We continue recent woik on the matching problem for firms and workers, and show that, for a suitable ordering, the set of stable matchii^^ is a lattice. 1. Introduction. Gale and Shapley [2] considered the problem of matching finite sets of men and women to form couples. A matching was caUed stable if there did not exist a man in one couple and a woman in another who preferred each other to their current partners. [2] showed that for any preference orderings, there always exists at least one stable matching. A striking feature of this model was pointed out in Knuth [4, pp. 92-93, attributed to J. H. Conway]. A natural partial ordering on the set of stable matchings has matching / > matching g if every man is at least as happy with his partner in / as with his partner in g. [4] showed that this partial ordering gives a distributive lattice. In addition, (i) replacing "man" by "woman" in the definition has the effect of replacing " > " by " < ", (ii) the l.u.b. and g.l.b. operations in the lattice are simple: the l.u.b. of /, g is obtained by giving each man whichever of the partners he likes better, g.l.b. gives each man the partner he likes less. In addition to its intrinsic interest, the lattice structure provides insight into the existence of stable outcomes that are simultaneously optimal either for the set of men or for the set of women. Such outcomes are surprising since the players of the same sex are competing against each other. This issue is discussed further in Roth [5]. Kelso and Crawford [3] considered a matching model in which the players on one side have multiple partners, e.g., a firm hires sets of workers, with each worker allowed to work for only one firm. Roth [5,6] generalized this model to allow a worker to be employed by a set of firms, thus treating firms and workers symmetrically. In each of these models, existence of a stable multi-partner matching is established, provided the preferences of each player satisfy a "substitutability condition." TTiis paper considers the most general of these models, model III of [5], and investigates the extent to which the lattice structure present in the monogamous Gale-Shapley model is preserved. [5] showed that the l.u.b. and g.l.b. operations did not generalize. We show that the partial ordraing in the monogamous case does generalize to give a lattice structure for the multi-partner case. However, the lattice is not necessarily distributive and the l.u.b. operation is nontrivial. [1] showed that the set of lattices which could occur in the monogamous case is precisely the set of finite distributive lattices. The problem of characterizing the lattices which can occur here is open.

We show, using small examples, that two algorithms previously published for the Bilevel Linear Programming problem (BLP) may fail to find the optimal solution and thus must be considered to be heuristics. A proof is given that solving BLP problems is NP-hard, which makes it unlikely that there is a good, exact algorithm.

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