REVIEW 2 minor 2 cited by
Temporal Fair Division of Indivisible Goods with Scheduling
T0 review · 0 major / 2 minor · reviewed 2026-05-16 · grok-4.3
Pith's one-line read A scheduling buffer of size at least n/2 achieves TEF1 for identical days in temporal fair division of indivisible goods.
desk verdict The paper shows a scheduling buffer of n/2 gets TEF1 for identical days while TEFX and TMMS stay impossible even with scheduling. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The scheduling buffer, which permits agents to receive goods in staggered time slots within each round to balance cumulative allocations.
What would settle it
An explicit instance with non-identical days in which no finite buffer size yields TEF1 for any algorithm would falsify the scheduling positive result.
Extended reading notes
Core claim
The central claim is that temporal fair division without scheduling cannot guarantee any constant-factor α-TEFX in unrestricted domains, yet achieves a 1/2-approximation for generalized binary valuations with identical days and two agents; separately, a scheduling buffer of size at least n/2 suffices to guarantee TEF1 for identical days, while TEFX and TMMS remain impossible across both scheduled and unscheduled models even under domain restrictions.
Load-bearing premise
The positive results for the 1/2-approximation and for TEF1 require identical days together with generalized binary valuations.
Editorial extensions
If this is right
- Constant-factor α-TEFX is impossible in general domains without scheduling.
- A 1/2-approximation to TEFX is achievable for generalized binary valuations, identical days, and two agents.
- TEF1 becomes attainable once a scheduling buffer of size at least n/2 is introduced for identical days.
- TEFX and TMMS remain impossible even with scheduling or domain restrictions.
- The boundary between possibility and impossibility is fully drawn for the studied fairness notions.
Reading between the lines
- Real-world repeated allocation problems such as cloud-resource assignment over days could adopt the buffer mechanism to reach cumulative fairness with modest extra flexibility.
- The n/2 buffer size may be tight; checking whether a smaller buffer suffices under additional symmetry would test the result directly.
- The impossibility results for TEFX suggest that online variants of the problem will also require approximation or further restrictions on arrival patterns.
- Extending the buffer analysis to non-binary valuations could reveal whether the same n/2 threshold continues to work.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies temporal fair division of indivisible goods over multiple rounds with cumulative fairness requirements. It analyzes Temporal Envy-Freeness Up to One Good (TEF1), Temporal Envy-Freeness Up to Any Good (TEFX) and its constant-factor approximations α-TEFX, and Temporal Maximin Share (TMMS). Without scheduling, it proves constant-factor α-TEFX is impossible in general domains but shows a 1/2-approximation is achievable under generalized binary valuations, identical days, and two agents. With scheduling, a buffer of size at least n/2 enables TEF1 for identical days, while TEFX and TMMS remain impossible in most cases even with scheduling or domain restrictions.
Significance. If the results hold, the work usefully maps the boundary between impossibility and possibility for temporal fairness notions, providing both general negative results and positive approximation guarantees under explicit restrictions such as identical days and binary valuations. These delineate practical trade-offs (e.g., scheduling buffers) and could guide algorithm design in repeated allocation settings.
minor comments (2)
- [Abstract] Abstract: the statement that 'a scheduling buffer of size at least n/2 enables TEF1 for identical days' should explicitly note whether this holds only for two agents or general n, to match the scope of the 1/2-approximation result.
- The paper should include a dedicated section or table summarizing all impossibility and possibility results across the different models (no scheduling vs. scheduling, general vs. restricted valuations) for quick reference.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our work and the recommendation for minor revision. The referee's summary accurately reflects our main contributions in mapping the boundaries between possibility and impossibility for temporal fairness notions (TEF1, TEFX, and TMMS) under various domain restrictions and with scheduling buffers.
Circularity Check
No significant circularity
full rationale
The paper consists of direct mathematical proofs establishing impossibility results in general domains and existence/approximation results under explicitly restricted settings (generalized binary valuations, identical days, two agents, scheduling buffer of size n/2). No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; all claims are derived from the stated model definitions and standard proof techniques without circular reduction.
Assumptions & free parameters
assumptions (1)
- domain assumption Generalized binary valuations and identical days in restricted settings
Cite this review
Pith. "Pith review of Temporal Fair Division of Indivisible Goods with Scheduling." pith.science (2026). https://pith.science/paper/2601.12835
@misc{pith2026260112835,
author = {Pith},
title = {Pith review of: Temporal Fair Division of Indivisible Goods with Scheduling},
year = {2026},
howpublished = {\url{https://pith.science/paper/2601.12835}},
note = {Machine review of arXiv:2601.12835}
}
abstract
We study temporal fair division, where agents receive goods over multiple rounds and cumulative fairness is required. We investigate Temporal Envy-Freeness Up to One Good (TEF1) and Up to Any Good (TEFX), its approximation $\alpha$-TEFX, and Temporal Maximin Share (TMMS). Motivated by known impossibilities in standard settings, we consider the model in various restricted settings and extend it by introducing scheduling. Our main contributions draw the boundary between possibility and impossibility. First, regarding temporal fair division without scheduling, we prove that while constant-factor $\alpha$-TEFX is impossible in general, a $1/2$-approximation is achievable for generalized binary valuations and identical days with two agents. Second, regarding temporal fair division with scheduling, we demonstrate that a scheduling buffer of size at least $n/2$ enables TEF1 for identical days. However, we establish that TEFX and TMMS remain largely impossible even with scheduling or restricted domains. These results highlight the inherent difficulty of strict temporal fairness and quantify the trade-offs required to achieve approximation guarantees.
Forward citations
Cited by 2 Pith papers
-
Online Fair Division Meets Reordering Buffers
With buffers of size linear in k and number of agents, algorithms achieve EF1 at every step and EF at most steps for personalized k-value mixed manna instances, extending to general additives with ratio dependence.
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The Coordination Gap: Multi-Agent Alternation Metrics for Temporal Fairness in Repeated Games
In a multi-agent resource race, independent Q-learning agents alternate less often than random policies, while conventional fairness/efficiency metrics hide the deficit.
Reference graph
Works this paper leans on
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= 0) g1,g 2,g 3 (v(A1
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= 3) Without loss of generality, assume agent1gets the good valued0and1, and agent2gets the good valued3. Lemma 6.Ont= 2, the good valued1must be allocated to agent2, and the good valued3must be allocated to agent1. Proof.Case 1:Assume agent1does not get any good, and agent2gets the good valued1and3. Now, v1(A1) = 0 + 1 = 1andv 1(A2) = 3 + 1 + 3 = 7. It c...
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= 1 + 3 = 4) g2 (v(A2
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= 3 + 1 = 4) Case 1 ∅(v(A 2
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= 3 + 4 = 7) Case 2 g2 (v(A2
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= 1 + 1 = 2) g3 (v(A2
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= 3 + 3 = 6) Case 3 g2,g 3 (v(A2
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= 1 + 1 + 3 = 5) ∅(v(A 2
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Proof.If an allocation is TEFX,∀j∈[N], v i(Ai)≥v i(Aj\{g}),∀g∈A j
= 3) Proposition 1.The addition of a good where it is zero-valued for every agent in a previously non-TEFX allocation does not make it TEFX. Proof.If an allocation is TEFX,∀j∈[N], v i(Ai)≥v i(Aj\{g}),∀g∈A j. Assume agentienvies agent j, andv i(Ai)< v i(Aj\{g}),∀g∈A j. If the g...
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= 4 + 4 = 8) Case 2 g2 (v(A3
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= 4 + 1 = 5) g3 (v(A3
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= 4 + 3 = 7) Case 3 g3 (v(A3
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= 4 + 3 = 7) g2 (v(A3
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= 4 + 1 = 5) Case 4 g2,g 3 (v(A3
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= 4 + 4 = 8) ∅(v(A 3
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Thus, our result follows
= 4) Then, by Proposition 1, after adding the zero-valued goods, Lemma 7 still holds true. Thus, our result follows. A.6 Algorithm and Proof of Theorem 7 Proof.The polynomial runtime of Algorithm 8 is easy to verify, given that there are only nestedForloops, which run in polyn...
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Roundf 1 tof 2 −1:This period of rounds is invalid
=v 2(At 1)−b. Roundf 1 tof 2 −1:This period of rounds is invalid. Roundf 2 toT:LetPandQbe a set of goods such that∀p∈P, and∀q∈Q,v 1(p) =v 2(q) =b, and v1(q) =v 2(p) = 0. Note that∀f 2 ≤t≥T, v 1(At 2\Q) =v 1(At 2), andv 2(At 1\P) =v 2(At 1). Therefore, this period is equivalent...
2020
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same bundle
We have vi(Ai)≥α·v i(Aj\{g}),∀g∈A j. Denoteγ= 1 + β·vi(g) vi(Ai) . Notice that γ·v i(Ai)≥β·v i(Aj). Therefore, MAX-ECE returns a β γ -TEF allocation. Lemma 9.The minimum value of β γ is whenv i(Ai) =min g∈Ovi(g)andv i(g) =max g∈O,vi(g)vi(g). Proof.Denotex=v i(Ai),y=v i(g), and...
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22 In addition, the value ofgcan be described as v1(g)≤v 1(Aeven 2 )· 1 2
First, for even roundst even, we have v1(Aeven 1 )≥v 1(Aeven 2 \{g}), g=argmin g′∈Aeven 2 v1(g′). 22 In addition, the value ofgcan be described as v1(g)≤v 1(Aeven 2 )· 1 2. Ifv 1(g)> v 1(Aeven 2 )· 1 2, it either meansA even 2 ={g}or there exists a goodg ∗ s.t.v 1(g)> v 1(g∗)....
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RR-round
= 10. In round1, in order to obtain a TMMS allocation, each agent must be allocated either{g1 1, g1 3}or{g 1 10}. In round2, each agent must be allocated either{g 1 1, g1 3, g1 10}or{g 2 1, g2 3, g2 10}to obtain a TMMS allocation. In order for the allocation at the end of roun...
2025
Reviewed May 16, 2026 · model on record in the stance chip above.
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