Who gets what if money can't buy it? Increasing welfare in allocation problems

Deciding how scarce resources should be assigned to people when monetary transfers are inappropriate is a common problem in our society (e.g. assigning students to schools, patients to organ donors, ...). In this proposal, we aim to improve existing allocation mechanisms by enforcing that the resulting lottery results in a higher (and guaranteed) welfare for society, however, while keeping as much as possible the beneficial properties (e.g. efficiency, fairness, ...) of the allocation mechanisms for the individuals involved.