Weak Stability and Pareto Efficiency in School Choice

We study the trade-off between stability and students' welfare in school choice problems. We call a matching weakly stable if none of its blocking pairs can be matched in a more stable matching–one with a weakly smaller set of blocking pairs. A matching is said to be self-constrained efficient if for students it is not Pareto dominated by any more stable matching, and it is self-constrained optimal if it weakly Pareto dominates all such matchings. We show that the following are equivalent for any matching: (i) it is weakly stable and self-constrained efficient; (ii) it is self-constrained optimal; (iii) it is an efficiency-adjusted deferred acceptance mechanism (EADAM) outcome under some consenting constraint; and (iv) it is exactly the EADAM outcome when its own set of blocking pairs is used as consenting constraint