  We consider stability concepts for random matchings where agents have
preferences over objects and objects have priorities for the agents. When
matchings are deterministic, the standard stability concept also captures the
fairness property of no (justified) envy. When matchings can be random, there
are a number of natural stability / fairness concepts that coincide with
stability / no envy whenever matchings are deterministic. We formalize known
stability concepts for random matchings for a general setting that allows weak
preferences and weak priorities, unacceptability, and an unequal number of
agents and objects. We then present a clear taxonomy of the stability concepts
and identify logical relations between them. Furthermore, we provide no envy /
claims interpretations for some of the stability concepts that are based on a
consumption process interpretation of random matchings. Finally, we present a
transformation from the most general setting to the most restricted setting,
and show how almost all our stability concepts are preserved by that
transformation.
