Solidarity and Minimal Envy in Matching Problems

In two-sided matching problems, we formulate (i) a concept of eq-uity of matchings based on envy minimization, and (ii) a solidarity property of matching rules under “natural ” and “simple ” changes of preferences which represent enhancement of partnership of the pairs. We show that there exists no rule that selects an envy-minimizing matching in the set of stable matchings, and that also satisfies the solidarity property. In contrast, any rule with a certain separabil-ity condition that selects an envy-minimizing matching in the set of individually rational and Pareto efficient matchings satisfies solidarity