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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.

Axioms of Cooperative Decision Making provides a unified and comprehensive study of welfarism, cooperative games, public decision making, and voting and social choice theory - technically heterogeneous subjects that are linked by common axioms. Hervé Moulin studies these areas from an axiomatic perspective. Every axiom conveys a certain ethical principle (e.g. 'one man one vote', or 'to each according to his contribution'). Axiomatic theory examines the compatibility of various combinations of axioms. The book describes recent successes of this method. It gives many examples of the axiomatic approach, ranging from the construction of numerical indices for measuring inequality, to the pricing of a regulated monopoly, to the comparison of various voting rules. The book will be useful to mathematical modellers, especially those interested in economics and political science.