Solidality and Minimal Envy in Matching Problems

In two-sided matching problems, we formulate (i) a concept of equity 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 separability condition that selects an envy-minimizing matching in the set of individually rational and Pareto efficient matchings satisfies solidarity.