Compatibility of efficiency and stability in priority-augmented object allocation problems with weak priorities

In a priority-augmented object allocation problem with weak priorities, efficiency and stability are incompatible. To recover the compatibility, we identify two conditions on the priority-supply profile. The first condition that we call "equal treatment for scarce objects" is necessary and sufficient for the efficient correspondence to be stable. Hence, the condition is sufficient for "any" efficient subcorrespondence to be stable. The second condition that we call "no priority reversal" is necessary and sufficient for the stable correspondence to be efficient. Then, the condition is sufficient for "any" stable subcorrespondence to be efficient. For each of our conditions, we also provide an easy-to-verify equivalent condition on the priority-supply profile. Those two conditions are independent.