Worst case compromises in matroids with applications to the allocation of indivisible goods

We consider the problem of equitably allocating a set of indivisible goods to n agents with additive utilities so as to provide worst case guarantees on agents' utilities. Demko and Hill [6] showed the existence of an allocation where every agent values his share at least Vn(α)Vn(α), which is a family of nonincreasing functions of α, defined as the maximum value assigned by an agent to a single good. A deterministic algorithm returning such an allocation in polynomial time was proposed in [15]. Interestingly, Vn(α)Vn(α) is tight for some values of α, i.e. it matches the highest possible utility of the least happy agent. However, this is not true for all values of α  . We propose a family of functions WnWn such that Wn(x)≥Vn(x)Wn(x)≥Vn(x) for all x  , and Wn(x)>Vn(x)Wn(x)>Vn(x) for values of x   where Vn(x)Vn(x) is not tight. The functions WnWn apply on a problem that generalizes the allocation of indivisible goods. It is to find a base in a matroid which is common to n agents. Our results are constructive, they are achieved by analyzing an extension of the algorithm of Markakis and Psomas. We also present an upper bound on the utility of the least happy agent.