Game Theory
The Keynesian Beauty Contest: Guessing What Others Will Guess
The Keynesian Beauty Contest is the strategic situation in which the best action is not to pick what you think is best, but to anticipate what everyone else will pick — and then to anticipate that everyone else is doing the same anticipation about you. John Maynard Keynes used a 1930s newspaper photo contest as a metaphor for the stock market: winners guessed not the prettiest face, but the face the average reader would call prettiest. The idea is made concrete by theguess 2/3 of the average game, where iterated reasoning drives rational guesses down toward zero — yet almost nobody actually guesses zero.
- Named afterJohn Maynard Keynes (1936)
- First describedThe General Theory, Ch. 12
- Canonical gameGuess 2/3 of the average (0–100)
- Nash equilibriumEveryone guesses 0
- Typical human average≈ 20–35 (winner ≈ 13–23)
- Key ideaBeliefs about others' beliefs
Interactive visualization
Press play, or step through manually. The visualization is yours to drive — try it before reading on.
Watch the 60-second explainer
A condensed visual walkthrough — narrated, captioned, under a minute.
Keynes's original metaphor: beauty, not the beautiful
In Chapter 12 of The General Theory of Employment, Interest and Money (1936), Keynes described a newspaper competition in which readers were shown 100 photographs and asked to pick the six prettiest faces. The prize went to whoever's choices came closest to the average preference of all entrants. Keynes's insight was that a clever entrant does not choose the faces she finds prettiest, nor even the faces she thinks are objectively prettiest.
Instead, as Keynes put it, "we devote our intelligences to anticipating what average opinion expects the average opinion to be." There are, he noted, third, fourth, and higher degrees. You pick faces you expect others to pick — because you expect them to be doing the same. Keynes used this as a metaphor for professional investing: a stock's short-run price reflects not fundamentals but a crowd's forecast of the crowd's forecast. Buying a share can be less about whether the company is good and more about whether other people will soon think it is good.
The mechanism made precise: guess 2/3 of the average
The metaphor becomes a solvable game. Let N players each simultaneously pick a number xᵢ in the interval [0, 100]. Compute the average x̄ = (∑ xᵢ) ÷ N. The target is T = (2/3) × x̄, and the winner is whoever is closest to T (ties split the prize). More generally, the parameter p sets the target at T = p × x̄ — this is the p-beauty contest (here p = 2/3).
Why does rational reasoning collapse to 0? Because guesses above a threshold are dominated. The average can be at most 100, so T ≤ (2/3)×100 ≈ 66.7. Any guess above 66.7 can never win — it is farther from the target than 66.7 no matter what. So no rational player guesses above 66.7. But if everyone knows that, the average is at most 66.7, so T ≤ (2/3)×66.7 ≈ 44.4, and guesses above 44.4 are now dominated. Iterate this elimination of dominated strategies: 66.7 → 44.4 → 29.6 → 19.8 → … → 0. The unique Nash equilibrium is that everyone guesses 0 — the only fixed point where x = (2/3)×x.
A worked example with real numbers
The animation's core dynamic is the layered march downward. Formalized as levels of reasoning (level-k), each level is one more step of "I think that you think":
- Level 0 — non-strategic. Picks randomly, so on average guesses
≈ 50(midpoint of 0–100). - Level 1 — assumes everyone is level 0, so best-responds to an average of 50:
(2/3)×50 ≈ 33.3. - Level 2 — assumes everyone is level 1:
(2/3)×33.3 ≈ 22.2. - Level 3 —
(2/3)×22.2 ≈ 14.8. - Level ∞ — full common knowledge of rationality:
0.
Real data pins where humans stop. When the Financial Times ran the game for readers in 1997, the winning number was 13. In a large sample by Rosemarie Nagel (1995), the average was around 27–35 and guesses clustered near 33 and 22 — exactly the level-1 and level-2 focal points, with a spike of level-∞ players at 0. The lesson: guessing 0 (the "rational" answer) is a near-guaranteed loss in one-shot play, because you'd need everyone to be perfectly rational. The practical winning move is to reason one step deeper than the crowd — roughly two-thirds of ~33, i.e. aim near 22.
The critical assumption — and why it fails
The equilibrium at 0 rests on common knowledge of rationality: not just that every player is rational, but that everyone knows everyone is rational, knows that everyone knows, and so on infinitely. That is an extraordinarily strong assumption, and the beauty contest is the cleanest demonstration that it breaks in practice.
Two things go wrong. First, bounded rationality: people perform only a few iterations of reasoning before stopping. Second, and more subtly, even a fully rational player should not guess 0 if she believes others are boundedly rational — best-responding to a crowd that averages 30 means guessing 20, not 0. So deviating from equilibrium can itself be the smart move. This is why the beauty contest is a favorite in behavioral and experimental economics: it separates raw intelligence from a correct model of other people. Notably, when the game is played repeatedly with the same group and results announced each round, guesses do march down toward 0 — as players learn the crowd is getting sharper, exactly the collapsing cloud the animation depicts.
Where it shows up: bubbles, IPOs, and the greater-fool market
Keynes's target was speculation, and the metaphor names a real market failure mode. When an asset's price is driven by beliefs about others' beliefs rather than cash flows, you get the greater-fool dynamic: I'll overpay because I expect a greater fool to overpay more later. Historical episodes fit the pattern:
- Dutch Tulip Mania (1636–37) — bulbs traded at multiples of a craftsman's annual income because buyers anticipated other buyers, until the coordination snapped.
- The dot-com bubble (peaking March 2000) — the NASDAQ ran to ~5,048 on companies with no earnings; the trade was "others will keep buying," not discounted fundamentals.
- Meme stocks (GameStop, Jan 2021) — coordinated retail buying pushed GME from ~$17 to an intraday $483; the explicit game was guessing when the crowd would pile in, and when it would fold.
The same logic appears in currency attacks (Soros vs. the pound, 1992), bank runs, and IPO pricing, where underwriters set a price they expect the market to endorse. In each case the profitable question is Keynes's: not "what is this worth?" but "what will average opinion soon think average opinion thinks it's worth?"
A common misconception, and the subtle point
The most common error is treating "guess 0" as the right answer to win. It is the game-theoretically undominated equilibrium, but it is almost always a losing guess against real humans — you'd win only if literally everyone reasoned infinitely. The right practical answer depends on your model of the population's average depth of reasoning, not on pure logic. Overshooting the crowd's sophistication loses just as surely as undershooting it.
The subtle point is that the beauty contest is fundamentally about higher-order beliefs, not about beauty, numbers, or stocks. It is the same structure behind self-fulfilling prophecies, fashion and fads, social conventions (which side of the road to drive on), and the trader's adage that markets can "remain irrational longer than you can remain solvent" (an aphorism popularly but wrongly pinned on Keynes). Whenever the payoff to your choice depends on matching others' choices — which themselves depend on matching yours — you are in a beauty contest, and the winning move is to reason exactly one level deeper than the person you're trying to beat.
| Feature | Nash / rational-agent theory | Level-k / cognitive hierarchy (observed) |
|---|---|---|
| Predicted guess | 0 for everyone | Positive number, ~20–35 on average |
| Assumption about others | Everyone is fully rational and knows it (common knowledge) | Finite steps of reasoning; others assumed slightly dumber |
| How reasoning stops | Infinite iteration → fixed point at 0 | Stops after 1–3 levels (level-0 → 50, level-1 → 33, level-2 → 22) |
| Winning strategy | Undominated but almost never wins in practice | Guess ~2/3 of your estimate of others' average |
| Empirical fit | Poor for one-shot play | Strong; also predicts convergence toward 0 with repetition |
Frequently asked questions
Why is the Nash equilibrium 0 if almost no one wins by guessing 0?
Because 0 is the only fixed point of the game: it is the single number that equals two-thirds of itself, so if everyone guessed it, no one could do better by deviating. Reaching it requires infinite iterated deletion of dominated strategies and common knowledge that everyone is perfectly rational. In real one-shot play, people stop reasoning after a couple of steps, so the average lands near 20–35, and guessing 0 loses. Equilibrium describes the logical endpoint, not the smart bet against actual humans.
What is the actual winning number if I want to win?
Estimate the crowd's average and multiply by two-thirds. Empirically the average of naive-to-savvy players sits around 30–35, so the target is roughly (2/3)×33 ≈ 22. Winning numbers in famous runs have landed near 13–23 (the Financial Times reader contest in 1997 was won at 13). The trick is to out-reason the crowd by exactly one level — not to jump all the way to 0, which over-assumes everyone else's sophistication.
What does p do in the 'p-beauty contest'?
The target is p × average. With p = 2/3 (< 1) reasoning drives guesses down to 0. If p > 1 the pull reverses and guesses climb toward the upper bound (100), because you now want to beat the average upward — a nice model of upward-spiraling bubbles. At p = 1 any common number is an equilibrium (pure coordination). Changing p thus turns the same game into a model of deflationary races-to-the-bottom or inflationary manias.
How does this relate to Keynes's view of the stock market?
Keynes argued that short-horizon investing is not about estimating an asset's intrinsic value ('enterprise') but about forecasting the average forecast ('speculation'). A professional profits by anticipating changes in mass psychology just ahead of the crowd. This is why prices can detach from fundamentals for long stretches, why fads and momentum work, and why a market 'can remain irrational longer than you can remain solvent' (a trader's adage often misattributed to Keynes). The beauty contest is his compact model of that reflexivity.
What is level-k or cognitive hierarchy thinking?
It is a model of bounded strategic reasoning. A level-0 player acts non-strategically (e.g., guesses randomly, ~50). A level-1 player best-responds to level-0 (guesses ~33). A level-2 player best-responds to level-1 (~22), and so on. Cognitive hierarchy models assume a distribution over these levels — most people are level 1 or 2, few reach higher. It fits beauty-contest data far better than full Nash and is now a standard tool for predicting behavior in one-shot strategic games.
Does repeating the game change the outcome?
Yes. When groups play repeatedly and see each round's target announced, guesses fall steadily toward 0 over several rounds — the collapsing cloud in the animation. Players learn that the crowd is getting sharper and adjust downward. This is direct evidence that the equilibrium is reachable through learning even though it is not reached by pure introspection in a single shot, and it mirrors how speculative markets can grind toward or away from fundamentals as participants update.