Fair Division Under Boolean Valuations: Beyond Normalization

Nisarg Shah; Paritosh Verma

Home/Archive/arXiv

arXiv · 2608.29497

Fair Division Under Boolean Valuations: Beyond Normalization

Nisarg Shah·Paritosh Verma

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study fair division of indivisible items when agents have arbitrary two-level preferences: the value of each agent for any set of items is Boolean, which need not be monotone or additive. Notably, we do not impose the standard assumption of normalization, i.e., different agents may value the empty set at different Boolean levels. Since the preferences are nonmonotone, envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) each admit several variants, depending on which items are tested for removal and whether they are removed from the envious agent's bundle or the envied agent's bundle. This paper investigates the existence of these variants of EF1 and EFX, on their own and together with economic efficiency, incentive compatibility, feasibility constraints, and lottery-based randomization. Our results highlight that the existence landscape depends crucially on the number of normalized agents, who value the empty bundle at the lower Boolean level. The authors used significant assistance from GPT-5.6-Sol for deriving theoretical results, verified any AI-generated proofs for correctness, and expanded on the exposition and simplified arguments, with the aid of GPT-5.6-Sol and Claude Opus 5.

Explore related subjects

cs.GT

Keep this discovery

BibTeX RIS

Nisarg Shah, Paritosh Verma. 2026-08-30. Fair Division Under Boolean Valuations: Beyond Normalization. https://arxiv.org/abs/2608.29497

Cite the original work for its findings. Save a collection to share your selection of sources.

Discover connections

cs.GT subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related discoveries

arXiv2026

Peer Oversight in Collective Decision Making

This article introduces peer $k$-oversight, a property of sequential collective decision mechanisms requiring at least $k$ agents to be responsible for every harmful outcome. It is shown that whenever $k$-oversight can be achieved by redistributing control over the decisions in a mechanism, it can be achieved using just $k$ agents. A polynomial-time algorithm is also presented that determines whether such a redistribution exists and, when it does, constructs one. These results establish peer oversight as a tractable design principle for multiagent decision-making systems.

cs.GT↗

arXiv2026

Fully Distributed GNE Algorithms for Multi-Robot Placement without Consensus on Multipliers

Recent machine learning research has increasingly focused on equilibrium analysis in non-cooperative games rather than solely on optimal solutions. Many such problems involve shared constraints and can be formulated as Generalized Nash Equilibrium Problems (GNEPs). For strongly monotone games, existing methods compute consensus-based variational GNEs (v-GNEs) by exchanging Lagrange multipliers. We propose a fully distributed continuous-time algorithm for shared linear equality constraints that converges without multiplier exchange and reaches any GNE, reducing communication overhead and improving privacy. Discrete-time schemes are also provided, and the method is validated on a multi-robot placement task.

cs.LG↗

arXiv2026

From the Social Choice Problem to a Collusion-Proof Tendering Mechanism for Dynamic Stochastic Projects

The VCG family and the AGV mechanism are two classical approaches to efficient implementation in the static social choice problem. In 2024, Csóka et al. showed that AGV has critical weaknesses. In contrast, the transferable-utility Guaranteed Utility Mechanism (TU-GUM) retains all the standard desirable properties of AGV while adding further ones, including collusion-proofness, because it implements efficiency in Guaranteed Utility Equilibrium. TU-GUM also applies to a more general dynamic setting with multiple extensions. Moreover, TU-GUM is a special case of an even more general and robust mechanism that combines contingent first-price tendering with the coordinated execution of dynamic stochastic multi-agent projects through a surprisingly simple rule. This paper summarizes and connects existing results from a different perspective, with some minor new observations.

econ.TH↗