Efficient Random Assignment with Cardinal and Ordinal Preferences

We develop a finite random assignment model where players know either their cardinal or their ordinal preferences and may make cardinal or ordinal reports to an assignment mechanism. Under truthful reporting, we find that all mechanisms that disregard the cardinal information in players' reports (e.g., Deferred Acceptance and Probabilistic Serial) are utilitarian inefficient, as are classic mechanisms that make use of cardinal information (e.g., Pseudo-markets). Motivated by these negative results, we introduce a "Simple Mechanism" that makes use of cardinal information "in the right way." We establish that this mechanism is utilitarian efficient, treats equals equally, and makes truth-telling almost Bayesian incentive compatible.