The Complexity of Computing the Random Priority Allocation Matrix

Abstract

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 i receives 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.

Computational aspects of random serial dictatorship

“…Independently, Saban and Sethuraman [2013] have shown the same result for the assignment setting. Theorem 1.…”

“…(Aziz et al [2013a], Saban and Sethuraman [2013]) In the assignment setting, computing the RSD probabilities is #P-complete.…”

“…Similarly, in voting, the probabilities returned by RSD can be interpreted as fractions of time or other resources allotted to the alternatives. Saban and Sethuraman [2013] mentioned identifying the conditions under which the RSD probabilities can be computed in polynomial time as an open problem. In another paper, Mennle and Seuken [2013] propose random hybrid assignment mechanisms that hinge on the RSD probabilities.…”

Counting houses of Pareto optimal matchings in the house allocation problem

“…We are aware of four recent papers that considered similar results to our complexity results [16,3,4,7]. Their main motivation is to study the behavior of the randomized serial dictatorship also called randomized priority allocation.…”

“…The first paper is by Sabán and Sethuraman [16]. Their result, reformulated in our context, is NP-hardness of Problem 1, for arbitrary matrices.…”