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Ladder

Shapley value — the ladder

5 distinct arguments against 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.

    rung 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.

    rung 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.

    rung 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.

    rung 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.

    rung 5 · applied

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