Shapley–Shubik power index¶
A voter’s a priori power measured as the probability of being pivotal over all orderings in a monotone simple voting game.
Core Idea¶
The index assumes equiprobable voter orderings rather than observed coalition formation, weighted vote share need not equal power and quota or coalition restrictions change results. For each permutation, voters enter a coalition sequentially and the first whose addition changes it from losing to winning is pivotal; each voter’s fraction of pivotal permutations yields normalized power. 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.
Scope of Application¶
Shapley–Shubik power index belongs to cooperative game theory and is useful where the analyst can specify the typed cooperative game theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the voter set and monotone simple game, weights and quota if applicable, all permutations and probability assumption, predecessor coalition, pivotal condition, factorial weighting formula, normalized index sum and symmetry dummy and efficiency properties are explicit. The scope is broad within that domain but bounded by the need for the voter set and monotone simple game, weights and quota if applicable, all permutations and probability assumption, predecessor coalition, pivotal condition, factorial weighting formula, normalized index sum and symmetry dummy and efficiency properties are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the voter set and monotone simple game, weights and quota if applicable, all permutations and probability assumption, predecessor coalition, pivotal condition, factorial weighting formula, normalized index sum and symmetry dummy and efficiency properties are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
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–Shubik power index. Shapley–Shubik power index 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¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed cooperative game theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
- Lock the constitutive rule. Express the voter set and monotone simple game, weights and quota if applicable, all permutations and probability assumption, predecessor coalition, pivotal condition, factorial weighting formula, normalized index sum and symmetry dummy and efficiency properties are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of cooperative game theory because they reuse the typed cooperative game theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, For each permutation, voters enter a coalition sequentially and the first whose addition changes it from losing to winning is pivotal; each voter’s fraction of pivotal permutations yields normalized power., and type the carrier, state every parameter and convention in the definition, test that the voter set and monotone simple game, weights and quota if applicable, all permutations and probability assumption, predecessor coalition, pivotal condition, factorial weighting formula, normalized index sum and symmetry dummy and efficiency properties are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Shapley–Shubik power index Domain-specific
Parents (1) — outgoing typed relations
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Shapley–Shubik power index is a kind of Measurement Prime
It remains its own entry because its identity is fixed by the voter set and monotone simple game, weights and quota if applicable, all permutations and probability assumption, predecessor coalition, pivotal condition, factorial weighting formula, normalized index sum and symmetry dummy and efficiency properties.
Hierarchy path (1) — routes to 1 parentless root
- Shapley–Shubik power index → is a kind of Measurement
Neighborhood in Abstraction Space¶
Shapley–Shubik power index sits in a crowded region of the domain-specific corpus (16th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Cooperative Game Theory & Power Indices (12 abstractions)
Nearest neighbors
- Banzhaf power index — 0.94
- Shapley value — 0.94
- Non-cooperative game theory — 0.91
- Game form — 0.89
- Markov strategy — 0.89
Computed from structural-signature embeddings · 2026-10-25