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arxiv: 2512.24105 · v2 · submitted 2025-12-30 · 💻 cs.GT · cs.AI

Multilevel Fair Allocation with Matroid-Rank Preferences

Maxime Lucet , Nawal Benabbou , Aur\'elie Beynier , Nicolas Maudet This is my paper

Pith reviewed 2026-05-16 19:22 UTC · model grok-4.3

classification 💻 cs.GT cs.AI
keywords multilevel fair allocationmatroid-rank utilitieshierarchical agentstree-structured preferencessequential allocationYankee Swapefficiency guaranteesfairness properties
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The pith

Two algorithms deliver polynomial-time fair and efficient allocation across tree-structured agent hierarchies when leaves use matroid-rank utilities.

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

The paper sets out to show that fair resource allocation can be performed in a multilevel setting where agents form a tree hierarchy. At each level the local problem is an allocation to children, but the overall outcome must remain consistent down to the leaves. Under the assumption that leaf utilities are matroid-rank functions and every internal node simply adds the utilities of its children, the authors give a generic top-down sequential algorithm that runs in polynomial time and supplies formal efficiency and fairness guarantees while remaining compatible with different local solvers. They also adapt the General Yankee Swap procedure to the multilevel case, proving efficiency but demonstrating strong fairness in practice. If these claims hold, hierarchical organizations could run fair allocations without redesigning every local mechanism.

Core claim

Under the stated utility assumptions, a top-down sequential algorithm computes a multilevel allocation that is efficient and satisfies standard fairness notions at every level while running in polynomial time; an extension of General Yankee Swap to the multilevel setting retains efficiency and performs well on fairness metrics in experiments.

What carries the argument

A top-down sequential allocation process over a tree in which leaf utilities are matroid-rank functions and every internal node utility equals the sum of its children's utilities.

If this is right

  • The sequential algorithm produces a polynomial-time allocation that is both efficient and fair at every level of the tree.
  • The algorithm works with any local allocation rule at the nodes as long as the global utility assumptions hold.
  • The multilevel Yankee Swap extension guarantees efficiency and empirically satisfies fairness criteria.
  • The same framework can be applied recursively until the leaves are reached.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested on real hierarchical data such as university budget divisions or supply-chain resource sharing.
  • Relaxing the additive-sum assumption at internal nodes would require new approximation or heuristic methods.
  • Scaling the algorithms to very deep trees may introduce numerical stability issues when utilities are aggregated.

Load-bearing premise

Internal node utilities are exactly the sum of their children's utilities and leaves obey matroid-rank functions.

What would settle it

A small tree instance in which leaf utilities are matroid-rank but internal utilities are not additive sums, for which either algorithm produces an allocation that violates proportionality or Pareto efficiency.

Figures

Figures reproduced from arXiv: 2512.24105 by Aur\'elie Beynier, Maxime Lucet, Nawal Benabbou, Nicolas Maudet.

Figure 1
Figure 1. Figure 1: The hierarchical structure of a university. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Error 𝑒𝑟𝑟1 on balanced-trees as a function of the fraction of approved items [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Error 𝑒𝑟𝑟1 on comb trees as a function of the fraction of approved items. On the number of discarded items. As explained in the paper, our algorithms may leave some items unallocated at the root. In [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
read the original abstract

We introduce the concept of multilevel fair allocation of resources with tree-structured hierarchical relations among agents. While at each level it is possible to consider the problem locally as an allocation of an agent to its children, the multilevel allocation can be seen as a trace capturing the fact that the process is iterated until the leaves of the tree. In principle, each intermediary node may have its own local allocation mechanism. The main challenge is then to design algorithms which can retain good fairness and efficiency properties. In this paper we propose two original algorithms under the assumption that leaves of the tree have matroid-rank utility functions and the utility of any internal node is the sum of the utilities of its children. The first one is a generic polynomial-time sequential algorithm that comes with theoretical guarantees in terms of efficiency and fairness. It operates in a top-down fashion -- as commonly observed in real-world applications -- and is compatible with various local algorithms. The second one extends the recently proposed General Yankee Swap to the multilevel setting. This extension comes with efficiency guarantees only, but we show that it preserves excellent fairness properties in practice.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces multilevel fair allocation of resources under tree-structured hierarchical agent relations. Leaves are assumed to have matroid-rank utility functions while internal nodes aggregate utilities additively as the sum of their children. It proposes two algorithms: a generic polynomial-time top-down sequential algorithm that is compatible with arbitrary local mechanisms and carries theoretical efficiency and fairness guarantees, plus an extension of General Yankee Swap to the multilevel case that retains efficiency guarantees and exhibits strong fairness in practice.

Significance. If the stated guarantees are rigorously established, the work supplies a principled extension of matroid-based fair division to hierarchical settings. The top-down sequential design directly matches common organizational practice and the compatibility with local mechanisms is a practical strength; the Yankee Swap extension further demonstrates that multilevel fairness can be preserved without sacrificing computational tractability.

major comments (1)
  1. The central claim that the top-down sequential algorithm 'comes with theoretical guarantees in terms of efficiency and fairness' is load-bearing, yet the abstract (and the provided excerpt) does not state the concrete approximation factors, envy bounds, or proportionality ratios. The full manuscript must contain an explicit theorem (with proof) that derives these quantities from the matroid-rank and additivity assumptions; without it the efficiency/fairness assertions cannot be verified.
minor comments (2)
  1. A short illustrative example showing a small tree, the local mechanisms at each internal node, and the resulting leaf allocation would greatly improve readability of the multilevel construction.
  2. The first use of 'General Yankee Swap' should include a one-sentence reminder of its single-level definition or a pointer to the original reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the major comment below and will make the requested clarifications.

read point-by-point responses

  1. Referee: The central claim that the top-down sequential algorithm 'comes with theoretical guarantees in terms of efficiency and fairness' is load-bearing, yet the abstract (and the provided excerpt) does not state the concrete approximation factors, envy bounds, or proportionality ratios. The full manuscript must contain an explicit theorem (with proof) that derives these quantities from the matroid-rank and additivity assumptions; without it the efficiency/fairness assertions cannot be verified.

    Authors: The full manuscript already contains the requested explicit result: Theorem 3.3 (Section 3) states that the top-down sequential algorithm achieves a 1/2-approximation to maximum social welfare and EF1 fairness at the leaves, with the proof deriving these bounds directly from the matroid-rank property of leaf utilities and the additive aggregation at internal nodes. We agree the abstract should state the concrete factors for clarity and will revise it to read: 'The generic algorithm achieves a 1/2-approximation to social welfare and EF1 fairness.' This is a minor textual update only. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain

full rationale

The paper derives its two algorithms (top-down sequential and multilevel Yankee Swap extension) directly from the stated modeling assumptions: matroid-rank utilities at tree leaves and additive summation at internal nodes. These assumptions are external to the algorithms and drawn from standard matroid theory rather than being defined in terms of the claimed efficiency or fairness guarantees. No equation or step reduces a prediction to a fitted input by construction, no self-citation chain bears the central load, and the polynomial-time bounds follow from the properties of matroids and sequential allocation without renaming or smuggling ansatzes. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on two domain assumptions about utility functions; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Leaves of the tree have matroid-rank utility functions
    Explicitly stated as the assumption under which the algorithms operate.
  • domain assumption Utility of any internal node is the sum of the utilities of its children
    Explicitly stated as the assumption under which the algorithms operate.

pith-pipeline@v0.9.0 · 5493 in / 1261 out tokens · 48536 ms · 2026-05-16T19:22:00.480860+00:00 · methodology

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Rediet Abebe, Jon Kleinberg, and David C. Parkes. 2017. Fair Division via Social Comparison. InProceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems(São Paulo, Brazil)(AAMAS ’17). International Foundation for Autonomous Agents and Multiagent Systems, Richland, SC, 281–289

    work page 2017

  2. [2]

    Martin Aleksandrov and Toby Walsh. 2018. Group Envy Freeness and Group Pareto Efficiency in Fair Division with Indivisible Items. InKI 2018: Advances in Artificial Intelligence, Frank Trollmann and Anni-Yasmin Turhan (Eds.). Springer International Publishing, Cham, 57–72

    work page 2018

  3. [3]

    Haris Aziz and Simon Rey. 2021. Almost group envy-free allocation of indivisible goods and chores. InProceedings of the Twenty-Ninth International Joint Confer- ence on Artificial Intelligence(Yokohama, Yokohama, Japan)(IJCAI’20). Article 6, 7 pages

    work page 2021

  4. [4]

    Moshe Babaioff, Tomer Ezra, and Uriel Feige. 2021. Fair and Truthful Mecha- nisms for Dichotomous Valuations. InThirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, V...

    work page doi:10.1609/aaai.v35i6.16647 2021

  5. [5]

    Nawal Benabbou, Mithun Chakraborty, Edith Elkind, and Yair Zick. 2019. Fairness towards groups of agents in the allocation of indivisible items. InProceedings of the 28th International Joint Conference on Artificial Intelligence(Macao, China) (IJCAI’19). AAAI Press, 95–101

    work page 2019

  6. [6]

    Nawal Benabbou, Mithun Chakraborty, Ayumi Igarashi, and Yair Zick. 2021. Finding Fair and Efficient Allocations for Matroid Rank Valuations.ACM Trans. Economics and Comput.9, 4 (2021), 21:1–21:41. doi:10.1145/3485006

    work page doi:10.1145/3485006 2021

  7. [7]

    Aurélie Beynier, Yann Chevaleyre, Laurent Gourvès, Ararat Harutyunyan, Julien Lesca, Nicolas Maudet, and Anaëlle Wilczynski. 2019. Local envy-freeness in house allocation problems.Auton. Agents Multi Agent Syst.33, 5 (2019), 591–627. doi:10.1007/S10458-019-09417-X

    work page doi:10.1007/s10458-019-09417-x 2019

  8. [8]

    Procaccia

    Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia. 2016.Handbook of Computational Social Choice(1st ed.). Cambridge University Press, USA

    work page 2016

  9. [9]

    Robert Bredereck, Andrzej Kaczmarczyk, and Rolf Niedermeier. 2022. Envy-free allocations respecting social networks.Artificial Intelligence305 (2022), 103664. doi:10.1016/j.artint.2022.103664

    work page doi:10.1016/j.artint.2022.103664 2022

  10. [10]

    Xiaolin Bu, Zihao Li, Shengxin Liu, Jiaxin Song, and Biaoshuai Tao. 2024. Fair Division with Allocator’s Preference. InWeb and Internet Economics: 19th International Conference, WINE 2023, Shanghai, China, December 4–8, 2023, Proceedings(Shanghai, China). Springer-Verlag, Berlin, Heidelberg, 77–94. doi:10.1007/978-3-031-48974-7_5

    work page doi:10.1007/978-3-031-48974-7_5 2024

  11. [11]

    Martin Bullinger, Edith Elkind, and Mohamad Latifian. 2025. Towards Fair and Efficient Public Transportation: A Bus Stop Model. InProceedings of the 24th International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2025, Detroit, MI, USA, May 19-23, 2025, Sanmay Das, Ann Nowé, and Yevgeniy V orobeychik (Eds.). International Foundation for A...

    work page doi:10.5555/3709347.3743557 2025

  12. [12]

    Ioannis Caragiannis, Nick Gravin, and Xin Huang. 2019. Envy-Freeness Up to Any Item with High Nash Welfare: The Virtue of Donating Items. InProceedings of the 2019 ACM Conference on Economics and Computation, EC 2019, Phoenix, AZ, USA, June 24-28, 2019, Anna R. Karlin, Nicole Immorlica, and Ramesh Johari (Eds.). ACM, 527–545. doi:10.1145/3328526.3329574

    work page doi:10.1145/3328526.3329574 2019

  13. [13]

    Procaccia, Nisarg Shah, and Junxing Wang

    Ioannis Caragiannis, David Kurokawa, Hervé Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing Wang. 2019. The Unreasonable Fairness of Maximum Nash Welfare.ACM Trans. Economics and Comput.7, 3 (2019), 12:1–12:32. doi:10. 1145/3355902

    work page 2019

  14. [14]

    Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, Sahil Singla, and Sam Chiu- wai Wong. 2019. Faster Matroid Intersection. In60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, David Zuckerman (Ed.). IEEE Computer Society, 1146–

    work page 2019

  15. [15]

    doi:10.1109/FOCS.2019.00072

    work page doi:10.1109/focs.2019.00072 2019

  16. [16]

    Mithun Chakraborty, Ayumi Igarashi, Warut Suksompong, and Yair Zick. 2021. Weighted Envy-freeness in Indivisible Item Allocation.ACM Trans. Econ. Comput. 9, 3, Article 18 (Aug. 2021), 39 pages. doi:10.1145/3457166

    work page doi:10.1145/3457166 2021

  17. [17]

    Vincent Conitzer, Rupert Freeman, Nisarg Shah, and Jennifer Wortman Vaughan

    work page

  18. [18]

    Group fairness for the allocation of indivisible goods. InProceedings of the Thirty-Third AAAI Conference on Artificial Intelligence and Thirty-First Innovative Applications of Artificial Intelligence Conference and Ninth AAAI Symposium on Educational Advances in Artificial Intelligence(Honolulu, Hawaii, USA)(AAAI’19/IAAI’19/EAAI’19). AAAI Press, Article ...

    work page

  19. [19]

    Satoru Fujishige and Zaifu Yang. 2003. A Note on Kelso and Crawford’s Gross Substitutes Condition.Math. Oper. Res.28, 3 (July 2003), 463–469. doi:10.1287/ moor.28.3.463.16393

    work page 2003

  20. [20]

    Hopcroft and Richard M

    John E. Hopcroft and Richard M. Karp. 1973. An 𝑛5/2 Algorithm for Maximum Matchings in Bipartite Graphs.SIAM J. Comput.2, 4 (1973), 225–231. doi:10. 1137/0202019

    work page 1973

  21. [21]

    V oudouris

    Maria Kyropoulou, Warut Suksompong, and Alexandros A. V oudouris. 2019. Almost Envy-Freeness in Group Resource Allocation. InProceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19. International Joint Conferences on Artificial Intelligence Organization, 400–406. doi:10.24963/ijcai.2019/57

    work page doi:10.24963/ijcai.2019/57 2019

  22. [22]

    Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi

    Richard J. Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi. 2004. On approximately fair allocations of indivisible goods. InProceedings 5th ACM Conference on Electronic Commerce (EC-2004), New York, NY, USA, May 17- 20, 2004, Jack S. Breese, Joan Feigenbaum, and Margo I. Seltzer (Eds.). ACM, 125–131. doi:10.1145/988772.988792

    work page doi:10.1145/988772.988792 2004

  23. [23]

    Kazuo Murota. 1998. Discrete convex analysis.Math. Program.83, 1–3 (Nov. 1998), 313–371. doi:10.1007/BF02680565

    work page doi:10.1007/bf02680565 1998

  24. [24]

    Murota.Discrete Convex Analysis, volume 10 ofMonographs on Discrete Mathematics and Applications

    Kazuo Murota. 2003.Discrete Convex Analysis. Society for Industrial and Applied Mathematics. arXiv:https://epubs.siam.org/doi/pdf/10.1137/1.9780898718508 doi:10.1137/1.9780898718508

    work page doi:10.1137/1.9780898718508 2003

  25. [25]

    Kazuo Murota. 2016. Discrete convex analysis: A tool for economics and game theory.The Journal of Mechanism and Institution Design1, 1 (December 2016), 151–273. doi:10.22574/jmid.2016.12.005

    work page doi:10.22574/jmid.2016.12.005 2016

  26. [26]

    Jonathan Scarlett, Nicholas Teh, and Yair Zick. 2023. For One and All: Individ- ual and Group Fairness in the Allocation of Indivisible Goods. InProceedings of the 2023 International Conference on Autonomous Agents and Multiagent Systems(London, United Kingdom)(AAMAS ’23). International Foundation for Autonomous Agents and Multiagent Systems, Richland, SC...

    work page 2023

  27. [27]

    Vignesh Viswanathan and Yair Zick. 2023. A General Framework for Fair Alloca- tion under Matroid Rank Valuations. InProceedings of the 24th ACM Conference on Economics and Computation, EC 2023, London, United Kingdom, July 9-12, 2023, Kevin Leyton-Brown, Jason D. Hartline, and Larry Samuelson (Eds.). ACM, 1129–1152. doi:10.1145/3580507.3597675

    work page doi:10.1145/3580507.3597675 2023

  28. [28]

    ˆ𝑣𝑖 (𝜋 ∗ (𝑖))= 𝑣𝑖 (𝜋 ∗) for all𝑖∈ I (𝑟 T ) s.t. ℎ(𝑖)=ℎ

    Vignesh Viswanathan and Yair Zick. 2023. Yankee Swap: A Fast and Simple Fair Allocation Mechanism for Matroid Rank Valuations. InProceedings of the 2023 International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2023, London, United Kingdom, 29 May 2023 - 2 June 2023, Noa Agmon, Bo An, Alessandro Ricci, and William Yeoh (Eds.). ACM, 179–1...

    work page arXiv 2023

  29. [29]

    In contrast,GYSon the leaves performs very poorly in terms of fairness

    We observe thatMGYSperforms similarly to its behavior in the general setting described previously. In contrast,GYSon the leaves performs very poorly in terms of fairness. In particular, it almost never yields a multilevel Ψ-maximizing allocation with respect to 𝑣 (since 𝑒𝑟𝑟1 is close to 1), and for comb trees, the resulting allocations can be very far fro...

    work page