Banzhaf Power Index

A measure of a voter's ability to influence the outcome of a vote, calculated as the number of coalitions in which they are pivotal.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

In a weighted voting game $[q; w_1,\ldots,w_n]$ , a voter $i$ is critical (pivotal) for a coalition $S$ if adding $i$ to $S$ makes $S$ winning ( $w(S)<q \le w(S)+w_i$ ). The Banzhaf index is $\beta_i = \eta_i / \sum_j \eta_j$ , where $\eta_i$ is the number of coalitions in which $i$ is critical.

The absolute Banzhaf index is $\eta_i / 2^{n-1}$ (proportion of coalitions where $i$ is pivotal).

Intuition

A voter's power is not their share of votes but their chance of being the deciding vote.

Banzhaf (1965) used this to show that the weighted voting system of Nassau County's Board of Supervisors gave some districts zero power despite having votes.

Worked example

In the EU Council with triple majority rules, Banzhaf indices show that large countries have disproportionately high power relative to population.

In a shareholder election with 49% vs 51%, the 51% bloc has essentially all the power despite having only 2% more shares.

The math

Related to, but distinct from, the Shapley-Shubik index, which orders voters by random arrival rather than random coalitions.

Computing Banzhaf indices is #P-complete in general but tractable for small or structured games.

Where it is used

Constitutional analysis, corporate governance, shareholder voting, and international treaty ratification.

Used in legal challenges to voting systems on equal-protection grounds.

Go deeper

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Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.