Online Fair Division With Subsidy: When Do Envy-Free Allocations Exist, and at What Cost?
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arXiv
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| Format: | Preprint |
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2025
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| author | Kulkarni, Pooja Mehta, Ruta Narayan, Vishnu V. Ponitka, Tomasz |
| author_facet | Kulkarni, Pooja Mehta, Ruta Narayan, Vishnu V. Ponitka, Tomasz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Online Fair Division With Subsidy: When Do Envy-Free Allocations Exist, and at What Cost? Kulkarni, Pooja Mehta, Ruta Narayan, Vishnu V. Ponitka, Tomasz Computer Science and Game Theory 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. |
| title | Online Fair Division With Subsidy: When Do Envy-Free Allocations Exist, and at What Cost? |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2510.13633 |