Teacher Assignment and Minimal Envy Mechanisms

We investigate the teacher assignment problem where one has to assign both newly tenured teachers without an initial assignment and tenured teachers with an initial position. Each school has a priority ordering over the teachers. In this framework, there is no mechanism that is both i) individually rational (IR), i.e., that assigns each teacher to a school that he/she weakly prefers to his/her initial one, and ii) stable, i.e., there are no blocking pairs: teachers and schools that are not assigned together and would prefer to be. A mechanism has minimal envy if there is no other mechanism that always leads to a the set of blocking pairs included in the one of the former mechanism. We show that the modified Deferred Acceptance mechanism (DA*) is a minimal envy mechanism in the set of IR and strategy-proof mechanisms. We also show that, in a one-to-one setting, an extension of the Top Trading Cycle (TTC*) mechanism is also a minimal envy mechanism in the set of IR, strategy-proof and Pareto-efficient mechanisms. These results generalize the results in the school choice literature concerning the standard DA and TTC mechanisms