Matpic
mathematics, in pictures
Concept

Simple game

A cooperative game in which every coalition is worth either one or nothing, so the structure is entirely a matter of which coalitions win. A weighted vote is one, and a member's power in it is the fraction of arrival orders in which they are the one who carries a coalition over the quota.

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 Sampling the orders, and how fast the answer arrives. The largest error in the estimated shares against the number of orderings sampled, both on logarithmic axes, with the square-root rate drawn through the first point.

Too many orders to list

The rule is an average over every order the players could have arrived in. At seven players that is five thousand orders and at twenty it is more than there are seconds in the age of the universe — so the average is sampled, and the error falls at a rate that can be measured.

applied · shapley value

Named alongside it

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

ApproximationBanzhaf indexConvergence rateDummy playerIntractabilityMarginal contributionMonte CarloPermutationPivotal playerQuotaSamplingShapley shubik index

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