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We develop polynomialtime algorithms for the fair and efficient allocat...
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Computing the Shapley Value in Allocation Problems: Approximations and Bounds, with an Application to the Italian VQR Research Assessment Program
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Fair and Efficient Allocations under Subadditive Valuations
We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. EnvyFreeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely 12EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents' valuations) is one of the most commonly used measures of efficiency. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like EnvyFree up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomialtime algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an šŖ(n) approximation to the Nash welfare. Our result also improves the current bestknown approximation of šŖ(n log n) and šŖ(m) to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an šŖ(n) approximation to a family of welfare measures, pmean of valuations for pā (ā, 1], thereby also matching asymptotically the current best known approximation ratio for special cases like p =ā while also retaining the fairness properties.
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