  We provide a framework to unify classic models of two-sided matching with
recent models of recontracting. In the classic model, agents from two sides
must be matched; in models of recontracting, agents improve a status quo match.
We generalize the core (matches not blocked by any coalition) from cooperative
game theory to our setting by restricting the set of permissible coalitions to
coalitions containing neither or both agents in a status quo match, dubbed
"agreeable" coalitions. The agreeable core is the set of all weak improvements
of the status quo that are not blocked by any agreeable coalition. Our main
result is that the agreeable core is nonempty and can be found through a
computationally efficient and economically meaningful algorithm: our
Propose-Exchange algorithm. The applications of the agreeable core include
early decision, out-of-match agreements in the NRMP, matching with minimum
constraints, and efficiency in school choice.
