Online Fair Division With Subsidy: When Do Envy-Free Allocations Exist, and at What Cost?

Pooja Ravi Kulkarni, Ruta Mehta, Vishnu V. Narayan, Tomasz Ponitka

Keywords: Fair Division, Online Algorithms, Subsidy, Envy-Freeness, Resource Allocation

TL;DR: We study the problem of finding an envy-free allocation with subsidy for indivisible items arriving online, among $n$ (offline) agents.

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:
- **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 valuations and SPLC valuations.
- **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 $\Omega(mn)$, 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.

Area: Game Theory and Economic Paradigms (GTEP)

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Submission Number: 201