Jugal Garg

Jugal Garg, Eklavya Sharma

We consider the problem of fairly dividing indivisible goods among agents with additive valuations. It is known that an Epistemic EFX and $2/3$-MMS allocation can be obtained using the Envy-Cycle-Elimination (ECE) algorithm. In this work, we explore whether this algorithm can be randomized to also ensure ex-ante proportionality. For two agents, we show that a randomized variant of ECE can compute an ex-post EFX and ex-ante envy-free allocation in near-linear time. However, for three agents, we show that several natural randomization methods for ECE fail to achieve ex-ante proportionality.

Jugal Garg, Edin Husić, Wenzheng Li et al.

For any $\varepsilon>0$, we give a simple, deterministic $(4+\varepsilon)$-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents' valuations, and give an $e (ω+ 2 + \varepsilon)$-approximation if the ratio between the largest weight and the average weight is at most $ω$. We also show that the $1/2$-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time that is both $1/2$-EFX and a $(8+\varepsilon)$-approximation to the symmetric NSW problem under submodular valuations.

Jugal Garg, Yixin Tao, László A. Végh

We study the relationship between two central concepts in the allocation of divisible goods: competitive equilibrium (CE) and allocations that maximize Nash welfare, i.e., allocations where the weighted geometric mean of the utilities is maximal. When agents have homogeneous concave utility functions, these concepts coincide: the classical Eisenberg-Gale convex program that maximizes Nash welfare over feasible allocations yields a competitive equilibrium. However, they diverge for non-homogeneous utilities. From a computational perspective, maximizing Nash welfare amounts to solving a convex program for any concave utility functions, whereas computing CE becomes PPAD-hard already for separable piecewise linear concave (SPLC) utilities. We introduce the concept of Gale-substitute utility functions, an analogue of the weak gross substitutes (WGS) property for the so-called Gale demand system. For Gale-substitutes utilities, we show that any allocation maximizing Nash welfare provides an approximate-CE with surprisingly strong guarantees, where every agent gets at least half the maximum utility they can get at any CE, and is approximately envy-free. Gale-substitutes include utility functions where computing CE is PPAD hard, such as all separable concave utilities and the previously studied non-separable class of Leontief-free utilities. We introduce a broad new class of utility functions called generalized network utilities based on the generalized flow model. This class includes SPLC and Leontief-free utilities, and we show that all such utilities are Gale-substitutes. Conversely, although some agents may get much higher utility at a Nash welfare maximizing allocation than at a CE, we show a `price of anarchy' type result: for general concave utilities, every CE achieves at least $(1/e)^{1/e} > 0.69$ fraction of the maximum Nash welfare, and this factor is tight.

Michal Feldman, Jugal Garg, Vishnu V. Narayan et al.

We study fair mechanisms for the classic job scheduling problem on unrelated machines with the objective of minimizing the makespan. This problem is equivalent to minimizing the egalitarian social cost in the fair division of chores. The two prevalent fairness notions in the fair division literature are envy-freeness and proportionality. Prior work has established that no envy-free mechanism can provide better than an $Ω(\log m/ \log \log m)$-approximation to the optimal makespan, where $m$ is the number of machines, even when payments to the machines are allowed. In strong contrast to this impossibility, our main result demonstrates that there exists a proportional mechanism (with payments) that achieves a $3/2$-approximation to the optimal makespan, and this ratio is tight. To prove this result, we provide a full characterization of allocation functions that can be made proportional with payments. Furthermore, we show that for instances with normalized costs, there exists a proportional mechanism that achieves the optimal makespan. We conclude with important directions for future research concerning other fairness notions, including relaxations of envy-freeness. Notably, we show that the technique leading to the impossibility result for envy-freeness does not extend to its relaxations.