The maximum Nash welfare (MNW) solution—which selects an allocation that maximizes the product of utilities—is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy-free up to one good—a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and—even more so—in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation and demonstrate that it scales well on real data. These results establish MNW as a compelling solution for allocating indivisible goods and underlie its deployment on a popular fair-division website.
The maximum Nash welfare (MNW) solution -which selects an allocation that maximizes the product of utilities -is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is unexpectedly, strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy free up to one good -a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and -even more so -in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation, and demonstrate that it scales well on real data. These results lead us to believe that MNW is the ultimate solution for allocating indivisible goods, and underlie its deployment on a popular fair division website.
A new lanthanum calcium borate with the composition La2CaB10O19 has been synthesized. Single crystals have been grown from a melt close to the stoichiometric composition. The compound crystallizes in the monoclinic system, space group C2, with a = 11.043(3) Å, b = 6.563(2) Å, c = 9.129(2) Å, α = γ = 90°, β = 91.47°, and two formula units per cell. The crystal structure contains B5O12 double-ring pentaborate groups, which are linked together to form an infinite two-dimensional double layer. The layer runs almost perpendicular to the c axis of the crystal. The La atoms are located in layers, while the Ca atoms are located between two layers. La2CaB10O19 exhibits an optical second-harmonic generation effect about twice as large as that of KDP (KH2PO4).
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