Constrained Object Allocation Problems

The Object Allocation Problem (OAP) is a well studied problem which is about to allocate a set X of n objects to a set N of n agents. This paper deals with a generalization called Constrained Object Allocation Problem (COAP) where the set of objects allocated to the agents must satisfy a given feasibility constraint. The input is a set X of at least n elements and a collection S of subsets of X , each of size n. Every S ∈ S defines a set of elements that the agents can collectively possess and such that every agent is allocated exactly one element. In this article we first study the problem of a central authority who wants to maximize the social welfare defined in two ways: the sum of the agents’ utility for the item they receive (utilitarian) or the utility of the poorest agent (egalitarian). This optimization problem is shown NP-hard for COAP in general but polynomial time solvable when S is the base set of a matroid (for the utilitarian social welfare and the egalitarian social welfare). An allocation can be built by the agents without communicating their utilities to a central authority. They can use a mechanism like the famous Serial Dictatorship mechanism (SD). In SD, a permutation of the agents is given and, starting from scratch, the agents select in turn their most preferred element among the remaining items. We analyse the solutions produced by a version of SD adapted to COAP. There are instances of COAP such that SD fails to produce a socially optimal allocation, whatever the order on the agents. However, if S is the base set of a matroid, then we prove that SD produces a social optimum for at least one permutation (for the utilitarian social welfare and the egalitarian social welfare). Then, we give tight worst case bounds on the ratio between the social utility under SD and the optimal social utility. These bounds are valid for both OAP and COAP. We conclude with a proof showing that manipulating SD for inducing a socially good allocation in OAP is NP-hard even with 3-approval scores. Here, we retain two ways to indicate that an allocation is socially good: the sum of the agents’ utilities is maximum and the minimum utility of an agent is maximum. We end the paper by showing some special cases where manipulating is polynomial. In particular, we obtain a dichotomic complexity result of the manipulation problem for the egalitarian social welfare.