A Proofs for Section 4

This section contains further exposition (including proofs) for Section 4. A.1 Limitations of utility difference as an instability measure But its utility difference remains at 0. Minimum stabilizing subsidy equals Subset Instability for any market outcome. This is no larger than Subset Instability by definition. Let's take the maximum weight matching of We first formally define the unhappiness of a coalition, as follows. Recall that, in terms of unhappiness, Proposition 4.3 is as follows: Proposition 4.3. By Proposition 4.2, we know that Subset Instability is equal to Thus, it suffices to prove that the maximum unhappiness of any coalition is equal to (7).