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Maximin Share Guarantees for Few Agents with Subadditive Valuations

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper proves that for at most four agents with subadditive valuations, every instance admits an allocation giving each agent at least half her maximin share, and this factor is tight.

desk verdict Tight 1/2-MMS for four subadditive agents is a real result with a genuine but likely repairable gap in the four-agent case analysis. read the letter →

arxiv 2502.05141 v1 pith:DXE6ANLL submitted 2025-02-07 cs.GT

George Christodoulou , Vasilis Christoforidis , Symeon Mastrakoulis , Alkmini Sgouritsa This is my paper · ORCID
classification cs.GT MSC 91B32
keywords fairdivisionmaximinsharesubadditivevaluationssubmodularindivisiblegoodsapproximationalgorithmsalpha-MMS(d)
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that fair division of indivisible goods can be guaranteed at the half-maximin-share level for all subadditive valuation functions when there are at most four agents: every such instance has an allocation in which each agent receives value at least half of the share she could guarantee herself by splitting the goods into four parts. The factor 1/2 is tight, matching the known impossibility bound. The proof goes through a new parametrized notion, alpha-MMS(d), which relaxes the number of bundles each agent partitions into and the required threshold, and which the authors use as an inductive tool and analyze completely for three agents. The paper also shows that for any number of agents with only two distinct subadditive valuation functions a half-MMS allocation always exists, and it improves the existing impossibility for three submodular agents from 3/4 to 2/3.

What carries the argument

The technical engine is the Maximum Desired Half of an agent with respect to a cut. Given a partition P = (S_1, ..., S_r) of all goods and a subset C, the agent marks each S_i ∩ C as good if it is worth at least half of v(S_i), and likewise for S_i \ C; the Maximum Desired Half is whichever side, C or its complement, contains more good intersections. Subadditivity guarantees it contains at least half of the bundles, so it gives the proof a set worth at least 1/2 to one agent while leaving other agents' maximin bundles intact. The paper packages this in the $\alpha$-MMS(d) framework, where each agent i partitions goods into d_i bundles and is promised alpha_i times her minimum, a model that subsumes standard MMS, 1-out-of-d MMS, and ($\alpha$, $\beta$)-MMS and is used for the inductive and few-agent arguments.

What would settle it

To test the proof, run the exact four-agent protocol of Lemma 3 on small subadditive instances and check whether every allowed choice of the two sides of the three cuts forces two of the final bundles to overlap; an instance where the without-loss-of-generality reductions conflict would invalidate the proof. For the theorem itself, a counterexample would be a four-agent subadditive instance in which no allocation gives every agent half her maximin share.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1: for subadditive valuations and n <= 4, a 1/2-MMS allocation always exists. The authors establish it by proving the stronger Lemma 3, which gives a 1/2-MMS(3,3,4,4) allocation: with agents asked to partition into 3, 3, 4, and 4 bundles, the protocol builds up to four candidate partial allocations, using maximum-desired-half sets from adaptively chosen cuts, and argues that if the first candidate cannot be completed via a two-agent cut-and-choose lemma, the failure itself produces the structured partial allocations needed for the next candidate; after at most three failures the final allocation is forced. Tightness comes from the 1/2 upper bound of [16]. Additional results include a two-type valuation theorem for arbitrary n, complete characterizations of 1/2-MMS(d) and (1,1/2,1/2)-MMS(d) for three agents, and Theorem 5, a submodular 3-agent instance in which no (2/3 + epsilon)-MMS allocation exists for any epsilon > 0.

Load-bearing premise

The four-agent result depends on the assumption that every 'without loss of generality' choice of sides of the cuts can be made simultaneously while keeping the four candidate bundles pairwise disjoint; if these symmetry choices interact, the proof's case analysis collapses.

Editorial extensions

If this is right

  • For every subadditive instance with at most four agents, a 1/2-MMS allocation exists; combined with the matching upper bound of [16], this makes 1/2 the exact approximation factor for four-agent subadditive and XOS instances.
  • The four-agent guarantee automatically improves the known few-agent bounds for every class below subadditive, including submodular, XOS, gross-substitutes, and OXS valuations.
  • For any number of agents, if every valuation is one of two fixed subadditive functions, a 1/2-MMS allocation exists.
  • For three submodular agents, no (2/3 + epsilon)-MMS allocation can be guaranteed for any epsilon > 0, improving the previous upper bound of 3/4.
  • The complete three-agent characterizations of 1/2-MMS(d) and (1,1/2,1/2)-MMS(d) delimit exactly when such thresholded allocations exist under any choice of partition sizes d.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the proof's without-loss-of-generality reductions in Lemma 3 compose as written, the same maximum-desired-half machinery is a plausible route toward a constant-factor MMS guarantee for more than four agents; the paper does not claim this, and the open question remains.
  • The fact that four-agent subadditive instances already hit the 1/2 barrier suggests that any asymptotic improvement for general n would have to start with n at least 5 and might require a genuinely different idea than cut-and-choose over maximin bundles.
  • A direct computational check of Lemma 3, enumerating small subadditive instances to see whether the candidate allocations can always be made disjoint, would certify or break the fragile symmetry step in the proof; this is not something the paper does.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies approximate maximin share (MMS) allocations for subadditive valuations. Its main result, Theorem 1, claims that a 1/2-MMS allocation exists for at most four agents with subadditive valuations, and notes that this factor is tight by the upper bound of [16]. The paper also introduces a generalized notion, alpha-MMS(d), develops a technical framework around it, and uses this framework to prove supporting lemmas for two, three, and four agents. Further results include Theorem 2, a 1/2-MMS guarantee when all agents have one of two subadditive valuation functions, two characterization theorems for three agents (Theorems 3 and 4), and Theorem 5, an improved impossibility bound of 2/3 for three agents with submodular valuations.

Significance. If the main theorem is correct, it settles the four-agent subadditive MMS approximation at 1/2 and simultaneously improves the state of the art for XOS and other subclasses. The alpha-MMS(d) model is a useful unifying generalization of several existing MMS variants, and the two characterization theorems for three agents are a strong conceptual contribution. The proofs are constructive, the tightness claims rely on external upper bounds rather than fitted parameters, and there is no circularity or data-dependent selection in the argument. However, as detailed below, the four-agent proof contains an unverified w.l.o.g. reduction that is load-bearing for Theorem 1, and the subadditivity claim in Lemma 8 is not actually proved; these issues are repairable but currently block acceptance.

major comments (2)
  1. [Section 3.3, Lemma 3] The second-cut w.l.o.g. reduction is not demonstrated. After offering the cut C = S*_2 ∪ R4 to T, the proof assumes without loss of generality that X*_T(C) = X_T(M \ C) and dismisses the case X*_T(C) = X_T(C) in a footnote as 'simpler and can be handled the same way'. This is load-bearing because invariants (2), (3), (4), and (5) all rely on T*_j ⊆ M \ C, giving T*_j ∩ R4 = ∅ and T*_j ∩ S*_2 = ∅. In the omitted branch T*_j ⊆ C, both inclusions fail. Since different agents' MMS partitions can overlap arbitrarily as physical item sets, relabeling bundles such as swapping S*_1 with S*_2 and R3 with R4 requires an explicit verification that all later disjointness claims still hold; the manuscript does not provide that verification.
  2. [Section 4.1.3, Lemma 8] The assertion that 'Subadditivity is trivially guaranteed' for the constructed valuations is not justified. The valuation is a maximum over three components, and within each component values are assigned by cardinality thresholds and by the B* structure of 4- and 5-element sets. Claim 3 verifies only a monotonicity-type property about complements of B* sets, which is not a proof that v(A) + v(B) ≥ v(A ∪ B) for arbitrary A and B. This lemma is the impossibility construction used for d = (3,3,3) in Theorem 4, so the subadditivity of the valuation functions is essential and needs a complete proof.
minor comments (5)
  1. [Corollary 4 and Section 3.3] Corollary 4 says 'by Observation 1' but the reduction from Lemma 3 to the 1/2-MMS statement uses Observation 2; the same typo appears in the closing paragraph of Section 3.3.
  2. [Theorem 4] The final sentence of Theorem 4 says 'there exists an instance with no 1/2-MMS(d) allocation', but the theorem is about (1, 1/2, 1/2)-MMS(d); the statement should be corrected.
  3. [Lemma 8] The definition of v_R(B) for |B| = 5 uses the expression 1 - v_R(R_i \ B), but the values of singleton sets are not explicitly specified in equation (6); this ambiguity should be removed.
  4. [Proof of Claim 4, Theorem 5] The submodularity check uses the label 'Case 2' twice, and the line 'v(S) = v(S ∪ {g} = 2/3' has a missing parenthesis; renumber the cases and fix the typo.
  5. [Figures in Section 3.3] The figure captions say that 'we could construct the same allocation by renaming the bundles', but since bundles from different agents' partitions may overlap in items, 'renaming' is not a free operation unless the w.l.o.g. argument is made explicit; this presentation issue is closely connected to the first major comment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 1/2-MMS existence proofs are self-contained constructions, and the tightness references are external upper bounds.

full rationale

The paper's main theorem (Theorem 1, via Lemma 3) is a self-contained constructive proof: the candidate allocations are built from the agents' own MMS bundles and the maximum-desired-half definition, and every disjointness invariant (1)-(5) is derived from the cuts rather than assumed from a fitted parameter or a prior result. The alpha-MMS(d) framework is a definition, and Observation 2, which is used to pass from the stronger statements to ordinary 1/2-MMS, is proved directly. The tightness claims rely on the external upper bound of [16], not on any result proved in this paper. The only self-citation, [12], appears in the context of two-agent submodular tightness but is not load-bearing for any theorem here; Theorem 5 proves its own impossibility instance and checks submodularity in Claim 4. The asserted 'w.l.o.g.' cases inside Lemma 3 and the 'trivially guaranteed' subadditivity statement in Lemma 8 are potential proof gaps or exposition shortcuts, but they are not circular: they are omitted case checks, not reductions of a claimed output to an input by definition. No numerical fitting, no data-dependent filtering, and no self-referential benchmark are present. Accordingly, the derivation chain is not circular, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or postulated entities; it is a proof-driven theory paper. The main unproved burden is the asserted subadditivity of the Lemma 8 construction and the w.l.o.g. reductions in the 4-agent case analysis.

assumptions (5)
  • domain assumption Valuations are monotone, normalized, and scaled so mu^{d_i}_i(M)=1 for every agent.
    Standard normalization in MMS papers; scaling preserves approximation ratios and is used to set all MMS thresholds to 1.
  • domain assumption An agent's MMS-optimal partition into d_i bundles exists; the bundles in it each have value at least 1.
    Follows from finite item set and the definition of mu^{d_i}; used throughout as the benchmark partition.
  • standard math Subadditivity implies v(S cap C) + v(S \ C) >= v(S), used in Observation 1 and all cut-and-choose arguments.
    This is the defining inequality of subadditivity applied to the disjoint sets S cap C and S \ C.
  • standard math Restrictions of subadditive valuations to subsets of items are subadditive, used in Corollary 2 and the induction in Theorem 2.
    The restricted function inherits the defining inequality; the paper does not spell this out.
  • ad hoc to paper The step-function valuations defined in Lemma 8 are subadditive.
    The proof states 'Subadditivity is trivially guaranteed' and only verifies monotonicity (Claim 3), leaving the subadditivity check to the reader. This axiom is load-bearing for the (1,1/2,1/2)-MMS impossibility result.

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Pith. "Pith review of Maximin Share Guarantees for Few Agents with Subadditive Valuations." pith.science (2026). https://pith.science/paper/DXE6ANLL

@misc{pith2026250205141,
  author       = {Pith},
  title        = {Pith review of: Maximin Share Guarantees for Few Agents with Subadditive Valuations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXE6ANLL}},
  note         = {Machine review of arXiv:2502.05141}
}
abstract

We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an improved lower bound of $1/2$ (which is tight) for the case of subadditive valuations when the number of agents is at most four. We also provide a tight lower bound for the case of multiple agents, when they are equipped with one of two possible types of valuations. Moreover, we propose a new model that extends previously studied models in the area of fair division, which will hopefully give rise to further research. We demonstrate the usefulness of this model by employing it as a technical tool to derive our main result, and we provide a thorough analysis for this model for the case of three agents. Finally, we provide an improved impossibility result for the case of three submodular agents.

Figures

Figures reproduced from arXiv: 2502.05141 by the authors.

Figure 1
Figure 1. Let P = (S1, S2, S3) be a partition for agent S. The maximum desired half over set C is depicted by thick circle. For each bundle Si, plus bundles (+) attain value at least vS(Si)/2, while minus (-) attain value at most vS(Si)/2. Hence XS(C, P) = {S3 ∩ C}, XS(M \ C, P) = {S1 \ C, S2 \ C} and X ∗ S (C, P) = XS(M \ C, P). the superscript and simply denote it by µi(M). Also, when M and d are clear from the context, we … view at source ↗
Figure 2
Figure 2. We illustrate the partition for both d = (1, 2) and d = (2, 2). The set of items S1 can be divided into T1 and T2. If agent S values bundle T1 ∩ S1 (represented with a thick line in (a)) more than vS (S1) 2 then allocation (b) has the desired properties (the blue bundle for agent S and the red bundle for agent T ). If this is not the case, then due to the subadditivity, the same holds for agent S and bundle T2 ∩ S1;… view at source ↗
Figure 3
Figure 3. We use blue, red, and green to denote the bundles fro [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The first candidate allocation A = (S ∗ 2 , T ∗ 1 ) for four agents and d = (3, 3, 4, 4). We use blue to denote S ∗ 2 ⊆ S2 and red to denote T ∗ 1 ⊆ T1. We use a thick line to illustrate the cut C = {S ∗ 2 ∪ R4}. The allocation is valid and none of the bundles intersec…
Figure 5
Figure 5. Figure 5: The second candidate allocation A ′ = (T ∗ 2 , Q∗ 1). We use red for bundle T ∗ 2 ⊆ T2 and green for Q ∗ 1 ⊆ Q1. The cut C = {T ∗ 1 ∪ S ∗ 2 } is shown with a thick line. We try to apply Corollary 1 for agents S and R and the set of items M \ (T ∗ 2 ∪ Q ∗ 1). We could c…
Figure 6
Figure 6. Figure 6: The third candidate allocation A ′′ = (S ′ 1, Q∗ 2). We use blue for bundle S ′ 1 ⊆ S1 and green color for Q ∗ 2 ⊆ Q2. The cut C = {T ∗ 2 ∪ Q ∗ 1} is shown with a thick line. Note that the allocation is valid and none of the bundles intersects with R4. We try to apply …
Figure 7
Figure 7. Figure 7: The final allocation A ∗ = (S ′ 2, T ′ 1, Q∗ 3, R4). We use blue for bundle S ′ 2 ⊆ S2, red for bundle T ′ 1 ⊆ T1, and green for Q ∗ 3 ⊆ Q3. We use magenta to denote bundle R4. The cut is shown with a thick line C = {S ′ 1, Q∗ 3} The allocation is valid. at least 1/2 f…
Figure 8
Figure 8. Figure 8: In (a), with a thick line we show the cut [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: In (a), with a thick line we show the cut [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: In (a) and (b), with a thick line we show the cut [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: For cut C = {(Q1 \ S1) ∪ S3}, let T ∗ 2 ∈ XT (C) (as shown in (a)). Then, the allocations A = (S1, T ∗ 2 ) (b) and A ′ = (S4, T ′′ 1 ) (c) are disjoint on Q2. We use blue for agent S and red for agent T ’s bundles. In the following, we make the convention that i + 1 =…
Figure 12
Figure 12. Figure 12: The red bundle corresponds to T1 ∩ ((Q1 ∩ S2) ∪ S1) and the blue one to S3, as shown in (a). Agent Q values the remaining items at 1/2 − ǫ. In (b), the green bundle corresponds to Q1 ∩ ((T1 ∩ S1) ∪ S2) and the blue one to S1. Agent T values the remaining items at 1/2 …
Figure 13
Figure 13. Figure 13: If a bundle is large for some agent then either it co [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. From multi-allocations to allocations, with subadditive valuations

    cs.GT 2025-06 conditional novelty 7.0 of 10

    A d-multi-allocation with subadditive valuations can be converted to an allocation losing only a factor of about d, yielding an Omega(1/log log n)-MMS guarantee.

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