  We provide a framework to study stability notions for two-sided dynamic
matching markets in which matching is one-to-one and irreversible. The
framework gives centerstage to the set of matchings an agent anticipates would
ensue should they remain unmatched, which we refer to as the agent's
conjectures. A collection of conjectures, together with a pairwise stability
and individual rationality requirement given the conjectures, defines a
solution concept for the economy. We identify a sufficient
condition--consistency--for a family of conjectures to lead to a nonempty
solution (cf. Hafalir, 2008). As an application, we introduce two families of
consistent conjectures and their corresponding solution concepts:
continuation-value-respecting dynamic stability, and the extension to dynamic
markets of the solution concept in Hafalir (2008), sophisticated dynamic
stability.
