Weighted Envy-freeness in Indivisible Item Allocation

Abstract

We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their entitlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up to one item (WEF1): strong , where envy can be eliminated by removing an item from the envied agent’s bundle, and weak , where envy can be eliminated either by removing an item (as in the strong version) or by replicating an item from the envied agent’s bundle in the envying agent’s bundle. We show that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1, but it always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strong WEF1 and Pareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in which weighted fair division is richer and more challenging than its unweighted counterpart.

Almost Envy-Freeness for Groups: Improved Bounds via Discrepancy Theory

“…Indeed, the techniques from discrepancy theory that we used crucially rely on the additivity assumption; so does the result of Alon [1987] that established the existence of a consensus 1/k-division for divisible goods. Even in the case k = 2, where a consensus halving can be guaranteed for non-additive utilities [Simmons and Su, 2003], it is unclear whether such a halving can be rounded into a discrete allocation with a loss that is bounded only in terms of n. Beyond the setting of our paper, one could also consider allocating a mixture of indivisible and divisible goods [Bei et al, 2021] or allowing groups to have different entitlements which can correspond to the group sizes [Chakraborty et al, 2020] as well.…”

Tractable Fragments of the Maximum Nash Welfare Problem

“…When goods are indivisible and agents are symmetric, Caragiannis et al [7] showed the "unreasonable fairness" of an MNW allocation by proving that it satisfies a relaxation of EF called envy-freeness up to one good (EF1) and is PO. These desirable properties also carry over to the asymmetric case, where the MNW allocation is PO and satisfies a weighted relaxation of EF [9].…”

“…This idea has been used in models for bargaining in committees [25], maintaining fairness guarantees among ethnic groups [6], etc. From a computational viewpoint, fair division of goods to asymmetric agents is relatively less understood, but has gained interest in recent years [15,17,9].…”

Fair Division of Indivisible Goods: A Survey

“…However, there are se ings where the fairness of an allocation must be considered with respect to asymmetric entitlements, e.g., in many inheritance scenarios, closer relatives have higher entitlements o en determined by law. In order to capture fairness in the presence of arbitrary entitlements, one may generalize existing notions to their weighted counterparts, like weighted MMS [Farhadi et al, 2019] and weighted EF1 [Chakraborty et al, 2021], or tweek their definitions approprietly, like in the MMS-inspired ℓ-out-of-d share of Babaioff et al [2021c]. In a recent work, Babaioff et al [2021b] introduce the notion of AnyPrice share (APS) as the maximum value an agent can guarantee to herself if she has a budget equal to her entitlement and the goods are adversarially priced with prices that sum up to 1, and show how to efficiently compute an allocation where everyone gets value no less than 3/5 of her APS.…”