# Lag: A Mean-Field Optimal Policy for Online Assignment with Reusable Resources

arXiv: https://arxiv.org/abs/2610.19038 · HTML: https://mlsanity.com/paper/2610.19038/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 31 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 5 of 5)

- [On Strategyproof Conference Peer Review](https://mlsanity.com/paper/1806.06266/) (2018)
- [Allocation Problems in Ride-Sharing Platforms: Online Matching with Offline Reusable Resources](https://mlsanity.com/paper/1711.08345/) (2017)
- [Contextual Restless Multi-Armed Bandits with Application to Demand Response Decision-Making](https://mlsanity.com/paper/2403.15640/) (2024)
- [Lagrangian Index Policy for Restless Bandits with Average Reward](https://mlsanity.com/paper/2412.12641/) (2024)
- [Online Task Assignment Problems with Reusable Resources](https://mlsanity.com/paper/2203.07605/) (2022)

## Abstract (arXiv)

We study online task assignment with reusable resources. This problem arises in peer reviewer assignment, skill-based call routing, rideshare matching, emergency dispatch, and crowdsourcing: Tasks arrive according to a geometric process and must be immediately assigned to servers that become temporarily busy, yielding rewards that depend on the task-server pair. The goal is to maximize long-run average reward. We cast the problem as a contextual restless multi-armed bandit (CRMAB). We derive an occupancy-measure LP (OCC-LP) that upper-bounds the reward of any work-conserving policy. From its dual we obtain Lag, a closed-form index policy. We introduce a scaling regime in which the number of copies of each server and the number of arrivals per step grow together. This regime yields the first deterministic fluid limit for a contextual bandit. In it, Lag is asymptotically optimal. Finally, we evaluate our algorithms on peer review datasets and call center routing. The Lagrangian index achieves 83-99% of the LP upper bound across all load levels. Moreover, the Lagrangian index policy significantly outperforms the natural greedy assignment scheme, improving reward by up to 12.3% on journal data and 36% on structured synthetic instances.