Not all is lost: Sorting and self-stabilizing sets

It is well known that in the absence of transfers, stable matchings are positive assortative when agents' preferences are strictly monotonic in each other's types and types are commonly known. Instead, monotonicity is consistent with the existence of stable matchings that exhibit negative sorting when types on one side of the market are private information. This paper sheds light on the scope and meaning of this consistency by showing that within familiar monotonic markets we cannot conclude that a stable matching is negative, and not positive, assortative.