Selecting PhD Students and Projects with Limited Funding

In some universities, there is a fixed number of PhD grants, but a larger number of eligible projects and students, each student being allowed to apply on several projects, and a committee builds up a ranking (masterlist) over student-project pairs. The paper analyses three mechanisms to choose the pairs to be funded. The first one, used in some universities, is a greedy mechanism that gives a huge priority to the masterlist and very little to student's preferences. At the other extremity of the spectrum, we have a priority-list variant of student-oriented Gale-Shapley. It is strategyproof and optimal for students, but has one drawback: it pays too little attention to the masterlist, and thus to the committee. Inbetween, we have an intermediate mechanism which can be seen as a good tradeoff. Among the properties we study, one is specific to our setting: dynamic monotonicity, which deals with cases when a student suddenly leaves the system in the middle of the process.