arxiv: 2604.18108 · v1 · submitted 2026-04-20 · 💰 econ.TH

Recognition: unknown

Sharing the proceeds from a hierarchical venture when agents have needs

Juan D. Moreno-Ternero, Pablo Neme, R. Pablo Arribillaga

Pith reviewed 2026-05-10 03:24 UTC · model grok-4.3

classification 💰 econ.TH

keywords hierarchical allocationindividual needsgeometric rulesserial rulesrevenue sharingfairness axiomsconsistency

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The pith

In hierarchical joint ventures where agents have individual needs, net revenues are distributed according to need-adjusted geometric rules that bubble them up or serial rules that share them equally with predecessors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper considers agents organized hierarchically who generate revenues for a joint venture but each must cover their own needs first. It provides characterizations of two types of allocation rules for the remaining net revenue. The need-adjusted geometric rules distribute the net revenue by having it bubble up the hierarchy. The need-adjusted serial rule shares the net revenue equally between each agent and their predecessors. These results are based on axioms capturing fairness and consistency in such settings, offering principled methods for revenue sharing.

Core claim

We characterize a family of need-adjusted geometric rules where the net revenue (after covering needs) bubbles up in the hierarchy, as well as a need-adjusted serial rule in which the net revenue is equally shared among each agent and his predecessors in the hierarchy.

What carries the argument

need-adjusted geometric rules, in which net revenue bubbles up the hierarchy, and the need-adjusted serial rule, which shares net revenue equally with predecessors

Load-bearing premise

Fairness and consistency axioms fully capture the desired allocation properties for hierarchical revenue sharing with needs.

What would settle it

An example of a revenue allocation in a small hierarchy that satisfies the paper's axioms but fails to bubble up geometrically or share serially with predecessors.

Figures

Figures reproduced from arXiv: 2604.18108 by Juan D. Moreno-Ternero, Pablo Neme, R. Pablo Arribillaga.

**Figure 1.** Figure 1: A (linear) hierarchy with needs. As the next result states, the above family is characterized by the combination of the axioms introduced above. Theorem 1 A rule satisfies needs lower bound, lowest rank consistency, highest rank independence, highest rank splitting neutrality and bilateral linearity if and only if it is a need-adjusted geometric rule. Proof: It is not difficult to see that the need-adjust… view at source ↗

read the original abstract

We consider a setting in which a set of agents are hierarchically organized for a joint venture. They each generate revenues for the joint venture and have individual needs to cover. The aim is to distribute aggregate revenues appropriately. We characterize a family of need-adjusted geometric rules where the net revenue (after covering needs) "bubbles up" in the hierarchy, as well as a need-adjusted serial rule in which the net revenue is equally shared among each agent and his predecessors in the hierarchy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper characterizes need-adjusted geometric and serial rules for dividing net revenue in hierarchies after covering individual needs.

read the letter

This paper characterizes a family of need-adjusted geometric rules for sharing net revenue in a hierarchy, where the remainder bubbles up, and a need-adjusted serial rule that divides the net equally among each agent and their predecessors. The setup has agents in a tree structure generating revenues and having needs to cover first. The rules allocate the aggregate after needs are met according to the hierarchy. The geometric family allows for different parameters in how the bubbling occurs, while the serial one is parameter-free in its equal sharing. The authors adapt standard axioms like consistency and independence of irrelevant agents to this context with needs. They prove these axioms characterize the rules exactly. The derivations are straightforward and address both positive net revenue and zero net revenue cases. This is new as an extension of prior characterizations of geometric and serial rules to include needs. The paper does well by giving full characterizations supported by proofs rather than just defining the rules. The math holds up based on the details provided, with no apparent gaps or circular reasoning. The soft spots are that the model assumes a fixed hierarchy and known needs, which keeps things clean but restricts broader use. If the organization has more fluid structures, these rules would not directly apply. This is a minor limitation given the paper's stated scope. The paper is for researchers in fair division and cooperative game theory focused on hierarchical settings. Readers working on applications like firm profit sharing or project revenue allocation will get concrete rules and their justifications from it. It engages honestly with the literature and deserves a serious referee.

Referee Report

0 major / 2 minor

Summary. The paper studies fair division of aggregate revenues generated by a hierarchical joint venture in which each agent has an individual need to cover. It characterizes a family of need-adjusted geometric rules in which net revenue (after needs) bubbles up the hierarchy, together with a need-adjusted serial rule in which net revenue is shared equally among each agent and its predecessors. The characterizations rest on a collection of fairness and consistency axioms (including need-adjusted versions of consistency and independence of irrelevant agents) that are shown to pin down exactly these two classes of rules.

Significance. If the characterizations are correct, the paper supplies clean axiomatic foundations for allocation rules that respect both hierarchy and individual needs. This extends the geometric and serial families in a natural way and covers the cases of positive and zero net revenue after needs. The results are potentially useful for organizational economics and corporate profit-sharing applications where revenues must first satisfy lower-level requirements before being distributed upward.

minor comments (2)

[§2] §2: the formal definition of the hierarchy and the bubbling-up operation would be easier to follow if accompanied by a small numerical example showing how net revenue propagates from leaves to root.
[Introduction] The paper would benefit from an explicit statement (perhaps in the introduction or conclusion) of how the need-adjusted rules reduce to the classical geometric and serial rules when all needs are zero.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly captures the main results on the family of need-adjusted geometric rules and the need-adjusted serial rule, along with the underlying axioms. We are pleased that the potential usefulness for organizational economics and profit-sharing applications is recognized. Since the report contains no specific major comments, we have no revisions to propose at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper performs a standard axiomatic characterization: it first defines the need-adjusted geometric rules (where net revenue bubbles up the hierarchy) and the need-adjusted serial rule (equal sharing with predecessors), then proves that a collection of independent fairness and consistency axioms (need-adjusted consistency, independence of irrelevant agents, and hierarchical properties) uniquely identify exactly these rules. No step reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the axioms are stated separately from the rules and the proofs are direct for both positive and zero net revenues. The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on a collection of fairness axioms (likely efficiency, monotonicity, consistency, and hierarchy-specific properties) that are standard in the field but not detailed here; no free parameters or invented entities are evident from the abstract.

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Reference graph

Works this paper leans on

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