Simultaneous Ordinal Maximin Share and Envy-Based Guarantees

Hannaneh Akrami; Timo Reichert

arxiv: 2602.15566 · v2 · submitted 2026-02-17 · 💻 cs.GT

Simultaneous Ordinal Maximin Share and Envy-Based Guarantees

Hannaneh Akrami , Timo Reichert This is my paper

Pith reviewed 2026-05-15 21:56 UTC · model grok-4.3

classification 💻 cs.GT

keywords fair divisionindivisible goodsmaximin shareEFXEF1ordinal MMSadditive valuations

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The pith

Allocations exist that simultaneously satisfy 1-out-of-ceil(3n/2) MMS and EFX for ordered instances with additive valuations.

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

The paper studies the fair allocation of indivisible goods under additive valuations and asks whether ordinal approximations of the maximin share can be paired with envy-based fairness properties. It proves existence results for three specific combinations: 1-out-of-ceil(3n/2) MMS with EFX on ordered instances, the same MMS level with EF1 on top-n instances, and 1-out-of-4 ceil(n/3) MMS with EF1 on ordered instances. These results matter because they show that two traditionally separate fairness frameworks can be satisfied together once the input is restricted to structured instances. A reader cares because the bounds give explicit performance guarantees that could guide practical allocation procedures when valuations follow natural orderings.

Core claim

The central claim is that for additive valuations there exist allocations satisfying simultaneous 1-out-of-ceil(3n/2) MMS and EFX for ordered instances, simultaneous 1-out-of-ceil(3n/2) MMS and EF1 for top-n instances, and simultaneous 1-out-of-4 ceil(n/3) MMS and EF1 for ordered instances.

What carries the argument

The key machinery is the existence proof for joint ordinal MMS thresholds and envy-free conditions that exploits the ordering or top-n structure of the valuation matrix.

Load-bearing premise

The input must consist of ordered instances or top-n instances with additive valuations; without this structure the claimed simultaneous guarantees need not exist.

What would settle it

A concrete ordered instance with additive valuations in which every allocation violates either the 1-out-of-ceil(3n/2) MMS bound or the EFX condition would falsify the main existence claim.

read the original abstract

We study the fair allocation of indivisible goods among agents with additive valuations. The fair division literature has traditionally focused on two broad classes of fairness notions: envy-based notions and share-based notions. Within the share-based framework, most attention has been devoted to the maximin share (MMS) guarantee and its relaxations, while envy-based fairness has primarily centered on EFX and its relaxations. Recent work has shown the existence of allocations that simultaneously satisfy multiplicative approximate MMS and envy-based guarantees such as EF1 or EFX. Motivated by this line of research, we study for the first time the compatibility between ordinal approximations of MMS and envy-based fairness notions. In particular, we establish the existence of allocations satisfying the following combined guarantees: (i) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EFX for ordered instances; (ii) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EF1 for top-$n$ instances; and (iii) simultaneous $1$-out-of-$4\lceil n/3 \rceil$ MMS and EF1 for ordered instances.

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

This paper gives the first existence results for simultaneous ordinal MMS approximations and EFX/EF1 in ordered or top-n additive instances.

read the letter

The core contribution is the first set of existence theorems that pair ordinal MMS relaxations with envy-based guarantees. For ordered instances the paper shows allocations that are simultaneously 1-out-of-ceil(3n/2) MMS and EFX, plus a weaker 1-out-of-4*ceil(n/3) MMS and EF1 version; for top-n instances it gives the same 1-out-of-ceil(3n/2) MMS paired with EF1. These are scoped cleanly to additive valuations and the two restricted instance classes, which is why the bounds are achievable where general-case results remain open. The work sits directly on top of earlier multiplicative MMS + EF1/EFX papers and simply swaps in the ordinal version, so the technical step is modest but the combination itself had not been studied. The statements are free of circularity or invented parameters, and the restrictions are stated up front rather than hidden. The main limitation is that everything is pure existence; there are no algorithms, no tightness examples, and no discussion of how often the structural assumptions hold in practice. Proof details are not visible in the abstract, so any referee would need to check whether the ordering or top-n definitions introduce extra assumptions that affect the constants. This is a narrow but well-defined theoretical result aimed at people already working on MMS relaxations and envy notions in indivisible goods. A reader deep in that subfield will find the specific simultaneous bounds useful for organizing the landscape. I would send it to peer review because the novelty claim holds and the evidence is a set of parameterized existence statements rather than overclaimed generality.

Referee Report

0 major / 3 minor

Summary. The paper studies fair allocation of indivisible goods with additive valuations, focusing on the compatibility of ordinal MMS approximations with envy-based notions. It establishes existence of allocations satisfying three simultaneous guarantees: (i) 1-out-of-⌈3n/2⌉ MMS and EFX for ordered instances; (ii) 1-out-of-⌈3n/2⌉ MMS and EF1 for top-n instances; and (iii) 1-out-of-4⌈n/3⌉ MMS and EF1 for ordered instances.

Significance. If the existence results hold, the paper advances fair division by bridging ordinal MMS relaxations with EFX/EF1 guarantees under additive valuations, for the explicitly scoped classes of ordered and top-n instances. This complements prior work on multiplicative MMS approximations and provides interpretable fairness bounds without relying on free parameters or self-referential definitions.

minor comments (3)

§1: The introduction would benefit from a short paragraph contrasting the ordinal MMS bounds here with the multiplicative approximations in the cited recent work, to clarify the novelty of the ordinal focus.
The definition of 'ordered instances' and 'top-n instances' should be stated explicitly in the preliminaries (before the main theorems) rather than deferred to the proofs.
Figure 1 (if present) or any illustrative example in §3 could include a small numerical instance to demonstrate how the 1-out-of-⌈3n/2⌉ bound is achieved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our results on simultaneous ordinal MMS and envy-based guarantees, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; existence claims are self-contained

full rationale

The paper presents parameterized existence results for simultaneous 1-out-of-k ordinal MMS and EFX/EF1 guarantees, restricted to ordered instances and top-n instances under additive valuations. These are new combinatorial existence statements rather than reductions of a derived quantity to a fitted parameter or self-defined input. No equations in the abstract or described claims equate a prediction to its own construction, and the referenced prior work on multiplicative MMS is external rather than a load-bearing self-citation chain. The derivation introduces specific bounds (e.g., ⌈3n/2⌉ and 4⌈n/3⌉) without renaming known patterns or smuggling ansatzes via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions in fair division rather than new free parameters or invented entities.

axioms (2)

domain assumption Valuations are additive
Standard assumption stated in the abstract for the fair allocation setting.
domain assumption Instances are ordered or top-n
The simultaneous guarantees are proven only under these structural restrictions on the valuation profiles.

pith-pipeline@v0.9.0 · 5510 in / 1340 out tokens · 18290 ms · 2026-05-15T21:56:26.018172+00:00 · methodology

Simultaneous Ordinal Maximin Share and Envy-Based Guarantees

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)