# Online Nash Social Welfare via Promised Utilities

### Abstract

We consider the problem of allocating a set of divisible goods to $N$ agents in an online manner over $T$ periods, with adversarially-chosen normalized valuations in each period. Our goal is to maximize the Nash social welfare, a widely studied objective which provides a balance between fairness and efficiency. On the positive side, we provide an online algorithm that achieves a competitive ratio of $O(\log N)$ and $O(\log T)$, but also a stronger competitive ratio of $O(\log k)$ in settings where the value of any agent for her most preferred item is no more than $k$ times her average value. We complement this by showing this bound is essentially tight: no online algorithm can achieve a competitive ratio of $O((\log N)^{1−\epsilon}) or$O((\log T)^{1-\epsilon})$for any constant$\epsilon >0\$.

Publication
Preprint arXiv_id:2008.03564
##### Siddhartha Banerjee
###### Assistant Professor

Sid Banerjee is an assistant professor in the School of Operations Research at Cornell, working on topics at the intersection of data-driven decision-making, market design, and algorithms for large-scale networks.