Cooperative game
Named by 9 essays across one field — each of them below, with the objects they name alongside it.
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.
A split nobody can walk away from
Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.
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.
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.
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.
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.
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.
Each user pays for its own last link
Several users must be connected to a source, and the cheapest network that does it is a tree. Dividing its cost so that no group of users would rather build its own looks like a hard search, and it has a one-line answer: each user pays for the link that joins it to the tree on its way to the source. No group is ever overcharged — while the average over orders of arrival, the rule that settles so much else, can charge a pair more than its own connection costs.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Shapley valueMarginal contributionCoalitionExhaustive searchFairnessSymmetryCoreAllocationCounterexampleEfficiencyPolytopeUniqueness