Efficiency and stability of probabilistic assignments in marriage problems

We study marriage problems where two groups of agents, men and women, match each other and probabilistic assignments are possible. When only ordinal preferences are observable, stochastic dominance efficiency (sd-efficiency) is commonly used. 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. Our second result pertains to stability. We show that when the preferences are strict, for each ordinal preference profile and each probabilistic assignment that is ex-post stable, there is a von NeumannMorgenstern utility profile consistent with the ordinal preferences for which the assignment belongs to the core: No coalition can deviate to another probabilistic assignment among its members and achieve a higher total expected utility. JEL Classification Numbers: C60, C71, C78, D61