Non-vetoed matching with status quo

We consider a matching market where there is an initial matching a priori. Both sides of the market can veto the new matching whenever they would rather keep their initial assignment. We propose a stability notion, called “conditional stability”. Our first mechanism—“Non-Vetoed Deferred Acceptance” (NDA)—is non-vetoed, conditionally stable, and strategy-proof. These three properties are incompatible with (two-sided) conditional constrained efficiency; hence NDA is not conditionally constrained efficient. However, no mechanism that is non-vetoed, conditionally stable, and strategy-proof improves on NDA. Lastly, we propose a non-vetoed, conditionally stable, and conditionally constrained efficient mechanism.