too, which rarely happens — real guesses usually average between 20 and 35.