Beauty contest game
Also known as: Keynesian beauty contest, Guess two-thirds of the average
Win by guessing not what's true, but what everyone thinks everyone thinks.
What it means
A game, inspired by Keynes's metaphor for the stock market, in which players choose a number and the winner is whoever lands closest to a fraction (often two-thirds) of the group's average guess. The rational equilibrium through iterated reasoning is zero, but reaching it requires infinite steps of 'I think that you think,' which real people truncate after a round or two. Average guesses therefore cluster well above zero, revealing limited depth of strategic reasoning rather than irrationality. It matters because it operationalizes higher-order beliefs, links directly to speculative bubbles, and is the canonical test of level-k thinking.
From Keynes's newspaper to Nagel's lab
Keynes described a 1930s newspaper contest in which readers picked the six prettiest faces from a hundred photographs and won if their choice matched the average preference. The clever entrant, he noted, picks not the faces he finds prettiest but those he expects others to favour, and some even reach the fourth, fifth and higher degrees of such reasoning. He offered it as an analogy for share prices, where you buy what you think others will buy. For sixty years it stayed a metaphor. Rosemarie Nagel turned it into a clean experiment in 1995 by asking subjects to guess two-thirds of the average, giving the intuition a number, a prediction, and a way to count how many steps of reasoning people actually take.
What the experiments show
Nagel's guesses did not scatter randomly. They piled up at focal points: near 50, where players treat the average as a coin toss; near 33, one best response below 50; and near 22, two steps down, with almost none at the equilibrium of zero. Average first-round guesses across studies cluster around 20 to 35, implying most people reason only one or two steps deep. The pattern is strikingly robust. Bosch-Domenech and colleagues ran the game as newspaper and magazine contests in Britain (the Financial Times), Spain (the business daily Expansión) and Germany (the science magazine Spektrum der Wissenschaft), drawing thousands of self-selected entrants, and recovered the same spikes. Repeating the game with feedback pushes guesses steadily toward zero, so the shallow reasoning reflects a starting point rather than a fixed ceiling on ability.
Counting the steps: level-k and cognitive hierarchy
The game became the workhorse for models of bounded strategic reasoning. In level-k theory a level-0 player chooses without strategizing, often assumed to pick at random and so anchor near 50; a level-1 player best-responds to level-0, a level-2 to level-1, and so on. Camerer, Ho and Chong's cognitive-hierarchy model refines this: a step-k thinker best-responds to a whole distribution of lower types rather than assuming everyone sits exactly one step below. Fitting many games, they estimate an average of roughly 1.5 steps of reasoning. These models earn their keep because they predict out of sample, and because estimated depth behaves like a stable individual difference, correlating with cognitive ability and, in imaging work, with activity in the medial prefrontal cortex.
Where higher-order beliefs bite
The contest's real subject is higher-order belief: not what is true, but what everyone thinks everyone thinks. That structure recurs wherever value depends on others' choices. In asset markets a trader may hold a stock she considers overpriced because she expects the crowd to keep buying, which was Keynes's original point and a mechanism behind bubbles. Currency pegs, bank runs and IPO pricing share the logic, since the payoff to acting hinges on forecasting others' forecasts. The caveat is interpretive. A high guess can mean shallow reasoning, or an accurate read that others reason shallowly, and the two are hard to separate; a sophisticated player deliberately guesses above the equilibrium because guessing zero against non-zero opponents loses. Depth of reasoning and beliefs about others' depth stay entangled.
Examples
Asked to pick two-thirds of the average of everyone's number from 0 to 100, most groups average around 20-30, not the rational 0.
Fund managers buy a stock they privately think is overpriced, betting others will buy it too — picking not the best company but the one everyone expects everyone else to pick.
Judging a bake-off, you don't simply pick your favourite cake; you pick the one you think the other judges will pick, and every judge is doing exactly the same thing.
In a party primary, voters often back the contender they judge most 'electable' — not their own favourite, but the one they expect other voters to expect everyone else to rally behind.
Speculators attack a shaky currency peg not when fundamentals finally give way, but the moment each expects the others to sell, since a devaluation pays off only if the crowd moves together.
First described in Keynes (1936); Nagel (1995).
Key references
- Coricelli, G., & Nagel, R. (2009). Neural correlates of depth of strategic reasoning in medial prefrontal cortex. Proceedings of the National Academy of Sciences, 106(23), 9163-9168. doi.org/10.1073/pnas.0807721106
- Camerer, C. F., Ho, T.-H., & Chong, J.-K. (2004). A cognitive hierarchy model of games. Quarterly Journal of Economics, 119(3), 861-898. doi.org/10.1162/0033553041502225
- Bosch-Domenech, A., Montalvo, J. G., Nagel, R., & Satorra, A. (2002). One, two, (three), infinity, ...: Newspaper and lab beauty-contest experiments. American Economic Review, 92(5), 1687-1701. doi.org/10.1257/000282802762024737
- Ho, T.-H., Camerer, C. F., & Weigelt, K. (1998). Iterated dominance and iterated best response in experimental 'p-beauty contests'. American Economic Review, 88(4), 947-969. authors.library.caltech.edu/records/2wqmd-n6102
- Nagel, R. (1995). Unraveling in guessing games: An experimental study. American Economic Review, 85(5), 1313-1326. ideas.repec.org/a/aea/aecrev/v85y1995i5p1313-26.html