Everyone writes down a number from 0 to 100. The winner is whoever lands closest to 2/3 of the average. Logic says everyone should write 0. People who play it with real crowds almost never win with 0. Both facts come from the same chain of reasoning, depending on how many links of it you expect other people to follow.
The average can be at most 100, so the target p × average is at most 100p, which is 66.67 when p = 2/3. No guess above that can ever be the best one, so a sensible player crosses those out. If everyone is sensible, the average is now at most 66.67 and the target at most 44.44, so the next band goes. The amber mark is the matching level-k player, who assumes everyone else reasons exactly one step less.
How to read this: each row (k = 0, 1, 2…) is one more round of "nobody sensible would guess above this". The green part is still possible, and the green number at its edge is the new ceiling.
Try this: press Play from 0, then set p to 1 and see nothing get crossed out.
With real-number guesses the bound 100·pk goes to 0 for any p < 1, so the only profile that survives every round is everyone choosing 0. That is the unique Nash equilibrium: if everyone says 0, the target is 0 and nobody can get closer by moving.
With whole-number guesses there is a second one. If everyone says 1, the target is 2/3, and a lone switch to 0 only ties the 1s when there are 4 or more players. A brute-force search over every profile (prize split among the closest) found exactly two equilibria, all-0 and all-1, for 4 to 7 players, and only all-0 for 2 or 3.
At p = 1 nothing is ever eliminated. The target is the average itself, so any number that everyone picks is an equilibrium. This is a pure coordination game.
Equilibrium play is only the best move if everyone else plays it too. Here you face a simulated crowd: level-0 players pick uniformly at random (average 50), level-k players pick near 50·pk, and Nash players pick 0. Set the mix, pick your number, and check the lower chart. It replays 200 fresh crowds with the same mix and shows how often each guess would win.
How to read this: top: what the crowd guessed this round, stacked by player type. Bottom: how often each possible guess would win against many crowds like this one; the peak is the best guess.
Try this: press "Play the best response", then "New crowd" a few times.
With n players in total and S the sum of everyone else's guesses, setting your guess x equal to the target gives the exact best response x*. In a big crowd it is almost exactly p times the others' average. In a small group your own number pulls the target noticeably.
John Maynard Keynes used this idea in 1936: picking stocks is like a newspaper beauty contest where you win by choosing the faces other entrants will choose, so you have to model what the average opinion expects the average opinion to be. Rosemarie Nagel's 1995 experiments with this exact game found first-round averages far above 0, with clusters near the level-1 and level-2 numbers.
Against a crowd like that, 0 is the one guess guaranteed to be far from the target. The winning move is to be one step deeper than the crowd, and no deeper.
Now play the same game again and again with the same group. After each round everyone sees the target. In this toy model every player aims at last round's target (or p times it, if they think one step ahead) and misses by some random noise. The crowd never reasons its way to 0 in one go, yet the average still shrinks by a factor of p each round.
Without noise the average falls exactly by pd+1 each round and heads to 0, the equilibrium from Lab 1. With noise, guesses can't go below 0, so the misses near the bottom only push upward. The average then settles at a small positive level that grows with the noise.
Repeated play reaches the equilibrium by feedback. Nobody needs to do the infinite chain of reasoning. Each player only reacts to the last result. Other games in this section where equilibrium logic and real play pull apart: 022 · Prisoner's dilemma and 024 · Centipede game.