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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.

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.

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 higher_order_beliefs, bounded_rationality, common_knowledge, and the cognitive_hierarchy family.

Relationships to Other Abstractions

Local relationship map for Guess 2/3 of the AverageParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Guess 2/3 ofthe AverageDOMAINDomain-specific abstraction: Beauty Contest Game — is a kind ofBeautyContest GameDOMAIN

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

Parents (1) — more general patterns this builds on

  • 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

Neighborhood in Abstraction Space

Guess ⅔ of the Average sits in a crowded region of the domain-specific corpus (0th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Strategic Interaction & Game Theory (23 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-07-12