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

TL;DR

This work investigates online fair division of $m$ indivisible goods among $n$ offline agents with subsidies to achieve envy-freeness. It establishes that envy-freeness via local efficiency can be maintained online for additive, SPLC, and $k$-demand valuations, while proving impossibility results for budget-additive, binary submodular, and binary supermodular valuations. The study characterizes both upper and lower bounds on the minimum subsidy, finding that online subsidies can be as large as $Ω(mn)$ in some cases but remain constant in $m$ for several structured valuation classes (e.g., rank-one, restricted additive, identical monotone). Across multiple valuation families, the authors provide (nearly) tight subsidy bounds and highlight a separation between online and offline regimes, showing substantial online costs compared to offline counterparts. The results have practical implications for streaming fair division and chore division, and they open several avenues for future work on extending online EF guarantees to broader valuation classes.

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.

Figures (1)

• Figure 1: Illustration of the lower bound construction for rank-one valuations in the proof of Theorem \ref{['thm:single_param']}. Each blue bar represents an item, with its length proportional to the item’s base value and the label inside indicating its arrival order. The allocation shown is the subsidy-minimizing one that maintains envy-freeability at every step. The red bar lengths are proportional to each agent's envy toward agent $1$.

Theorems & Definitions (30)

• Definition 2.1: Envy-Freeness with Subsidy, Envy-Freeability
• Definition 2.2: Local Efficiency
• Theorem 2.3: aragones1995derivationHaake2002Hartline2008halpern2019fair
• Definition 2.4: Total Subsidy
• Lemma 2.5: halpern2019fairBrustleDNSV20
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• proof