The Complexity of Computing the Random Priority Allocation Matrix

The Complexity of Computing the Random Priority Allocation Matrix

By
Daniela Saban, Jay Sethuraman
Mathematics of Operations Research. June
17, 2015, Vol. 40, Issue 4, Pages 1005-1014

The random priority (RP) mechanism is a popular way to allocate n objects to n agents with strict ordinal preferences over the objects. In the RP mechanism, an ordering over the agents is selected uniformly at random; the first agent is then allocated his most-preferred object, the second agent is allocated his most-preferred object among the remaining ones, and so on. The outcome of the mechanism is a bi-stochastic matrix in which entry (i, a) represents the probability that agent i is given object a. It is shown that the problem of computing the RP allocation matrix is #P-complete. Furthermore, it is NP-complete to decide if a given agent ireceives a given object a with positive probability under the RP mechanism, whereas it is possible to decide in polynomial time whether or not agent i receives object a with probability 1. The implications of these results for approximating the RP allocation matrix as well as on finding constrained Pareto optimal matchings are discussed.