Priorities over agents are crucial primitives in assignment problems of indivisible objects without monetary transfers. In this paper, we introduce a prioritization problem: there are several exogenous attributes and each agent is equipped with one such attribute, and the priority is initially determined only for agents with the same attribute. This leads to a partial priority order. Our problem is to construct a complete priority, which is needed to implement a known mechanism such as the serial dictatorship. We propose a simple prioritization rule called the relative position rule. We formulate three equity axioms and an invariance property; the priority preservation law, the equal treatment of equal positions, the equal split, and the attribute-wise consistency. We show that the relative position rule is characterized by these equity axioms. The result is applicable to general assignment problems with partial priorities. In the context of college students’ exchange programs, the rule levels the playing field in the sense that inequality across attributes is partially reduced.