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The stable allocation problem generalizes the 0,1 stable matching problems (one-to-one, one-to-many, and many-to-many) to the allocation of real valued hours or quantities. A strongly polynomial algorithm proves the existence of “stable allocations.” The set of stable allocations is shown to be a distributive lattice in general, but in the “nondegenerate” case it is a complete linear order. Indeed, in the generic case, when a problem is “strongly nondegenerate,” there exists a single stable allocation. A simple algorithm finds “row-optimal” and “column-optimal” stable allocations, given any stable allocation. When a problem is nondegenerate it finds all stable allocations.

The stable allocation problem generalizes the 0,1 stable matching problems (one-to-one, one-tomany, and many-to-many) to the allocation of real valued hours or quantities. A strongly polynomial algorithm proves the existence of "stable allocations."The set of stable allocations is shown to be a distributive lattice in general, but in the "nondegenerate" case it is a complete linear order. Indeed, in the generic case, when a problem is "strongly nondegenerate," there exists a single stable allocation.A simple algorithm finds "row-optimal" and "column-optimal" stable allocations, given any stable allocation. When a problem is nondegenerate it finds all stable allocations.Introduction. The stable marriage (or stable one-to-one) problem is the simplest example of a two-sided market. There are two distinct sets of agents, e.g., men and women, and each agent on one side of the market has preferences over the opposite set. Matchings between men and women are sought that are "stable" in the sense that no man and woman not matched could both be better-off by being matched (Gale and Shapley 1962). The stable admissions (or stable one-to-many) problem is a more general example of a two-sided market, again with two sets of agents each having preferences over the opposite set. On one side of the market there are individuals, e.g., prospective students or interns or employees, and on the other there are institutions, e.g., universities, hospitals, or firms, each seeking to enroll some given number of individuals (Gale and Shapley 1962). An even more general case is the stable polygamous polyandry (or stable many-to-many) problem where every agent seeks to enroll given numbers of agents of the opposite set (Baïou and Balinski 2000). All of these are problems of assignment: agents are matched with agents (Gale and Shapley 1962, Gusfield and Irving 1982, Roth and Sotomayor 1990.To date, however, no one has considered the stable allocation (or ordinal transportation) problem. This is a two-sided market with distinct sets of agents where each agent has strict preferences over the opposite set; but instead of assigning or matching, the question is how to allocate hours of work. For example, one set of agents is composed of employees, each having a certain number of available hours to work, and the other set of agents is composed of employers, each seeking a certain number of hours of work. "Stability" simply asks that no pair of opposite agents can increase their hours together either due to unused capacity or by giving up hours with less desirable partners.In the transportation interpretation, one set consists of suppliers, each offering a known number of units of a good, and the other consists of acquirers, each seeking a known number of units of the same good. Instead of having costs of transporting units, each acquirer (each