The Coalition

When cooperation creates a gain, the proposed division has to survive the alternatives that cooperation itself makes possible. The earlier Collected Bargain placed two parties before a surplus and asked where they would draw the line. The Division traced the range of settlements that their alternatives could sustain. Add more participants and the alternatives include one another: two dissatisfied members may leave the larger undertaking, form a smaller alliance and share what they can make together. The question becomes which divisions can withstand every such departure, and whether a rule for rewarding contribution will select one of them. Cooperative game theory gives those questions distinct mathematical forms. The core tests an allocation against the gains available to breakaway groups. The Shapley value assigns shares by averaging what each participant adds. Understanding their different tasks is part of understanding what an alliance can promise its members.

The first step is to specify what every possible coalition can achieve. In the version considered here, each group has a worth that can be divided among its members through transfers; a pound assigned to one reduces by a pound what remains for the others. The characteristic function lists these worths, including what each person can obtain alone and what everyone can obtain together. That compact representation carries assumptions. A group’s stated opportunity must be available without the help of outsiders, and the model must adequately describe what happens when it acts separately. Pollution, rivalry or other effects crossing the boundary may require a richer account. The ability to make and keep the promised division matters too. Calling the game cooperative permits analysis of joint arrangements; it does not establish that the participants are altruistic, that a binding agreement already exists, or that every proposed group will form.¹

An allocation belongs to the core when it divides the available total without leaving any coalition less than it can obtain for itself. If a group is assigned less than its worth, transferable gains let its members arrange a departure that makes each better off. The test includes single people and the whole group, so it respects individual fallbacks and exhausts the total available to everyone. It need not leave any allocation standing. Consider an invented three-person venture in which nobody can earn anything alone, any pair can earn six, and all three together can also earn six. To prevent each pair from departing, the amounts assigned to each of the three pairs must be at least six. Adding those demands counts every person’s allocation twice and requires at least eighteen. Yet twice the available total is only twelve. The core is empty: every division leaves some pair an opportunity to improve.²

The Shapley value answers the allocation question by a different discipline: consider each person’s contribution in every possible order of arrival. Let the group assemble one member at a time, crediting each newcomer with the increase in worth produced by joining those already present. Average that credit over all orders, weighting them equally. In each order the successive increments add to the whole group’s worth, so their averages divide it too. In his original formulation, Shapley derived the unique value from symmetry, distribution of the full yield to the players who can affect it, and addition of values when games are combined. A player who adds nothing receives nothing. These requirements supply a reasoned way to assign shares; their authority rests on accepting the requirements. The imagined order of arrival is a calculation and, in Shapley’s proposed procedure, a stipulated lottery. It is no assertion that real alliances assemble randomly.³

A contribution rule can distribute the whole gain and still leave an alliance vulnerable to departure. In the symmetric venture just considered, the Shapley value gives two to each person; every pair receives four while it could make six. A second invented venture shows that the difference persists even when the core exists. Let A hold an indispensable resource and let B and C be interchangeable partners. A with either partner can make six; B and C together can make nothing; the three together still make six; nobody produces anything alone. Across the six orders of arrival, A adds six in four orders and zero in two, receiving an average of four. Each partner receives one. Yet A and B could divide six as five and one, or 4.5 and 1.5; the latter improves both on the proposed four and one. In fact the core requires all six to A, because either partner can be replaced at no loss of output.⁴

Those sharply different divisions expose the judgments that an allocation rule cannot settle by itself. In the last example the core rewards control of the indispensable resource, while the Shapley value also credits a partner’s contribution when that partner arrives before the substitute. Which principle should govern is a question about the purpose of the arrangement. Nor does the difference establish that the two concepts must conflict. Change the technology so that all three are necessary to make six and every smaller group makes nothing: two each is then both the Shapley allocation and a member of the core. The change is in what the groups can do, and it changes the constraints on division. Bargaining costs, rules about joining and leaving, and enforceable commitments can further alter the opportunities that a realistic account must represent. The values entered in the model carry much of the substantive question.⁵

The coalition thus carries the division of the surplus into a world where the people dividing it can choose different company. The core asks whether a proposed settlement leaves any group a strictly better attainable arrangement. The Shapley value asks what each participant adds when every possible place in the formation of the group is given equal weight. One can be empty while the other remains defined; one can admit a harsh division while the other assigns a share to someone the threat of replacement can exclude. Neither alone predicts the negotiations by which people reach an agreement, and neither supplies a complete moral account of what they deserve. Together they make the alternatives and the chosen rule visible. There remains a prior difficulty: before anyone can divide the great common prize, the necessary participants must dare to act together. That is the assurance problem of the next plate, The Hunt.