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Concept

Banzhaf index

A measure of a voter's power in a weighted vote: the share of all coalitions in which that voter's switch would change the outcome. It differs from the Shapley value in counting coalitions rather than orderings, and the two can rank the same voters differently.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A share of the votes, and a share of the power. Three weighted assemblies, each with the members' share of the votes beside their share of the power counted two independent ways.

A share of the votes is not a share of the power

Give three members four, four and one vote, with five needed to pass. Every winning coalition needs exactly two of them, so all three have equal power — and one of them holds a ninth of the votes.

applied · Shapley value A block of shares in an ocean of small holders. a=0.02: exact 0.0249, limit 0.0204; a=0.06: exact 0.0647, limit 0.0638; a=0.1: exact 0.1144, limit 0.1111; a=0.14: exact 0.1642, limit 0.1628; a=0.18: exact 0.2139, limit 0.2195; a=0.22: exact 0.2836, limit 0.2821; a=0.26: exact 0.3532, limit 0.3514; a=0.3: exact 0.4229, limit 0.4286; a=0.34: exact 0.5124, limit 0.5152; a=0.38: exact 0.6119, limit 0.6129; a=0.42: exact 0.7214, limit 0.7241; a=0.46: exact 0.8507, limit 0.8519; a=0.5: exact 0.9950, limit 1.0000; a=0.54: exact 1.0000, limit 1.0000; a=0.58: exact 1.0000, limit 1.0000

A block in an ocean of small holders

A shareholder with 30% of a company whose other shares are scattered among thousands of small holders has, by the average over every order of arrival, 43% of the power to decide — 3/7 exactly, in the limit. The reason is a window of moments: if everyone arrives at a random time, the dispersed shares trickle in at a steady rate, and the block decides every vote in which it turns up while the trickle sits within 30% of the quota. The same picture, run along a diagonal, turns the average over orders into an integral and puts the two classic power indices on one curve.

applied · Shapley value

Named alongside it

The objects these essays reach for when they reach for this one.

Weighted votingCoalitionDummy playerIntegralMarginal contributionPivotal playerProbabilityQuotaShapley shubik indexShapley valueSimple gameSwing

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