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Series

Shapley value — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every order of arrival for three partners, and what each player adds. A table with one row per order in which the players could arrive, giving what each adds to the group already present, and the average of each column as that player's share.

    The order everybody arrives in

    Three people jointly earn nine, and the question is what each is owed. Ask instead what each adds on walking into a room the others are already in, average that over every order they could have arrived in, and four modest conditions leave no other answer.

    part 1 · applied
  2. Drop one condition, and something else satisfies the rest. A column for each of the four conditions, holding a sharing rule that breaks that one and keeps the other three, with the split each rule gives on a stated four-player game.

    None of the four conditions is spare

    Four conditions pick out one sharing rule. The half that is usually shown is that they are enough; the other half is that each is needed — drop any one and a different rule satisfies the rest, so the list cannot be shortened.

    part 2 · applied
  3. 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.

    part 3 · applied
  4. 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.

    part 4 · applied
  5. Every order of arrival for three users of one shared capacity, and what each player adds. A table with one row per order in which the players could arrive, giving what each adds to the group already present, and the average of each column as that player's share.

    Sharing a cost that is not the sum of its parts

    Three users need capacities three, six and twelve of one shared thing, and serving any group costs the largest of them. Averaging what each adds over every order of arrival divides the bill — and for this family the average collapses to a rule anybody could apply by hand.

    part 5 · applied
  6. Credit for one prediction, three ways of leaving an input out. Groups of bars, one group for each way of filling in the inputs that are not known, each bar one input's share of the difference between the prediction and the starting value.

    What a missing input is worth

    A model prices a house at 180 from its size, its garden and its bedrooms, and the question is how much of the price each input is responsible for. Make the inputs the players and the average over orders answers it — once somebody decides what the model says when an input is not known. Three reasonable decisions give bedrooms nothing, nothing, and sixteen, for a model that never reads them.

    part 6 · applied
  7. Shares on two triangles and a go-between. A network of players with each node labelled by its share of what the whole network earns, and its number of links beneath it.

    Cutting a link costs both of its ends the same

    Three players, any two of whom can earn 1 together — but only if they are linked. Link all three and each is due a third. Remove one link and the player holding both of the others is due two thirds. Averaging over orders on the game the network allows is the one rule under which breaking any link costs the two players it joined exactly the same, and it pays go-betweens more than their links.

    part 7 · applied
  8. Luxembourg's votes and Luxembourg's power, 1958 to 1995. Paired bars for Luxembourg at five enlargements of the Council: vote share falling and power share rising from nothing in 1958 to 0.95% in 1973 and 3.02% in 1981.

    A vote worth nothing until it was outnumbered

    From 1958 to 1973 Luxembourg held one vote of seventeen in the Council of the European Communities and could never once change an outcome. When the Council grew, Luxembourg's share of the votes fell and its share of the power rose from nothing. Power is not a quantity a member holds; it is a property of the whole assembly, and changing the assembly moves it in directions nobody would guess.

    part 8 · applied

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