Abstract

In the random assignment problem, objects are randomly assigned to agents keeping in view the agents ’ preferences over objects. A random assignment specifies the probability of an agent getting an object. We examine the struc-tural and computational aspects of ex post efficiency of random assignments. We first show that whereas an ex post efficient assignment can be computed easily, checking whether a given random assignment is ex post efficient is NP-complete. Hence implementing a given random assignment via deterministic Pareto opti-mal assignments is NP-hard. We then formalize another concept of efficiency called robust ex post efficiency that is weaker than stochastic dominance effi-ciency but stronger than ex post efficiency. We present a characterization of robust ex post efficiency and show that it can be tested in polynomial time if there are a constant number of agent types. It is shown that the well-known random serial dictatorship rule is not robust ex post efficient. Finally, we show that whereas robust ex post efficiency depends solely on which entries of the as-signment matrix are zero/non-zero, ex post efficiency of an assignment depends on the actual values.