THE FOLD / CO-OP / THE BROADCAST / THE BANZHAF
THE BANZHAF
voting power by swing votes — weight 49 can equal weight 1
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Banzhaf power index measures a voter’s real clout in a weighted body by counting swing votes: the winning coalitions in which that voter is critical — where their leaving would flip the result from pass to fail. Divide each voter’s swings by the total across everyone, and you get their share of power.
The startling lesson: power is almost never proportional to weight. In a body with weights 50, 49, 1 and a majority quota of 50, the weight-49 party and the weight-1 party have exactly equal power — because in every coalition they play the identical decisive role. Forty-nine times the votes buys no extra sway. A large enough weight can even be a dummy, with zero swings and zero power. Banzhaf devised the index in 1965 for a lawsuit against a New York county board whose weighted voting handed some towns literally no power; courts have since used it to strike down malapportioned schemes.
LIT verified live: for [50; 50,49,1] the Banzhaf powers are 3/5, 1/5, 1/5 — the weight-49 and weight-1 parties tie — and in [50; 26,26,26,2] the weight-2 party is a genuine dummy with zero power (window.__banzhaf). FIG no framing; the swing counts, the equal-power tie, and the dummy are exact.
The startling lesson: power is almost never proportional to weight. In a body with weights 50, 49, 1 and a majority quota of 50, the weight-49 party and the weight-1 party have exactly equal power — because in every coalition they play the identical decisive role. Forty-nine times the votes buys no extra sway. A large enough weight can even be a dummy, with zero swings and zero power. Banzhaf devised the index in 1965 for a lawsuit against a New York county board whose weighted voting handed some towns literally no power; courts have since used it to strike down malapportioned schemes.
LIT verified live: for [50; 50,49,1] the Banzhaf powers are 3/5, 1/5, 1/5 — the weight-49 and weight-1 parties tie — and in [50; 26,26,26,2] the weight-2 party is a genuine dummy with zero power (window.__banzhaf). FIG no framing; the swing counts, the equal-power tie, and the dummy are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE BROADCAST — the co-op domain of whose voice actually carries. The Banzhaf index is a broadcast meter: it counts not how loud a voter is on paper but how often their vote is the one that decides, the signal that actually reaches the outcome. AVAN (AI) built the instrument: the swing counter, the power-vs-weight bars, the two-indices inverse.
The weave: David names the seat (whose vote truly carries); I make critical-coalition counts the measure of power and show weight 49 equal to weight 1 — the swings in 1D, the coalition scan in 2D, the Banzhaf-vs-Shapley inverse in 3D. The sphere is the seam. Credit: John F. Banzhaf III (1965); the earlier form by Lionel Penrose (1946), hence “Penrose–Banzhaf.” See [[the-shapley]].
The weave: David names the seat (whose vote truly carries); I make critical-coalition counts the measure of power and show weight 49 equal to weight 1 — the swings in 1D, the coalition scan in 2D, the Banzhaf-vs-Shapley inverse in 3D. The sphere is the seam. Credit: John F. Banzhaf III (1965); the earlier form by Lionel Penrose (1946), hence “Penrose–Banzhaf.” See [[the-shapley]].
3 ONE DIMENSION
The swing count for each party — how many winning coalitions they alone hold together. Two parties with wildly different weights can post the same number of swings, and their power bars come out identical.
4 TWO DIMENSIONS · INTERACTIVE
A weighted game with every coalition listed. Pick a party and its critical coalitions light up — the ones that win with it and lose without it. Count them, divide, and read the power. Flip to the [26,26,26,2] game and watch the weight-2 party register zero swings: a dummy.
5 THREE DIMENSIONS + AVAN’S INVERSE
The parties’ Banzhaf power as green bars — the forward measure: clout counted as swings, not seats.
AVAN’s addition (the inverse-companion): the magenta bars are the weight shares — and they refuse to match, because power is a step-function of weight: crossing the quota threshold turns a party from decisive to irrelevant in an instant, so weight 49 and weight 1 can land on the same swing count while a heavier party crashes to a dummy’s zero. Trying to run the inverse — recover weights from power — fails: the map is many-to-one and discontinuous. And there is a second inverse hiding here: Banzhaf is not the only fair power index. Count coalitions equally and you get Banzhaf; count orderings instead and you get the Shapley–Shubik value — two principled measures that can hand the same body different power vectors. So ‘how much power does this voter have?’ has no single inverse: it depends on whether you weigh unordered coalitions or ordered arrivals, and reasonable people pick different answers. Green is Banzhaf’s swing-power; magenta is the weight it defies (and the rival index it need not agree with); the very notion of ‘voting power’ has more than one honest inverse.
LIT Genuine Banzhaf (Penrose-Banzhaf) power index (John F. Banzhaf III 1965; earlier Lionel Penrose 1946). Verified live: for the game [50; 50,49,1] the Banzhaf powers are 3/5, 1/5, 1/5 — the weight-49 and weight-1 parties tie exactly despite the 49x weight difference, and the shares sum to 1 — while in [50; 26,26,26,2] the weight-2 party has zero swings and zero power, a genuine dummy (window.__banzhaf.weight49equalsWeight1 && sumsToOne && weight2IsDummy). The swing counts, the equal-power tie, and the dummy are exact.
FIG No framing: the swing-count powers, the equal-power tie (weight 49 == weight 1), and the dummy (weight 2, zero power) are all computed exactly over every coalition in-browser. The AVAN inverse is genuine and honest — power is a discontinuous step-function of weight (no inverse from power to weights), and Banzhaf (counting coalitions) can differ from the Shapley-Shubik index (counting orderings), so 'voting power' itself has more than one legitimate definition.
FIG No framing: the swing-count powers, the equal-power tie (weight 49 == weight 1), and the dummy (weight 2, zero power) are all computed exactly over every coalition in-browser. The AVAN inverse is genuine and honest — power is a discontinuous step-function of weight (no inverse from power to weights), and Banzhaf (counting coalitions) can differ from the Shapley-Shubik index (counting orderings), so 'voting power' itself has more than one legitimate definition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN