Efficiency and Stability of Probabilistic Assignments in Marriage Problems

We study marriage problems where two groups of agents match to each other and probabilistic assignments are possible. When only ordinal preferences are observable, an extensively studied efficiency notion is stochastic dominance efficiency (sd-efficiency). First, we provide a characterization of sd-efficient allocations in terms of a property of an order relation defined on the set of man-woman pairs. Then, using this characterization, we constructively prove that for each probabilistic assignment that is sd-efficient for some ordinal preferences, there is a von Neumann-Morgenstern utility profile consistent with the ordinal preferences for which the assignment is Pareto efficient. Next, we analyze stability of probabilistic assignments. We show that, when the preferences are strict, for each probabilistic assignment that is ex-post stable for some ordinal preferences, there is a von Neumann-Morgenstern utility profile consistent with the ordinal preferences such that the assignment belongs to the core: no coalition can deviate to another probabilistic assignment among themselves and achieve a higher total expected utility