| Ahmet Alkan, Sabanci University |
| On Preferences over Subsets and the Lattice Structure of Stable Matchings |
| Session: C-13-13 Wednesday 16 August 2000 by Alkan, Ahmet |
| The set of (individually rational and pairwiae) stable matchings is known to be nonempty and a lattice in the general symmetric multipartner case provided agents have path-independent choice functions (Blair(1988)). There exists in particular a stable matching which is optimal (among all stable matchings) for all agents on one side of the market. Other than this fact, however, the lattice does not have the features and properties, in particular distributivity, of the lattice in the single-agent case or the special case where the preferences of an agent respect some underlying order on his potential partners (Alkan(1999) , Baiou and Balinski(1999).) Here we consider a model where each agent has a quota and a "quotafilling" choice function. Quotafillingness requires that one chooses quota-many elements from any set of cardinality no less than his quota. It is shown that, under this assumption in addition to path independence, the lattice of stable matchings has all the fine features and properties: (i) Given any two stable matchings, their supremum, namely the matching where each agent (on one side) is allowed to choose from the union of his two stable sets, is also stable. (ii) So is the infimum which gives each agent the partners he has not picked for the supremum, united with those in the intersection of his two stable sets. (iii) If an agent has unfilled quota at a stable matching then he has the same partnets at all stable matchings. (iv) Supremum choice always includes all partners in the intersection of the two stable sets. (v) Supremum and infimum are distributive. |