Guess ⅔ of the Average¶

A single-shot game where each player picks a number closest to two-thirds of the group's mean — its Nash equilibrium of zero is never reached, and because each level of iterated best-response leaves a distinct numerical signature, the modal guess reads off a population's depth of strategic reasoning.

Version: v3 · 2026-09-28 · History · 1 correction
Domain-specific #: 475
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Game Theory, Behavioural Game Theory → Mathematics

Core Idea¶

In the guess-⅔-of-the-average game, each participant picks a number from 0 to 100 and the winner is closest to two-thirds of the mean. Iterated elimination of dominated strategies gives a unique Nash equilibrium of zero. The structural content is the gap between that and actual play: real participants perform only finitely many steps, so level-1 players submit ~33 and level-2 ~22. Each reasoning level leaves a distinct numerical signature, making the game an instrument for the depth of iterated strategic reasoning.

How would you explain it like I'm…

Guess What They Guess

Everyone picks a number from 0 to 100. The winner is whoever is closest to two-thirds of everyone's average. If everyone kept thinking "but they'll guess lower, so I'll go lower," they'd all end up at zero. Real people stop thinking after a step or two, so the guesses stay much higher.

How Many Steps Ahead?

In this game, each player picks a number between 0 and 100. The winner is the one closest to two-thirds of the average of all picks. Someone who doesn't think ahead might pick about 50. Someone who thinks one step ahead picks about 33, two-thirds of 50. Someone who thinks two steps picks about 22. If everybody thought forever, everyone would pick 0, but in real games people usually stop after one or two steps. So the game is a way to measure how many steps ahead a group of people thinks.

Two-Thirds Beauty Contest

Guess 2/3 of the Average is a one-shot game: everyone picks a number from 0 to 100, and the closest to two-thirds of the mean wins. The average can never exceed 100, so the target never exceeds about 67, making any guess above 67 pointless. If everyone knows that, the target can't exceed about 44, and repeating this reasoning leads all the way to 0, the game's Nash equilibrium. But in experiments, people don't go all the way. A 'level-0' player guesses around 50, level-1 around 33, level-2 around 22, and most people land at levels 1 or 2. So the average guess tells researchers how many steps of reasoning a group typically does.

The guess-2/3-of-the-average game is a one-shot game: players choose numbers in [0,100], and the one closest to two-thirds of the mean wins. Two-thirds of the maximum possible mean is about 67, so any choice above 67 is weakly dominated. Under common knowledge of rationality, iterated elimination of weakly dominated strategies shrinks the bounds to 44, then lower, converging to the unique Nash equilibrium of zero. The interesting content is the gap between that prediction and actual behavior. In level-k terms, level-0 players choose near the focal 50, level-1 players best-respond with about 33, and level-2 players with about 22. Experiments consistently find clustering at levels 1 and 2. Experiments include newspaper contests in the Financial Times and Spektrum der Wissenschaft and lab studies. The mean guess thus serves as an instrument for inferring the depth of iterated reasoning. It distinguishes populations by education, game-theory training, and financial expertise, and it calibrates cognitive-hierarchy models. Moulin formalized the game, and Nagel shaped its experimental use, linking it to Keynes's beauty contest.

Scope of Application¶

Because the game is a named protocol, it applies wherever a population of strategic participants can submit a number under the winner rule.

Behavioral game theory — the home laboratory, the canonical instrument for measuring level-k reasoning.
Cognitive-hierarchy modeling — the workhorse data calibrating Camerer-Ho-Chong models.
Large-scale population surveys — the famous Financial Times and Spektrum runs.
Pedagogical demonstration — classroom use making higher-order beliefs tangible.
Financial-market microstructure — the metaphorical edge of the Keynesian beauty contest.

Clarity¶

The game converts "depth of strategic reasoning" from a slogan into something read off a single number. Higher-order beliefs — what you think others think — are notoriously hard to observe, and one action is usually consistent with many depths. The ⅔ game collapses that ambiguity because each level maps to a distinct signature (50, 33, 22, 14), so the modal guess pins the modal step-count. It replaces the binary "irrational" verdict with a quantitative one and distinguishes cannot-iterate from rationally-stops-short.

Manages Complexity¶

Higher-order belief is an unbounded recursion, and across most settings one action is consistent with many depths, leaving the analyst an underdetermined internal quantity. The ⅔ game compresses that tower into a single scalar: because each best-response step maps to one number, the recursion becomes a labeled sequence and the modal submission pins the modal step-count. A whole family of beliefs-about-beliefs questions collapses to "read the mean guess," with the gap from Nash zero measuring where the recursion truncates.

Abstract Reasoning¶

As a measurement instrument, the game licenses diagnostic reasoning (inferring depth from the modal guess, the gap from Nash as data not failure), boundary-drawing (separating cannot-iterate from rationally-stops-short, bounding when zero is even correct), interventionist reasoning (varying population or multiplier and predicting the shift, making depth a dependent variable), and calibration (fitting cognitive-hierarchy models to the numerical clusters).

Knowledge Transfer¶

The ⅔ game is a measurement instrument, not a causal mechanism, so it transfers literally wherever its precondition holds — across classrooms, panels, lab studies, and newspaper runs — the same protocol reading modal depth off the mean guess. The over-readings to guard against: treating a reading as a human constant (depth is population- and condition-specific), treating the gap from Nash as irrationality rather than data, and calling any beliefs-about-beliefs setting "a beauty contest" where no number is submitted. The truncation it reveals travels onward under Theory Of Mind, Bounded Rationality, Common Knowledge, and the cognitive-hierarchy family.

Relationships to Other Abstractions¶

Current abstraction Guess 2/3 of the Average Domain-specific

Parents (1) — outgoing typed relations

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

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

Guess 2/3 of the Average → is a kind of 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¶

Guess ⅔ of the Average sits in a crowded region of the domain-specific corpus (20^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