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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3409
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Cooperative Game Theory and Collective Choice → Economics & Finance

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.

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Whose Vote Flips It?

Imagine a club where some kids get more votes than others. To see how powerful you really are, look at every way the club could vote and count the times your vote alone would flip the answer. Having lots of votes does not always mean you are the one who decides.

Counting Swing Votes

In some groups, members have different numbers of votes, and a plan passes if it gets enough votes, called the quota. The Banzhaf power index measures real voting power by checking every possible group of yes-voters and counting how often a member is the swing voter: if that member switched, the result would change. A member with many votes can still rarely be the swing voter, and a member with few votes can matter a lot. The counts can be turned into shares so everyone's power adds up to a whole. The answer depends on the rules, like the quota and how the groups are counted.

Pivotal Voting Power Index

The Banzhaf power index measures voting power in a weighted voting system, where each voter has a weight and a proposal passes if the supporting weight reaches a quota. It separates formal weight from pivotal influence. The method goes through all possible coalitions, or all possible patterns of yes/no votes, and for each voter counts how many times removing or flipping that voter changes the outcome; those are the times the voter is critical, or a 'swing'. The counts can be reported as an absolute index (often read as a probability of being decisive under a stated assumption, like every coalition being equally likely) or normalized so everyone's shares add to 1. A voter who is never critical is a null player with zero power, even if they hold votes. The answer depends on the quota, the probability model, and which version is reported, so those must be stated.

The Banzhaf power index quantifies a voter's a priori influence in a voting game by counting how often that voter is critical. Given players with voting weights, a quota and winning rule, and a sample space of coalitions (or binary vote profiles) with a declared probability model, one enumerates the cases and tests for each player whether removing them from a winning coalition, or flipping their vote, changes the outcome. The number of such swings is then reported either in absolute form, typically normalized by the number of coalitions of the other players so it can be read as a probability of decisiveness under independence assumptions, or in relative form, normalized so the indices of all players sum to one. The index makes precise the gap between formal weight and pivotal influence: weights can be very unequal while powers are equal, and a player with positive weight can be a null player who is never critical. Because results depend on the quota, coalition model, normalization, and independence assumptions, a valid analysis states them all, along with the computational method, since exact enumeration grows exponentially with the number of players.

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.
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) — outgoing typed relations

Banzhaf power index is a kind of Counterfactual Reasoning Prime

It remains its own entry because its identity is fixed by 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.

Hierarchy path (1) — routes to 1 parentless root

Banzhaf power index → is a kind of Counterfactual Reasoning

Neighborhood in Abstraction Space¶

Banzhaf power index sits in a crowded region of the domain-specific corpus (25^th 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

Computed from structural-signature embeddings · 2026-10-25