The p-beauty contest guessing game, explained

The p-beauty contest is a guessing game in which each player picks a number from 0 to 100, and the winner is the player closest to p times the average of all choices. Rosemarie Nagel's December 1995 American Economic Review paper set the standard design: p is common knowledge, the prize is fixed, and tied winners split it. For p from 0 up to but not including 1, the only Nash equilibrium is for every player to choose 0. Nagel found first-round choices far from 0, clustered one or two steps of reasoning below 50. Named for a passage by John Maynard Keynes, the game tests how far a player, human or machine, reasons about others' reasoning.

Keynes's newspaper contest

In the 1936 General Theory, Keynes likened the stock market to a newspaper contest in which entrants pick the faces that others will judge prettiest. Camerer, Ho and Chong quote the passage in their August 2004 Quarterly Journal of Economics paper. Keynes calls the decisive level "the third degree," in which entrants anticipate "what average opinion expects the average opinion to be." Keynes adds that some practice higher degrees, which the authors read as a belief that few investors go beyond three steps.

Nagel notes that Keynes's thought experiment has p equal to 1, where steps of reasoning cannot be told apart. With p below 1, each step lands on a different number. Numbers above 100p are weakly dominated, and repeating that logic leads to 0. If only integers are allowed, all choosing 1 is also an equilibrium. Nagel learned the game from Roger Guesnerie, who used it as a classroom demonstration.

Nagel's 1995 sessions

Nagel ran three sessions with p of 1/2, four with p of 2/3, and three with p of 4/3. Each group of 15 to 18 subjects played four rounds. After every round, all numbers, the mean, and p times the mean went on the blackboard. Each round's winner received 20 German marks (DM), about $13.

First-round means were 27.05 with p of 1/2 (median 17), 36.73 with p of 2/3, and 60.12 with p of 4/3. No subject chose 0 in the sessions with p of 1/2 or 2/3, and only 6 percent chose numbers below 10.

Choices then moved toward equilibrium. With p of 1/2, more than half the fourth-round choices were below 1, yet only 3 of 48 subjects chose 0. With p of 4/3, where all choosing 100 is an equilibrium, 68 percent chose 100 in round four.

Levels of reasoning

Nagel's model starts from 50, the expected choice of a random player. A step 1 player chooses 50p, the best reply to random play, a step 2 player 50p squared, and so on. In the p of 1/2 and 2/3 sessions, almost 50 percent of first-round choices fell near step 1 or step 2, and only 6 to 10 percent at step 3 or higher. In later rounds, measured from the previous mean, the modal choice stayed at step 2, so Nagel rejected the idea that depth of reasoning rises with experience.

Camerer, Ho and Chong generalize the idea. In their cognitive hierarchy, step 0 players randomize, and a step k player best-responds to a Poisson-distributed mix of players at steps 0 through k minus 1. The parameter tau is the average number of steps. Across 24 beauty contest data sets, the median estimate of tau was 1.61, which the authors say explains why averages stop around 30. They report that an average of 1.5 steps fits data from many games.

First-round means differ by group

The 2004 paper's table includes these groups, all in games whose equilibrium is 0:

Group	Sample	Mean choice
Caltech board	73	42.6
Corporate CEOs	20	37.9
German students (Nagel 1995)	66	37.2
Portfolio managers	26	24.3
Newspaper contest entrants	7,884	23.0
Game theorists	136	19.1

The sources differ on Nagel's data: the 2004 table lists 66 subjects and a mean of 37.2, while the 1995 paper reports 36.73 and counts 67 subjects with p of 2/3.

The newspaper row pools contests of 3,696, 1,460, and 2,728 entries. Bosch-Domènech, Montalvo, Nagel and Satorra analyze such contests from newspapers in the United Kingdom, Spain, and Germany, and report Level-1, Level-2, and Level-3 reasoning in all samples, with the equilibrium choice prominent only among newspaper readers and theorists.

Language models in the guessing game

Published tests disagree, and the split follows model generation. In Siting Estee Lu's 2024 preprint, agents built on nine models, including GPT3.5 and GPT4, reasoned between level 0 and level 1, below human results, though their choices converged toward equilibrium in repeated play.

Alekseenko, Dagaev, Paklina and Parshakov replayed five human studies with five 2024-2025 models, including GPT-4o and Claude-Sonnet-4, taking 50 one-shot responses per scenario. In Nagel's p of 1/2 setting, model means ran from 2.00 to 20.42, against the human 27.05. With p of 2/3 they ran from 2.80 to 22.24, against 36.73. Model and human averages matched in 7 percent of comparisons. In two-player games, no response explained that 0 is weakly dominant. The prompts gave no payoff structure, a limit the authors state.

Kader and Lee compared standard and reasoning models with 11 players and p of 2/3. Standard models reasoned worse than typical human subjects. Reasoning models ran 1 to 2 levels deeper and often matched or exceeded people. With p of 4/3, only reasoning models showed non-random behavior, which the authors suggest reflects training on games that iterate downward.

What to check

• Which p, group size, and number rule applied, given that all choosing 1 is also an equilibrium with integers?
• Is a model's guess compared with the human first-round mean for the same design, such as 27.05 for p of 1/2?
• Does the test score winning against the actual group or only closeness to 0? Against Nagel's p of 1/2 subjects, the best reply was about 13.5.
• Was p also set above 1, where Kader and Lee found only reasoning models behaved non-randomly?

Azcuna's CALIBER puts AI agents in practice markets with play money and a news feed whose truth is known, and measures from their trades whether they are gullible, follow the crowd, overpay, or can be steered. It was proposed to DARPA under SBIR Phase I topic DPA26BZ06-DV026, and the proposal was submitted on October 21, 2026.

Sources

1. 1Unraveling in Guessing Games: An Experimental Study · Rosemarie Nagel, American Economic Review 85(5), December 1995 (RePEc record)
2. 2A Cognitive Hierarchy Model of Games · Camerer, Ho and Chong, Quarterly Journal of Economics 119(3), August 2004 (RePEc record)
3. 3One, Two, (Three), Infinity, ...: Newspaper and Lab Beauty-Contest Experiments · Bosch-Domènech, Montalvo, Nagel and Satorra, American Economic Review 92(5), December 2002 (RePEc record and abstract)
4. 4Strategic Interactions between Large Language Models-based Agents in Beauty Contests · arXiv (Siting Estee Lu), version 2, October 2024
5. 5Strategizing with AI: Insights from a Beauty Contest Experiment · arXiv (Alekseenko, Dagaev, Paklina and Parshakov), version 3, October 2025
6. 6The Emergence of Strategic Reasoning of Large Language Models · arXiv (Kader and Lee), version 4, October 2025

Where this shows up in our work

• CALIBER, the market testbed for AI agents measures from an AI agent's trades whether it is gullible, herds, overpays, or can be steered.
• LEDGER, auditable Army acquisition decisions turns a trade study into a package where every number cites its source and AI drafts but never decides.