Shapley value

The unique cooperative-game allocation that assigns each player their average marginal contribution over all coalition-entry orders under efficiency, symmetry, dummy and additivity axioms.

Version

v1 · 2026-09-08 · History

Domain-specific #

6700

Origin domain

cooperative game theory

Subdomain

payoff allocation

Core Idea

The Shapley value distributes a cooperative game's total value by averaging each player's incremental contribution across every possible coalition order. For each permutation, a player receives the value added when joining predecessors; averaging over permutations balances all coalition contexts and satisfies the axiomatic fairness characterization. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of cooperative game theory. It is axiomatically unique marginal-contribution allocation over the complete coalition lattice.

Scope of Application

Shapley value belongs to cooperative game theory and is useful where the analyst can specify a finite player set, characteristic value for every coalition, permutations of player entry, marginal contribution, factorial weights, total surplus or cost and allocation vector, then evaluate the allocation is computed from one characteristic function and exactly satisfies efficiency, symmetry, null-player and additivity under the standard transferable-utility model. The scope is broad within that domain but bounded by the need for the allocation is computed from one characteristic function and exactly satisfies efficiency, symmetry, null-player and additivity under the standard transferable-utility model. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the allocation is computed from one characteristic function and exactly satisfies efficiency, symmetry, null-player and additivity under the standard transferable-utility model the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Shapley value can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Shapley value. Shapley value compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: a finite player set, characteristic value for every coalition, permutations of player entry, marginal contribution, factorial weights, total surplus or cost and allocation vector. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the allocation is computed from one characteristic function and exactly satisfies efficiency, symmetry, null-player and additivity under the standard transferable-utility model independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of cooperative game theory because they reuse a finite player set, characteristic value for every coalition, permutations of player entry, marginal contribution, factorial weights, total surplus or cost and allocation vector, For each permutation, a player receives the value added when joining predecessors; averaging over permutations balances all coalition contexts and satisfies the axiomatic fairness characterization., and type the carrier, state every parameter and convention in the definition, test that the allocation is computed from one characteristic function and exactly satisfies efficiency, symmetry, null-player and additivity under the standard transferable-utility model, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Shapley value Domain-specific

Parents (1) — more general patterns this builds on

• Shapley value is a kind of Allocation Prime

  The proposed strict upward parent is prime:allocation.

Hierarchy path (1) — routes to 1 parentless root

• Shapley value → Allocation → Scarcity → Constraint

Neighborhood in Abstraction Space

Shapley value sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Fair Division & Cooperative Power (7 abstractions)

Nearest neighbors

• Shapley–Shubik power index — 0.94
• Banzhaf power index — 0.90
• Strategy-stealing argument — 0.88
• Truthful cake-cutting — 0.88
• Nash equilibrium computation — 0.88

Computed from structural-signature embeddings · 2026-09-08