Abstract
The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a “fair” manner, where every agent has a valuation for each subset of goods. We assume monotone valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is “envy-freeness up to any good,” EFX, where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$, where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have (1) envy-freeness up to any good, (2) no agent values $P$ higher than her own bundle, and (3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive and leads to a pseudopolynomial time algorithm to find such an allocation. When agents have additive valuations and $|{P}|$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation that is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. (Very recently and independently, Amanatidis, Ntokos, and Markakis [Theoret. Comput. Sci., 841 (2020), pp. 94--109], also showed the existence of a 4/7-GMMS allocation.)
Highlights
Fair division of goods among competing agents is a fundamental problem in Economics and Computer Science
A minor variant of our algorithm shows the existence of an allocation which is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS
We show our main existence result for general valuation functions, i.e., the only assumptions we make on any valuation function vi is that (i) it is normalized, i.e., vi(∅) = 0, and (ii) it is monotone, i.e., S ⊆ T implies vi(S) ≤ vi(T )
Summary
Fair division of goods among competing agents is a fundamental problem in Economics and Computer Science. It was shown in [26] that exponentially many value queries may be needed to determine EFX allocations even in the restricted setting where there are only two agents with identical submodular valuation functions1 It is not known if an EFX allocation always exists even when there are only three agents with additive valuations. Caragiannis et al [12] introduced a more relaxed notion of EFX called EFX-with-charity This is a partial allocation that is EFX, i.e., the entire set of goods need not be distributed among the agents. Xn∗ be an optimal Nash social welfare allocation on the entire set of goods It was shown in [12] that there always exists an EFX-with-charity allocation X = X1, . We seek EFX-with-charity allocations with bounds on the set given to charity, i.e., a bound on the size and a bound on the value of the set of goods donated to charity
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