Strategy-Proof and Minimally Wasteful Random Assignment
We study random assignment of indivisible objects among a set of agents with strict preferences and outside options. When agents may rank some objects as unacceptable, we consider different notions of measuring waste of object(s) from an ex-ante perspective. The most natural one is $q$-agent-object-waste whereby both one agent and one of his acceptable objects are unassigned with at least probability $q$. On the one hand, we show that any mechanism satisfying equal treatement of almost equals (whereby any two agents with identical rankings over objects receive the same probability shares for objects they both regard acceptable), strategy-proofness and ex-post weak non-wastefulness (whereby in any assignment in the support we cannot have that both one agent and one of his acceptable objects are unassigned) must be $q$-agent-object-wasteful with $q\geq \frac{1}{6}$. On the other hand, we show that random serial dictatorship (RSD) attains the minimal bound of $\frac{1}{6}$ in this class for four agents or three objects. In addition, we consider $q$-object-wastefulness where an object remains unassigned with probability $q$, while agents, who consider it acceptable, remain unassigned with aggregate probability $q$. We again show that RSD attains the minimal bound of $\frac{1}{4}$ in this class of mechanisms with respect to $q$-object-wastefulness for three objects. Finally, we show that random deferred acceptance (RDA) may be strictly less agent-object-wasteful than RSD (but at the cost of violating equal treatment of almost equals).