On extensions of partial priorities in school choice

We consider a school choice matching model where the priorities for schools are represented by binary relations that may not be weak order. We focus on the (total order) extensions of the binary relations. We introduce a class of algorithms to derive one of the extensions of a binary relation and characterize them by using the class. We show that if the binary relations are the partial orders, then for each stable matching for the profile of the binary relations, there is an extension for which it is also stable. Moreover, if there are multiple stable matchings for the profile of the binary relations that are ranked by Pareto dominance, there is an extension for which all of those matchings are stable. We provide several applications of these results.