# Online Fair Division: Towards Ex-Post Constant MMS Guarantees

arXiv: https://arxiv.org/abs/2503.02088 · HTML: https://mlsanity.com/paper/2503.02088/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2025.

## Built on by (most cited first, 1 of 1)

- [Online Fair Division for Personalized 2-Value Instances](https://mlsanity.com/paper/2505.22174/) (2025)

## Abstract (arXiv)

We investigate the problem of fairly allocating m indivisible items among n sequentially arriving agents with additive valuations, under the sought-after fairness notion of maximin share (MMS). We first observe a strong impossibility: without appropriate knowledge about the valuation functions of the incoming agents, no online algorithm can ensure any non-trivial MMS approximation, even when there are only two agents. Motivated by this impossibility, we introduce OnlineKTypeFD (online k-type fair division), a model that balances theoretical tractability with real-world applicability. In this model, each arriving agent belongs to one of k types, with all agents of a given type sharing the same known valuation function. We do not constrain k to be a constant. Upon arrival, an agent reveals her type, receives an irrevocable allocation, and departs. We study the ex-post MMS guarantees of online algorithms under two arrival models: 1- Adversarial arrivals: In this model, an adversary determines the type of each arriving agent. We design a frac1k-MMS competitive algorithm and complement it with a lower bound, ruling out any Omega(frac1√k)-MMS-competitive algorithm, even for binary valuations. 2- Stochastic arrivals: In this model, the type of each arriving agent is independently drawn from an underlying, possibly unknown distribution. Unlike the adversarial setting where the dependence on k is unavoidable, we surprisingly show that in the stochastic setting, an asymptotic, arbitrarily close-to-frac12-MMS competitive guarantee is achievable under mild distributional assumptions. Our results extend naturally to a learning-augmented framework; when given access to predictions about valuation functions, we show that the competitive ratios of our algorithms degrade gracefully with multiplicative prediction errors.