Banzhaf showed in 1968 that voting power is not proportional to vote weight — a nonlinear function of the quorum structure, not of the votes held

The lawyer-engineer John F. Banzhaf III argued that the nominal weight of a vote and the actual power to affect an outcome diverge, using a shareholder meeting as his canonical illustration: "[I]n a stockholders' meeting where members cast votes in proportion to their stock holdings, the ability of the voting members to affect the outcome is not equal and they do not have equal voting power." His method — counting the fraction of all possible voting combinations in which a given member can, by changing his vote, alter the outcome (the critical or swing vote), later named the Banzhaf power index — makes power a nonlinear function of the threshold/quorum structure rather than of raw votes. A member holding a large but sub-pivotal block can have far less power than its share suggests; a small block that is frequently pivotal can have far more.

The technique predates this article: Banzhaf introduced it in "Weighted Voting Doesn't Work: A Mathematical Analysis," 19 Rutgers L. Rev. 317 (1965), analyzing weighted county-board voting in New York, and in Iannucci v. Board of Supervisors (1967) New York's highest court adopted his approach — argued in his amicus curiae brief — holding that plans giving supervisors votes proportional to the populations they represent must be validated by mathematical computer analysis before "one person, one vote" is satisfied. Banzhaf's answer — that proportional weights do not produce proportional power — is a close cousin of the game-theoretic Shapley–Shubik index.

The same weight-versus-power gap recurs in modern token-based governance. Bongaerts, Lambert, Liebau & Roosenboom's 2025 "Vote Delegation in DeFi Governance" working paper (RSM; arXiv:2503.11940) reports that in Uniswap's DAO, parties holding fewer of their own tokens and those affiliated with the venture firm a16z draw outsized delegated votes despite decentralization branding — which the authors read as possible "window-dressing." That is Banzhaf's 1968 stockholder point playing out on-chain fifty-seven years later. This lineage — from a distributed-systems primitive through Banzhaf's critique into binding crypto regulation — is traced in observation-weighted-voting-power-gap-recurs-across-cs-law-regulation; the regulatory endpoint is claim-mica-2024-decentralization-test-holder-can-meet-quorum-alone, and the shared "weighted voting" mechanism originates in claim-gifford-1979-weighted-voting-quorum-replicated-data.