A Review of the Existence of Stable Roommate Matchings

We compare different preference restrictions that ensure the existence of a stable roommate matching. Some of these restrictions are generalized to allow for indifferences as well as incomplete preference lists, in the sense that an agent may prefer remaining single to matching with some agents. We also introduce a new type of cycles and in greater detail investigate the domain of preferences that have no such cycles. In particular, we show how the absence of these cycles relates to the "symmetric utilities hypothesis" by Rodrigues-Neto (Journal of Economic Theory 135, 2007) when applied to roommate problems with weak preferences.