Beauty Contest Game¶

Have players simultaneously pick a number to land closest to p times the group average, so that iterated best-response contracts the target toward zero — turning the infinite tower of 'what others expect others to expect' into a single measurable reasoning-depth scalar read off the choices.

Version: v3 · 2026-09-28 · History
Domain-specific #: 91
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Game Theory → Economics & Finance

Core Idea¶

The beauty contest game — Keynes's 1936 metaphor, operationalized by Nagel (1995) — is a coordination game where each player picks a number and the payoff rewards proximity to p times the average, with p < 1. Iterating best-response from any belief contracts the target, reaching the unique Nash equilibrium at zero by iterated dominance. The game is a clean probe of iterated higher-order belief: how many rounds of "I think that you think that I think" a real population executes, with reasoning depth k as an observable scalar.

How would you explain it like I'm…

Guess What Others Guess

Everyone picks a number from 0 to 100, and the winner is whoever gets closest to two-thirds of the average of everyone's picks. So you have to guess what everybody else will guess. If everyone kept thinking "they'll pick lower, so I'll go even lower," they'd all end up at 0, but real people stop thinking after a step or two.

I Think That You Think

In the Beauty Contest Game, each player secretly picks a number in a range, like 0 to 100. The winner is the one closest to a fraction, such as two-thirds, of the average of all the picks. If you think others pick randomly, around 50, you'd pick about 33. But if they think the same way, you'd pick about 22, and so on down. If everyone kept thinking forever, everyone would pick 0. Real people usually stop after one or two steps, so the game shows how many steps of 'I think that you think' people really use. The name comes from an old example by the economist Keynes about guessing which face other people will vote for.

Levels of Strategic Reasoning

The Beauty Contest Game, based on Keynes's metaphor and tested in experiments by Nagel, has each player choose a number in a bounded range; whoever is closest to p times the average wins, with p less than 1. Best-responding repeatedly shrinks the target: p times the midpoint, then p² times it, and so on, reaching the only Nash equilibrium, zero. Reaching zero requires everyone to be rational and to know that everyone knows that, all the way up. Real players reason only a few levels: level-0 picks at random, level-1 best-responds to level-0, level-2 to level-1. With p = 2/3, first-time choices cluster near 33 and 22 and very few pick 0. Keynes's point was that in a newspaper beauty contest, or a stock market, you win by guessing what others will think others will think.

The Beauty Contest Game, Keynes's 1936 metaphor made into an experiment by Nagel (1995), has n players simultaneously choose numbers in a bounded interval, with the prize going to whoever is closest to p times the mean choice, p < 1. Iterated best response contracts the target from any starting belief: p times the expected mean, then p², and so on, and iterated elimination of dominated strategies yields the unique Nash equilibrium at zero. Reaching it requires common knowledge of rationality to all orders, which real populations do not display. The game therefore measures depth of iterated higher-order belief. In the level-k account, level-0 chooses randomly or uniformly, and level-k best-responds to level-(k−1), so each level makes a distinct prediction and the choice distribution reveals the distribution of reasoning depths. First-play experiments with p = 2/3 show modes near 33 (level 1) and 22 (level 2), with very few zeros. Keynes's original analogy was a newspaper contest rewarding those who guess which faces others will think others will choose, which he extended to speculative asset pricing, where prices reflect expectations of others' expectations terminating at finite depth rather than fundamentals.

Scope of Application¶

The beauty contest game lives within a single home discipline — behavioural game theory — restaged across its strategic-interaction settings, bounded to populations running iterated best response on a complementarity payoff.

Experimental economics — the home: the workhorse for measuring level-k reasoning.
Macroeconomic expectations — the analytic core of Morris-Shin coordination-under-public-information models.
Behavioural finance — the "Keynesian beauty contest" theory of speculative pricing (metaphorical).
Bounded-rationality modelling — the level-k apparatus exported to auctions and signalling games.

Clarity¶

The game's clarifying force is to turn "how many rounds of strategic recursion do people run?" into a single observable number, since each reasoning level k maps to a distinct choice (≈50·pᵏ): a cluster at 33 is level-1, at 22 level-2, and the near-absence at 0 says almost no one iterates to the limit. It also makes legible the distinction between individual rationality and common knowledge of rationality at every order, so finite-depth play measures how far up that tower a population climbs rather than evidencing irrationality.

Manages Complexity¶

The complexity tamed is the unbounded recursion of higher-order belief — an infinite tower that drives speculative markets yet is intractable as stated. The game collapses it onto a single scalar, the level of reasoning k, by exploiting its payoff: each round multiplies the target by p, so depth k maps to one predicted choice. Because each depth is a distinct, separated prediction, the analyst reads the population's whole sophistication distribution off the choices, one number per player, and reads its responsiveness to stakes and experience off the same ladder.

Abstract Reasoning¶

The game licenses a diagnostic move reading reasoning depth off the choices via the depth-k mapping, a strategic best-response move setting one's own level one step above the counterparty's (explicitly not playing Nash at 0), a boundary-drawing move localizing the equilibrium's failure to the common-knowledge-of-rationality assumption, a comparative-statics move predicting how depth shifts with stakes and experience, and a reframing move reading speculative prices as expectations about expectations terminating at finite depth.

Knowledge Transfer¶

Within behavioural game theory the game transfers as mechanism, carried by the level-k framework — the diagnostic, the best-respond-one-deeper prescription, the common-knowledge boundary, and the comparative statics all carry wherever strategic complementarities make higher-order beliefs determine play (auctions, signalling, market-entry, macro coordination). These are one substrate restaged, not cross-substrate transfer. Beyond strategic interaction the game is a measurement probe valid only where reasoning agents exist, and the general ideas it makes vivid are carried by the parents Common Knowledge, Bounded Rationality, Coordination Problem and Equilibrium Selection, and expectations. Even Keynes's asset-pricing extension travels as metaphor.

Relationships to Other Abstractions¶

Current abstraction Beauty Contest Game Domain-specific

Parents (1) — outgoing typed relations

Beauty Contest Game is a decomposition of Keynesian Beauty Contest Prime

Removing the bounded-number experiment leaves the Keynesian structure in which the best choice depends on beliefs about others' beliefs and choices.

Children (1) — abstractions linked here by typed relations

Guess ⅔ of the Average Domain-specific is a kind of Beauty Contest Game

Guess ⅔ of the Average is the Beauty Contest Game with p fixed to two thirds, the action interval fixed to 0–100, and its canonical level ladder exposed.

Hierarchy paths (9) — routes to 7 parentless roots

Beauty Contest Game → is a decomposition of Keynesian Beauty Contest → is a kind of Coordination Problem and Equilibrium Selection → presupposes Coordination → presupposes Concurrency

Show alternative paths (8)

Neighborhood in Abstraction Space¶

Beauty Contest Game sits in a crowded region of the domain-specific corpus (34^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Game-Theoretic Models & Paradoxes (38 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25