AbstractUsing the assignment of students to schools as our leading example, we study many-to-one two-sided matching markets without transfers. Students are endowed with cardinal preferences and schools with ordinal ones, while preferences of both sides need not be strict. Using the idea of a competitive equilibrium from equal incomes (CEEI, Hylland and Zeckhauser (1979)), we propose a new mechanism, the Generalized CEEI, in which students face different prices depending on how schools rank them. It always produces fair (justified-envy-free) and ex ante efficient random assignments and stable deterministic ones with respect to stated preferences. Moreover, if a group of students are top ranked by all schools, the G-CEEI random assignment is ex ante weakly efficient with respect to students' welfare. We show that each student's incentive to misreport vanishes when the market becomes large, given all others are truthful. The mechanism is particularly relevant to school choice since schools' priority orderings can be considered as their ordinal preferences. More importantly, in settings where agents have similar ordinal preferences, the mechanism's explicit use of cardinal preferences may significantly improve efficiency. We also discuss its application in school choice with affirmative action such as group-specific quotas and in one-sided matching.