Banzhaf power index¶
A voting-power measure based on how often a voter is critical in winning coalitions, normalized either across that voter's possible swings or across all voters' swings.
Core Idea¶
The index distinguishes formal voting weight from pivotal influence and depends on quota, coalition model, normalization, independence assumptions, and whether absolute or relative power is reported. All coalitions or vote profiles are enumerated under a declared probability model; removing or flipping one voter tests whether the outcome changes, and critical occurrences are counted and normalized. 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¶
Banzhaf power index belongs to cooperative game theory and collective choice and is useful where the analyst can specify the typed cooperative game theory and collective choice carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the players and voting weights, quota and winning rule, coalition or binary-profile sample space, critical-voter definition, probability assumptions, absolute or normalized formula, null players, computational method, and interpretation are explicit. The scope is broad within that domain but bounded by the need for the players and voting weights, quota and winning rule, coalition or binary-profile sample space, critical-voter definition, probability assumptions, absolute or normalized formula, null players, computational method, and interpretation are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the players and voting weights, quota and winning rule, coalition or binary-profile sample space, critical-voter definition, probability assumptions, absolute or normalized formula, null players, computational method, and interpretation 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 Banzhaf power index. Banzhaf 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 and collective choice carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the players and voting weights, quota and winning rule, coalition or binary-profile sample space, critical-voter definition, probability assumptions, absolute or normalized formula, null players, computational method, and interpretation are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of cooperative game theory and collective choice because they reuse the typed cooperative game theory and collective choice carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, All coalitions or vote profiles are enumerated under a declared probability model; removing or flipping one voter tests whether the outcome changes, and critical occurrences are counted and normalized., and type the carrier, state every parameter and convention in the definition, test that the players and voting weights, quota and winning rule, coalition or binary-profile sample space, critical-voter definition, probability assumptions, absolute or normalized formula, null players, computational method, and interpretation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Banzhaf power index Domain-specific
Parents (1) — more general patterns this builds on
-
Banzhaf power index is a kind of Counterfactual Reasoning Prime
The proposed strict upward parent is
prime:counterfactual_reasoning.
Hierarchy path (1) — routes to 1 parentless root
- Banzhaf power index → Counterfactual Reasoning
Neighborhood in Abstraction Space¶
Banzhaf power index sits in a moderately populated region (42nd 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
- Non-cooperative game theory — 0.90
- Shapley value — 0.90
- Game form — 0.89
- Rank aggregation — 0.89
Computed from structural-signature embeddings · 2026-09-08