2/3 Average Guessing Game

How to play

1. Every player secretly picks a whole number between 0 and 90.
2. Enter the guesses one after another; each one becomes the next player.
3. Press Calculate Winner: the target is two thirds of the average of all guesses.
4. The guess closest to the target wins. Equally close guesses share the win.
5. Add 5 random players fills in uniformly random guesses, which is what a completely naive population would do.

Iterated elimination

Since no average can exceed 90, the target can never exceed 60, so guesses above 60 are dominated. If everybody knows this, nobody guesses above 60 and the target cannot exceed 40; then 26.7, and so on. Repeating the argument drives every rational guess down to 0, the unique Nash equilibrium.

Levels of reasoning

Real groups rarely reach 0. A level-0 player guesses randomly (average about 45), a level-1 player best-responds with 30, a level-2 player with 20 and so on. Winning guesses in classroom experiments usually sit between 15 and 30: winning requires predicting how far the others reason, not just being fully rational.

Reading the screen

The number line shows every guess as a dot, the average (orange) and the target (red). The winning guess is highlighted in green.