The residual maximin share

We consider fair allocations of indivisible goods to agents with general monotone valuations. We observe that it is useful to introduce a new share-based fairness notion, the {\em residual maximin share} (RMMS). This share is {\em feasible} and {\em self maximizing}. Its value is at least as large as the MXS for monotone valuations, and at least as large as $\frac{2}{3}$-MMS for additive valuations. Known techniques easily imply the existence of partial allocations that are both RMMS and EFX, and complete allocations that are both RMMS and EFL. This unifies and somewhat improves upon several different results from previous papers.

@article{feige2025_2505.19961,
  title={ The residual maximin share },
  author={ Uriel Feige },
  journal={arXiv preprint arXiv:2505.19961},
  year={ 2025 }
}