A market design solution to a multi-category housing allocation problem

We study multi-category housing allocation problems: A finite set of objects, which is sorted into categories of equal size, has to be allocated to a finite set of individuals, such that everyone obtains exactly one object from each category. We show that, in the large class of category-wise neutral and non-bossy mechanisms, any strategy-proof mechanism can be constructed by simply letting individuals choose an object from each category one after another following some priority order. We refer to these mechanisms as multi-category serial dictatorships and advocate for selecting priority orders across categories as fairly as possible.