The Stable Roommates Problem with Choice Functions

The stable marriage theorem of Gale and Shapley states that for n men and n women there always exists a stable marriage scheme, that is, a set of marriages such that no man and woman mutually prefer one another to their partners. The stable marriage theorem was generalized in two directions: the stable roommates problem is the “one-sided” version, where any two agents on the market can form a partnership. The generalization by Kelso and Crawford is in the “two-sided” model, but on one side of the market agents have a so-called substitutable choice function, and stability is interpreted in a natural way. It turned out that even if both sides of the market have substitutable choice functions, there still exists a stable assignment. The latter version contains the “many-to-many” model where up to a personal quota, polygamy is allowed for both men and women in the two-sided market.