Three-dimensional stable matching with cyclic preferences
Introduction
The stable marriage problem is: Given a set of men and a set of women, find a matching that is stable in the sense that no man m and woman w who both prefer each other to their current partners in the matching. Gale and Shapley (1962) introduced this problem and gave a constructive proof of the existence of a stable matching for any combination of preferences. The theory of stable matchings has become an important subfield within game theory, as documented by the book of Roth and Sotomayor (1990), but the first book on the subject was written by famous computer scientist Donald E. Knuth (1976). Knuth lists a dozen suggested further directions for research, one of which is to investigate three-dimensional stable matching, say of women, men and dogs. Such a matching would be a partition of the agents into triples consisting of one agent of each type. A matching is stable if there is no blocking triple, i.e. a triple that all of its members would strictly prefer to the current matching. A matching is strongly stable if there is no weakly blocking triple, i.e. a triple strictly preferred by some member and weakly preferred by all members. We will follow Ng and Hirschberg (1991) and refer to a system of agents of three types, together with their preferences on triples, as an instance of 3GSM (Three Gender Stable Marriage problem).
Alkan (1988), who seems to have been the first who published a result on 3GSM, found an instance where no stable matching exists. Ng and Hirschberg (1991) showed that a number of instances of 3GSM do not have any strongly stable matchings, and proved that the decision problem is NP-complete. As an open problem, Ng and Hirschberg mention the cyclic (or circular) 3GSM, where women care only about which man is in the triple, and similarly men care only about dogs, and dogs care only about women. The origin of this problem is attributed to Knuth. Recently, Boros et al. (2004) proved stability for cyclic 3GSM whenever n ≤ 3, where n is the number of agents of each type. (More generally, their result says that for any integer s ≥ 2, cyclic s-GSM has a stable matching whenever n ≤ s.) They also show that their method of proof breaks down for larger values of n.
In this paper, we will extend the partial result of Boros et al. by proving stability of cyclic 3GSM for n = 4. The proof, given in Section 3, is a quite technical case-by-case analysis that does not easily generalize to larger n. Therefore we also investigate whether stability of cyclic 3GSM would be implied by any of a number of well-known general sufficient conditions for stability (balancedness, effectivity function stability). In Section 4 we show that none of these conditions apply to cyclic 3GSM.
Another possible approach is to find a suitable relaxation of the cyclicity condition. Danilov (2003) proved stability of 3GSM under a certain acyclic lexicographic preference rule, where men base their preferences on triples in the first place on the woman in the triple (and in the second place on the dog), and women similarly are interested primarily in men. Boros et al. (2004) studied the lexicographic relaxation of cyclic preferences, where women care in the first place about men (and in the second place about dogs), and cyclically for men and for dogs. However, under this rule they found instances of 3GSM where no stable matching exists. In this paper we propose another relaxation of cyclicity: “strongest link 3GSM” (defined in the next section). Evidence from computer search leads us to conjecture that strongest link 3GSM always allows stable matchings, see Section 5.
Section snippets
Problem definition
We consider three sets of agents: W, M, D (for women, men and dogs). Let n be the maximal number of agents in a set; e.g. n = 3 means that we have at most nine agents (three women, three men, three dogs). Without loss of generality we can assume that we have the maximal number n of agents of each gender, for otherwise we can just fill the ranks with dummy agents who everybody likes less than any real agent. A triple is an element of W × M × D, and a set of n disjoint triples is a matching.
Each woman w
Cyclic 3GSM is stable for n ≤ 4
As promised in the Introduction, we shall prove the following result. Theorem 1 If n ≤ 4, every instance of cyclic 3GSM has a stable matching.
The number of possibilities is too large for an exhaustive search, see Section 5. Instead we have found a way to reduce the possibilities to a few cases.
Sufficient conditions for stability
The core of a game is the set of all outcomes for which no coalition of players can enforce another outcome that is preferable to all members of the coalition. For matching games it is easy to see that existence of a stable matching is equivalent to nonemptiness of the core (cf. Roth and Sotomayor, 1990). There exist several general approaches (i.e. sufficient conditions) to proving that a game has a nonempty core. The classic one is to show that the game is balanced in the sense of Scarf (1967)
Evidence from computer search
In this paper we have investigated Knuth's problem on stable three-dimensional matching under cyclic preferences, and improved on the partial solution from n = 3 to 4 agents of each type. Presumably one reason that this problem is still open, after almost 30 years, is that it is difficult. We have seen that several general approaches to prove stability do not work.
For any given n, the number of instances is finite, and hence exhaustive search is theoretically possible. In practice, the number of
Acknowledgements
This research was partially supported by the Swedish Science Council. We are very grateful to an anonymous referee for many helpful comments on a previous version of this paper.
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