{"ID":2855261,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.13633","arxiv_id":"2510.13633","title":"Online Fair Division With Subsidy: When Do Envy-Free Allocations Exist, and at What Cost?","abstract":"We study the problem of fairly allocating $m$ indivisible items arriving online, among $n$ (offline) agents. Although envy-freeness has emerged as the archetypal fairness notion, envy-free (EF) allocations need not exist with indivisible items. To bypass this, a prominent line of research demonstrates that there exist allocations that can be made envy-free by allowing a subsidy. Extensive work in the offline setting has focused on finding such envy-freeable allocations with bounded subsidy. We extend this literature to an online setting where items arrive one at a time and must be immediately and irrevocably allocated. Our contributions are two-fold: 1. Maintaining EF Online: We show that envy-freeability cannot always be preserved online when the valuations are submodular or supermodular, even with binary marginals. In contrast, we design online algorithms that maintain envy-freeability at every step for the class of additive valuations, and for its superclasses including $k$-demand and SPLC valuations. 2. Ensuring Low Subsidy: We investigate the quantity of subsidy required to guarantee envy-freeness online. Surprisingly, even for additive valuations, the minimum subsidy may be as large as $Ω(mn)$, in contrast to the offline setting, where the bound is $O(n)$. On the positive side, we identify valuation classes where the minimum subsidy is small (i.e., does not depend on $m$), including $k$-valued, rank-one, restricted additive, and identical valuations, and we obtain (mostly) tight subsidy bounds for these classes.","short_abstract":"We study the problem of fairly allocating $m$ indivisible items arriving online, among $n$ (offline) agents. Although envy-freeness has emerged as the archetypal fairness notion, envy-free (EF) allocations need not exist with indivisible items. To bypass this, a prominent line of research demonstrates that there exist...","url_abs":"https://arxiv.org/abs/2510.13633","url_pdf":"https://arxiv.org/pdf/2510.13633v1","authors":"[\"Pooja Kulkarni\",\"Ruta Mehta\",\"Vishnu V. Narayan\",\"Tomasz Ponitka\"]","published":"2025-10-15T14:57:47Z","proceeding":"cs.GT","tasks":"[\"cs.GT\"]","methods":"[]","has_code":false}