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Concept

Marginal contribution

What a player adds to a coalition on joining it — the value of the group with them, less its value without. Averaging it over every order in which the players could have arrived is the Shapley value, and the four conditions that pick out that average are conditions on the averaging rather than on the contributions.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

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.

applied · Shapley value 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.

applied · Shapley value 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 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.

applied · Shapley value 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.

applied · Shapley value 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.

applied · Shapley value The core of a market with two houses and two buyers. The core of an assignment game drawn in the plane of buyer A's payoff against buyer B's, a polygon with vertices (0, 1), (0, 0), (1, 0), (4, 3), (2, 3); the buyers' best corner is (4, 3).

The prices nobody can break away from

When houses are sold to buyers who value them differently, there is a whole range of prices at which nobody wants to walk away, and it has a remarkable shape — one corner best for every buyer at once, one best for every seller at once, and the buyers' corner pays each buyer exactly what the market would lose without them.

applied · Assignment The core of a group worth the square of its size: the outline of its six arrival orders. Splits of 9 among three players; core corners (1, 5, 3), (5, 1, 3), (1, 3, 5), (5, 3, 1), (3, 1, 5), (3, 5, 1); arrival-order splits (1, 3, 5), (1, 5, 3), (3, 1, 5), (5, 1, 3), (3, 5, 1), (5, 3, 1); average 3, 3, 3.

The corners are the orders of arrival

Line the players up, let each join in turn, and pay each what it adds on arrival: every order gives a split. When a newcomer always adds at least as much to a bigger group, those splits are exactly the corners of the core — so the core is never empty, it is the outline of the orders, and the average over all of them lies inside it. For a group worth the square of its size the outline is a hexagon whose corners are the six orderings of 1, 3 and 5.

applied · The core 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.

applied · Shapley value What a left glove is worth as the market tips. For 20 players and L from 1 to 19: the core value of a left glove (1 below 10, any value at 10, 0 above) and the Shapley value, from 0.950 to 0.003.

One glove too many

Give some people left gloves and others right ones, and let any group sell the pairs it can make. If the two sides are equal, the core — the splits no group can beat by walking out — is every price at once. If one side has a single glove more, the core is one split: the scarce side takes the whole of every pair and the other side gets nothing, however large the market. The average over orders of arrival barely notices the difference, and the two rules disagree about almost everything a market is.

applied · The core

Named alongside it

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

Shapley valueCooperative gameCoalitionSymmetryCoreExhaustive searchFairnessDualityDummy playerEfficiencyQuotaUniqueness

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