# Online Nash Social Welfare with Predictions

Siddhartha Banerjee, Vasilis Gkatzelis, Artur Gorokh, Billy Jin

October 2021
### Abstract

We consider the problem of allocating a set of divisible goods to $N$ agents in an online manner, aiming to maximize the Nash social welfare, a widely studied objective which provides a balance between fairness and efficiency. The goods arrive in a sequence of $T$ periods and the value of each agent for a good is adversarially chosen when the good arrives. We first observe that no online algorithm can achieve a competitive ratio better than the trivial $O(N)$, unless it is given additional information about the agents’ values. Then, in line with the emerging area of “algorithms with predictions”, we consider a setting where for each agent, the online algorithm is only given a prediction of her *monopolist utility*, i.e., her utility if all goods were given to her alone (corresponding to the sum of her values over the $T$ periods). Our main result is an online algorithm whose competitive ratio is parameterized by the multiplicative errors in these predictions. The algorithm achieves a competitive ratio of $O(\log N)$ and $O(\log T)$ if the predictions are perfectly accurate. Moreover, the competitive ratio degrades smoothly with the errors in the predictions, and is surprisingly robust: the logarithmic competitive ratio holds even if the predictions are very inaccurate. We complement this positive result by showing that our bounds are essentially tight: no online algorithm, even if provided with perfectly accurate predictions, can achieve a competitive ratio of $O((\log N)^{1−\epsilon})$ or $O((\log T)^{1-\epsilon})$ for any constant $\epsilon >0$.

Publication

*2022 ACM Symposium on Discrete Algorithms*

Earlier version titled Online Nash Social Welfare via Promised Utilities used normalized valuations instead of predicted valuations.

###### Associate Professor

Sid Banerjee is an associate 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.