We consider the problem of fair allocation of indivisible goods to agents, with no transfers. When agents have equal entitlements, the well established notion of the maximin share (MMS) serves as an attractive fairness criterion, where to qualify as fair, an allocation needs to give every agent at least a substantial fraction of her MMS.In this paper we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. The AnyPrice share of an agent is the value she can guarantee to herself if she is given a budget equal to her entitlement, and she buys her highest value affordable set when items are adversarially priced with a total price equal to the total entitlements. Even for the equal entitlements case, this notion is new, and satisfies ≥ , where the inequality is sometimes strict. We also present an alternative definition for the APS as a maximization problem (a fractional version of the MMS), and provide comparisons between the APS and previous notions of fairness.Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomialtime algorithm that gives every agent at least a 3 5 -fraction of her APS. This algorithm can also be viewed as providing a strategy in a certain natural bidding game, and this strategy secures each agent that uses it at least a 3 5 -fraction of her APS, regardless of the strategies used by other agents.
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.
We provide prophet inequality algorithms for online weighted matching in general (nonbipartite) graphs, under two well-studied arrival models: edge arrival and vertex arrival. The weights of the edges are drawn from a priori known probability distribution. Under edge arrival, the weight of each edge is revealed on arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. To study these settings, we introduce a novel unified framework of batched-prophet inequalities that captures online settings where elements arrive in batches. Our algorithms rely on the construction of suitable online contention resolution scheme (OCRS). We first extend the framework of OCRS to batched-OCRS, we then establish a reduction from batched-prophet inequality to batched-OCRS, and finally we construct batched-OCRSs with selectable ratios of 0.337 and 0.5 for edge and vertex arrival models, respectively. Both results improve the state of the art for the corresponding settings. For vertex arrival, our result is tight. Interestingly, a pricing-based prophet inequality with comparable competitive ratios is unknown.
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). In contrast, even dropping all fairness requirements, there is no truthful allocation mechanism (without money) that maximizes welfare when players have XOS valuations with dichotomous marginals.To gauge the robustness of our positive results, we also study ǫ-dichotomous valuations, in which the added value of any item to a set is either non-positive, or in the range [1, 1 + ǫ]. We show several impossibility results in this setting, and also a positive result: for players that have additive ǫ-dichotomous valuations with sufficiently small ǫ, we design a randomized truthful mechanism with strong ex-post guarantees. For ρ = 1 1+ǫ , the allocations that it produces generate at least a ρ-fraction of the maximum welfare, and enjoy ρ-approximations for various fairness properties, such as being envy-free up to one item (EF1), and giving each player at least her maximin share.
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