Fair and Truthful Mechanisms for Dichotomous Valuations

Abstract

We consider the problem of allocating a set on indivisible items to players with private preferences in an efficient and fair way. We focus on valuations that have dichotomous marginals, in which the added value of any item to a set is either 0 or 1, and aim to design truthful allocation mechanisms (without money) that maximize welfare and are fair. For the case that players have submodular valuations with dichotomous marginals, we design such a deterministic truthful allocation mechanism. The allocation output by our mechanism is Lorenz dominating, and consequently satisfies many desired fairness properties, such as being envy-free up to any item (EFX), and maximizing the Nash Social Welfare (NSW). We then show that our mechanism with random priorities is envy-free ex-ante, while having all the above properties ex-post. Furthermore, we present several impossibility results precluding similar results for the larger class of XOS valuations.

A General Framework for Fair Allocation with Matroid Rank Valuations

“…Benabbou et al [2021] first show that it is possible to compute a welfare maximizing envy free up to one good (EF1) allocation in polynomial time. Babaioff et al [2021a] extend this result and show that it is possible to compute Lorenz dominating allocations in polynomial time; Lorenz dominance is a stronger notion of fairness than leximin and it implies a host of other fairness properties. Recently, Viswanathan and Zick [2022] show that Lorenz dominating allocations can be computed using a simple Yankee Swap based mechanism.…”

“…This implies that X is Lorenz dominating and Y is leximin. Babaioff et al [2021a] introduce and study the concept of prioritized Lorenz dominating allocations. Each agent is given a priority which is represented using a permutation π :…”

“…When agents have MRF valuations {v i } i∈N , prioritized Lorenz dominating allocations are defined as Lorenz dominating allocations for the fair allocation instance where valuations are defined as Babaioff et al [2021a] show that when agents have MRF valuations, a prioritized Lorenz dominating allocation is guaranteed to exist and satisfies several desirable fairness properties such as leximin, envy freeness up to any good (EFX) and maximizing Nash welfare.…”

“…If agents have binary submodular valuations, then Yankee Swap is guaranteed to output a Lorenz dominating allocation. As shown by Babaioff et al [2021a], Lorenz dominating allocations satisfy a broad range of fairness and efficiency criteria: they are leximin fair, which implies that they maximize social welfare, Nash welfare, envy-freeness up to one item, and guarantee every agent at least half of their maximin share. In addition, they are truthful: no agent has an incentive to misreport their preferences over items.…”

“…In addition, they are truthful: no agent has an incentive to misreport their preferences over items. While Babaioff et al [2021a] present a poly-time algorithm for computing Lorenz dominating allocations, Yankee Swap is significantly simpler to implement, and has a faster running time.…”