Approximating the Nash Social Welfare with Indivisible Items

Abstract

We study the problem of allocating a set of indivisible items among agents with additive valuations, with the goal of maximizing the geometric mean of the agents' valuations, i.e., the Nash social welfare. This problem is known to be NPhard, and our main result is the first efficient constant-factor approximation algorithm for this objective. We first observe that the integrality gap of the natural fractional relaxation is exponential, so we propose a different fractional allocation which implies a tighter upper bound and, after appropriate rounding, yields a good integral allocation.An interesting contribution of this work is the fractional allocation that we use. The relaxation of our problem can be solved efficiently using the Eisenberg-Gale program, whose optimal solution can be interpreted as a market equilibrium with the dual variables playing the role of item prices. Using this market-based interpretation, we define an alternative equilibrium allocation where the amount of spending that can go into any given item is bounded, thus keeping the highly priced items under-allocated, and forcing the agents to spend on lower priced items. The resulting equilibrium prices reveal more information regarding how to assign items so as to obtain a good integral allocation.

Fairness and Incentive Compatibility via Percentage Fees

“…Our Results I: Incentive Compatible Mechanisms that Maximize the NSW The Nash Social Welfare has been heavily studied recently in Algorithmic Game Theory. Its game theoretic properties have been analyzed (e.g., [8]) and the possible approximation ratios achievable in different settings have been studied (e.g., [11,1,5,27]). Unfortunately, as discussed earlier, no dominant strategy mechanisms can maximize the NSW in the traditional model.…”

Approximations for Indivisible Concave Allocations with Applications to Nash Welfare Maximization

“…As discussed earlier, there is a recent line of work that has extensively examined the Nash welfare objective for indivisible items. This interest was sparked by the seminal result of Cole and Gkatzelis (Cole and Gkatzelis 2015), who obtained the first constant approximation for symmetric agents with additive valuations, i.e., for all agents i we have η i = 1 and v i (u i ) = u i . This bound has been subsequently improved and extended to more general valuation functions in the symmetric agent case.…”

“…Unfortunately, even when agents have additive valuations, the approximability of the asymmetric case still remains a key open problem in the area, where the best known approximation bound is currently O(n) (Garg, Kulkarni, and Kulkarni 2020). We note that the original breakthrough for the symmetric objective in (Cole and Gkatzelis 2015) was directly aimed at circumventing the Ω(n) integrality gap of the assignment convex program for the Nash objective. Subsequent work has either generalized these techniques or utilized other approaches that exploit the symmetry of the agents' valuations.…”

On the Max-Min Fair Stochastic Allocation of Indivisible Goods

“…For the fair allocation of indivisible goods with additive valuations, Caragiannis et al [7] proved that an MNW allocation is envy-free up to one good (EF1), that is, each agent i does not envy another agent j if some indivisible good is removed from the bundle of agent j. Since computing an MNW allocation is hard in general [25], there is a series of research to design an approximation algorithm [1,8,9,10,20]. Benabbou et al [5] proved that the set of MNW allocations coincides with that of minimizers of any symmetric strictly convex function, even when the utility of each agent is represented by a matroid rank function.…”