Finding Fair and Efficient Allocations for Matroid Rank Valuations

Abstract

In this article, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions . This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e., utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.

On Fair Division under Heterogeneous Matroid Constraints

“…This allocation is not EF1, since v B (X A ) = 5 so Bob envies Alice by more than one item (note that F-EF1 and EF1 are equivalent in this case, since both agents have the same constraints). Note that Benabbou et al (2021) prove that MNW always implies EF1 for submodular valuations with binary marginals. However, they consider clean allocations, where items with 0 marginal value are not allocated.…”

“…Proof. Benabbou et al (2021) prove that, for agents whose valuations are matroid-rank functions, there always exists an EF1 allocation that maximizes social welfare. When agents have unary valuations, the social welfare of every complete feasible allocation is exactly m (the number of items).…”

“…Since a complete feasible allocation exists, the maximum attainable social welfare is m. The agents' feasible-valuation functions vi are matroid-rank functions. Therefore, by Benabbou et al (2021), there exists an F-EF1 allocation with social welfare m. This allocation must be a complete feasible allocation.…”

“…These valuations are equivalent to matroid rank functions -functions that evaluate a bundle by the size of the largest independent set of a certain matroid contained in that bundle. In this setting, Benabbou et al (2021) and Babaioff et al (2021) present allocation mechanisms that are Pareto-efficient, EF1, EFX and strategyproof. Barman and Verma (2021) present a polynomial-time algorithm that finds an allocation satisfying maximinshare fairness.…”

“…The valuation functions constructed this way are no longer additive, but are submodular (Oxley, 2006). Recently, Babaioff, Ezra, and Feige (2021) and Benabbou, Chakraborty, Igarashi, and Zick (2021) have independently proved the existence of EF1 allocations in the special case in which agents have submodular valuations with binary marginals (where adding an item to a bundle adds either 0 or 1 to its value). Such an allocation can be converted to a fair and feasible allocation by giving each agent the best feasible subset of his/her allocated bundle, and disposing of the other items.…”