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Ask a room to guess two-thirds of the average, and the winning number is almost never zero, the “rational” answer. It clusters around 22 and 33. People reason one or two steps, then stop, and the theory that captures this predicts real play far better than full rationality.

Limited reasoning: level-k

Equilibrium assumes reasoning goes all the way down: everyone is rational, knows others are, and so on without end. Real people truncate. Level-k models say each person best-responds to a belief that others reason one step less than they do, and that simple idea fits laboratory play remarkably well.

From earlier lessons

Two quick ones before we start.

The Nash equilibrium of the two-thirds-of-the-average game is...

Common knowledge of rationality requires reasoning that goes...

Predict, then play

Guess two-thirds of the average against an illustrative six-bot crowd with levels 0, 0, 1, 1, 2, and 3. The level-0 guesses vary between 30 and 70, and your own guess is included in the average.

Why did that happen?

Take a guess, then reveal the reason

Getting to 0 requires believing everyone reasons infinitely far, and that they believe it too. Real crowds do not. Most people take one or two steps: “if others guess randomly (average 50), I should say 33; if they say 33, I should say 22.” Then they stop. Knowing the crowd stops early, the winning guess is not 0 but a bit below the average the crowd actually produces. Modeling the distribution of reasoning depths, rather than assuming it is infinite, is what predicts real behaviour.

The math

Definition, level-k and cognitive hierarchy

In a level-k model, a level-0 player acts non-strategically (here, guesses uniformly, averaging 50). A level- $k$ player best-responds to the belief that everyone else is level- $(k-1)$ . In the beauty contest, level-1 guesses $\tfrac23 \cdot 50 = 33$ , level-2 guesses $\tfrac23 \cdot 33 = 22$ , and so on. The cognitive-hierarchy variant assumes a level- $k$ player best-responds to a mix of all lower levels, with the depth distribution often Poisson with mean around $1.5$ .

Full rationality iterates the two-thirds map to the fixed point $0$ . Empirically the modal guesses are $33$ and $22$ , exactly one and two steps down from $50$ (Nagel, 1995). Because the crowd is a mixture of shallow reasoners, the average lands well above $0$ , and the best response to that crowd, around $20$ to $22$ , wins. The equilibrium prediction of $0$ is not just wrong; it is a losing guess.

Work it out

Climb the ladder, then stop early

Level-0 guesses average $50$ . A level-1 reasoner best-responds: $\tfrac23 \cdot 50 = 33$ . A level-2 reasoner assumes others are level-1 and guesses $\tfrac23 \cdot 33 = 22$ . Level-3 would guess $\tfrac23 \cdot 22 \approx 15$ . Iterating forever reaches $0$ , the Nash equilibrium.

But suppose the crowd is mostly level-1 and level-2, with a few level-0 and level-3. The average of their guesses is roughly in the low $30$ s, so two-thirds of it is around $20$ to $22$ . Best-responding to the actual crowd, not to an imagined infinitely-deep one, you should guess about $20$ . Playing $0$ assumes everyone else reasoned to the limit, which they did not, so it loses.

Key idea

People reason a bounded number of steps, not to infinity. Level-k and cognitive-hierarchy models best-respond to shallow-reasoning crowds and predict real play, where the equilibrium of 0 both mispredicts and loses.

Check yourself

Check yourselfapply

A level-2 reasoner in the beauty contest guesses about what?

Check yourselfanalyse

Why does the Nash prediction of 0 fail in practice?

Check yourselfbrings back: common knowledge

How does level-k relate to common knowledge of rationality?

Sources

Nagel, R. (1995). Unraveling in guessing games: an experimental study. American Economic Review, 85(5), 1313-1326.
Camerer, C. F., Ho, T.-H., and Chong, J.-K. (2004). A cognitive hierarchy model of games. Quarterly Journal of Economics, 119(3), 861-898.