A Solution to the Random Assignment Problem with a Matroidal Family of Goods

Problems of allocating indivisible goods to agents in an efficient and fair manner without money have long been investigated in the literature. The random assignment problem is one of them, where we are given a fixed feasible (available) set of indivisible goods and a profile of ordinal preferences over the goods, one for each agent, and we determine an assignment of goods to agents in a randomized way using lotteries. A seminal paper of Bogomolnaia and Moulin (2001) shows a probabilistic serial mechanism to give an efficient and envy-free solution to the assignment problem. In this paper we consider an extension of the random assignment problem to the case where we are given a family B of feasible sets of indivisible goods. In particular we consider the case where B is a family of bases of a matroid. Under the agents’ ordinal preferences over goods we show an extension of the probabilistic serial mechanism to give an efficient and envy-free solution that probabilistically makes a choice of a member (a base) of the family and its assignment to agents. The theory of submodular optimization plays a crucial rôle in the extension.