p-Beauty Contest

The p-Beauty Contest is a game about iterative reasoning, inspired by Keynes’ analogy of a newspaper beauty contest for behavior in stock markets. Players guess not what they themselves think is right, but what average opinion expects average opinion to be. The game reveals how deeply people reason about others’ expectations — the cognitive hierarchy.

All players simultaneously pick a number from 0 to 100. The winner is whoever is closest to p times the average of all numbers — often with p = 2/3. The iterative reasoning drives numbers down: if everyone picked 50, the target would be 33.3; anticipating that, you pick 33.3 → anticipating that, 22.2 → and so on. Carry the iteration all the way and the unique Nash equilibrium is 0. Yet almost no one lands there.

For understanding financial speculation and market bubbles, forecasting, and anywhere success depends on anticipating others’ expectations rather than some “true” value. Keynes’ point: in the market you don’t buy the stock you find beautiful, but the one you believe everyone else will find beautiful.

Gain your edge from understanding the levels of reasoning (cognitive hierarchy): naive players (“level 0”) pick randomly or 50; “level-1” players best-respond to level 0; and so on. Whoever correctly estimates the typical reasoning depth of the field — usually just one or two steps deep — beats both the naive players and the “perfectly rational” pickers of 0. Anchoring and shared reference points shift the result predictably.

In large public runs of the game, the winning number typically lands around 21–23 — far above the equilibrium of 0, because real people reason only a few steps deep. In markets: a stock that keeps rising because everyone buys expecting others to buy too — until the shared expectation flips and the bubble bursts.