Order Stable Solutions for Two-sided Matching Problems

We concern the problem of matching the agents from two disjoint sets, such that the agents from the first set have preferences over the agents from the second set and vice versa. Typical problems of this sort are: college admissions problem, matching workers with firms, men with women (in a matrimonial agency) and so on. Classical approach to this problem comes from Gale and Shapley (1962). Recently, a far reaching generalization of Gale-Shapley approach was proposed by Alkan and Gale (2003). In our paper we present another generalization of the Gale -Shapley method, in which we use different concept of stability than the one in Alkan and Gale (the so-called order stability). We show that in some cases the order-stable solutions may be treated as more “fair” than in the Alkan and Gale’s approach. We formulate conditions under which the generalized Gale-Shapley algorithm leads to order-stable and optimal solutions and prove that these conditions are independent.