Improving efficiency in school choice under partial priorities

We provide a school choice model where the student priority orders for schools are allowed not to be total. We introduce a class of algorithms, each of which derives a student optimal stable matching once we have an initial stable matching, when the priorities are represented by partial orders. Since a method to derive a stable matching exists when the priorities are partial orders, we can use it to derive a student optimal stable matching. Moreover, we show that any student optimal stable matchings that Pareto dominate the starting stable one are obtained via an algorithm within this class. For the problem of improving efficiency by allowing some priorities to be violated, the algorithms can also be applied, with a weaker assumption on the violations than in the previous study. Finally, we suggest some specific rules of priorities that can be introduced by weakening the requirement of total ordering.