When agents choose bundles autonomously: guarantees beyond discrepancy

TL;DR

The paper studies fair division of indivisible items under additive valuations where agents autonomously select from a predesigned partition. It shows that universal partitions cannot beat a $Θ(\sqrt{n})$ discrepancy barrier, and introduces a dynamic rebundling framework that, in polynomial time, delivers per-agent guarantees of $\mathsf{PROP}_i - O(\log n)$, a substantial improvement over classical discrepancy bounds. The authors develop multiple algorithmic regimes: static partitions near unanimity yield strong guarantees in ordered-additive settings, while dynamic rebundling achieves near-proportional guarantees for general valuations with bounded-proportional shares, bounded influence (hypergraph) cases, and bounded indifference; they also provide a way to compute fair arrival orders. They present tight analyses showing how the number of stages, transfers, and rebundlings control the loss from the proportional share, and extend results to structured valuation classes (linearly separable, laminar swaps, Lipschitz valuations). An impossibility result confirms the intrinsic limit of universal guarantees, while the organized framework enables high-prominence per-agent outcomes in a range of realistic contexts, with practical implications for sequential, autonomy-driven allocations.

Abstract

We consider the fair division of indivisible items among $n$ agents with additive non-negative normalized valuations, with the goal of obtaining high value guarantees, that is, close to the proportional share for each agent. We prove that partitions where \emph{every} part yields high value for each agent are asymptotically limited by a discrepancy barrier of $Θ(\sqrt{n})$. Guided by this, our main objective is to overcome this barrier and achieve stronger individual guarantees for each agent in polynomial time. Towards this, we are able to exhibit an exponential improvement over the discrepancy barrier. In particular, we can create partitions on-the-go such that when agents arrive sequentially (representing a previously-agreed priority order) and pick a part autonomously and rationally (i.e., one of highest value), then each is guaranteed a part of value at least $\mathsf{PROP} - \mathcal{O}{(\log n)}$. Moreover, we show even better guarantees for three restricted valuation classes such as those defined by: a common ordering on items, a bound on the multiplicity of values, and a hypergraph with a bound on the \emph{influence} of any agent. Specifically, we study instances where: (1) the agents are ``close'' to unanimity in their relative valuation of the items -- a generalization of the ordered additive setting; (2) the valuation functions do not assign the same positive value to more than $t$ items; and (3) the valuation functions respect a hypergraph, a setting introduced by Christodoulou et al. [EC'23], where agents are vertices and items are hyperedges. While the sizes of the hyperedges and neighborhoods can be arbitrary, the influence of any agent $a$, defined as the number of its neighbors who value at least one item positively that $a$ also values positively, is bounded.

Figures (4)

• Figure 1: An illustration of the different types of swaps we consider.
• Figure 2: An illustration of the execution of \ref{['alg:all-pos-val']}. At the $i$-th step of the picking phase, the top item $g_i$ from the donor part $M[\texttt{mid}+i]$ in the second half (stage $\ge 2$) is moved to the corresponding receiver part $M[i]$ in the first half (stage 1). The shaded green region indicates the remaining items that form the sub-instance for the recursive step.
• Figure 3: Illustration of the rebundling technique when no positive transfer is possible.
• Figure 4: Structure of the partition created in \ref{['alg:no-large-ties']}. Each part $M[i]$ is composed of a core $M_1[i]$ (Phase 1) and a Round-Robin part $R_i$ (Phase 2). The core ensures strict preference over subsequent parts.

Theorems & Definitions (62)

• theorem 1.1
• theorem 1.2
• theorem 1.3
• theorem 1.4
• corollary 1
• theorem 1.5
• theorem 1.6
• theorem 1.7
• proposition 1: DBLP:conf/sigecom/ConitzerF017DBLP:journals/teco/CaragiannisKMPS19
• remark 2.1: Power of rebundling