pith. sign in

arxiv: 2605.03436 · v1 · submitted 2026-05-05 · 🧮 math.OC

Promoting Fair Online Resource Allocation with Indivisible Units

Igor Averbakh , Hongyi Jiang , Samitha Samaranayake , Akang Wang , Jianghua Wu This is my paper

Pith reviewed 2026-05-07 15:40 UTC · model grok-4.3

classification 🧮 math.OC
keywords fair online resource allocationindivisible unitsfairness guaranteesrandom cyclic blocksstationary arrivalspriority-weighted loadonline policies
0
0 comments X

The pith

Online policies achieve optimal fairness of 1 over 1 plus priority-weighted load for any arrival sequence of indivisible resources.

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

The paper seeks to determine the best fairness guarantees possible when allocating a fixed stock of indivisible items to arriving demands from different groups in an online fashion. It defines fairness through the expected filling ratio that accounts for each group's priority weight and shows how to set acceptance probabilities dynamically based on remaining inventory. For unpredictable varying arrivals this yields an optimal guarantee of one divided by one plus the weighted load, while stationary arrivals allow the Random Cyclic Blocks policy to reach a higher exact expression that approaches one minus e to the minus load over the load. It also proves that policies limited to giving everything or nothing to a demand cannot attain these levels. A reader would care because many real allocation problems involve uncertainty and the risk of unfair outcomes for lower-priority or smaller groups.

Core claim

For arbitrary time-varying arrivals, online policies achieve the optimal universal fairness guarantee of 1/(1+R_beta) where R_beta is the priority-weighted system load. For time-invariant arrivals, the Random Cyclic Blocks algorithm achieves the exact finite-horizon guarantee [1-(1-R_beta/T)^T]/R_beta which is at least (1-e^{-R_beta})/R_beta and tight. All-or-nothing allocation policies cannot match these guarantees.

What carries the argument

Calibration of acceptance probabilities to remaining budget under the FE-FR-beta fairness criterion, implemented via the Random Cyclic Blocks algorithm for stationary arrivals.

If this is right

  • Stationarity in arrivals allows strictly improved fairness guarantees compared to arbitrary sequences.
  • Partial allocations are required to reach the optimal fairness levels.
  • The derived bounds are tight and cannot be improved without additional information.
  • The policies function without prior knowledge of the specific arrival sequence or distribution.

Where Pith is reading between the lines

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

  • The necessity of partial fulfillment suggests that systems allowing divisible units would improve equity outcomes in practice.
  • Forecasting demand patterns to detect stationarity could be integrated to switch between the two guarantee levels dynamically.
  • The same probability calibration approach might connect to fairness questions in related online problems like job assignment or inventory replenishment.

Load-bearing premise

Demands arrive as an unpredictable sequence from a fixed inventory and fairness is measured by the expected filling ratio weighted by group priorities.

What would settle it

Simulating the Random Cyclic Blocks policy on a known stationary arrival sequence with specific R_beta and T, then checking whether the realized fairness ratio equals exactly [1-(1-R_beta/T)^T]/R_beta would confirm or refute the exact finite-horizon guarantee.

read the original abstract

Allocating scarce, indivisible resources to diverse groups under uncertainty is a central challenge in operations research, where efficiency-focused methods often underserve marginalized populations. We study the Fair Online Resource Allocation with Indivisible Units (FORA-IU) problem, in which an unpredictable sequence of demands must be served from a strictly fixed inventory, and ask what fairness guarantees are achievable under different distributional and structural assumptions. We adopt a fairness criterion based on the expected filling ratio (FE-FR-beta), which balances each group's expected allocation against its expected demand and priority weight. We design online policies that calibrate acceptance probabilities to the remaining budget, analyze both arbitrary time-varying and stationary arrivals, introduce the Random Cyclic Blocks (RCB) algorithm tailored to the stationary case, and study the effect of restricting policies to all-or-nothing allocations. For arbitrary time-varying arrivals, our policy achieves the optimal universal fairness guarantee of 1/(1+R_beta), where R_beta denotes the priority-weighted system load. For time-invariant arrivals, RCB achieves the exact finite-horizon guarantee [1-(1-R_beta/T)^T]/R_beta, which is at least (1-e^{-R_beta})/R_beta and is also tight. We further show that all-or-nothing allocation policies cannot match these guarantees. These findings demonstrate that distributional stationarity strictly improves the fairness frontier, and that partial fulfillment is a necessary condition for attaining optimal fairness in online indivisible resource allocation.

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

0 major / 3 minor

Summary. The paper studies the FORA-IU problem of allocating indivisible resources online from a fixed inventory to demands arriving in an unpredictable sequence. It adopts the FE-FR-beta fairness criterion (expected filling ratio weighted by group priorities) and designs budget-calibrated online acceptance policies. For arbitrary time-varying arrivals the policy is shown to achieve the optimal universal guarantee 1/(1+R_beta) where R_beta is the priority-weighted load; for stationary arrivals the Random Cyclic Blocks (RCB) algorithm attains the exact finite-horizon value [1-(1-R_beta/T)^T]/R_beta (which is at least (1-e^{-R_beta})/R_beta and tight). The authors also prove that all-or-nothing policies cannot match these guarantees.

Significance. If the stated guarantees and tightness arguments hold, the work supplies a clean separation between the fairness frontier under general versus stationary arrivals and demonstrates that partial fulfillment is necessary for optimality. The explicit policy constructions, the RCB block-randomization technique, and the impossibility result for restricted policies constitute concrete contributions to online algorithms and fair resource allocation in operations research.

minor comments (3)
  1. [Abstract] The abstract introduces R_beta and T without a one-sentence gloss; a parenthetical definition would improve immediate readability.
  2. [Section 5] The finite-horizon expression [1-(1-R_beta/T)^T]/R_beta is stated to be tight; a brief remark on the matching lower-bound construction (e.g., which arrival pattern achieves equality) would help readers locate the argument.
  3. [Section 3.1] Notation for the remaining-budget calibration step is used before it is formally defined; a short forward reference or inline equation would reduce backtracking.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, the clear summary of our contributions on the FORA-IU problem, and the recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's central results derive explicit fairness bounds (1/(1+R_beta) for arbitrary arrivals; [1-(1-R_beta/T)^T]/R_beta for stationary arrivals) from the policy constructions (probability calibration to remaining budget and RCB block randomization) and the FE-FR-beta criterion. R_beta is defined externally as the priority-weighted system load based on fixed inventory and demand sequence, without fitting or self-reference. Optimality and tightness follow from separate impossibility arguments for all-or-nothing policies and comparison to the independent limit (1-e^{-R_beta})/R_beta. No step reduces by construction to its inputs, no load-bearing self-citation chain, and no ansatz or renaming of known results is used to force the claims. The derivation is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard probabilistic modeling of arrivals and the FE-FR-beta fairness definition; no new physical entities are postulated.

free parameters (1)
  • R_beta
    Priority-weighted system load parameter that appears in all stated guarantees; its value depends on demand and priority inputs.
axioms (2)
  • domain assumption Demands arrive in an unpredictable sequence from a strictly fixed inventory.
    Core setup of the FORA-IU problem stated in the abstract.
  • domain assumption Fairness is measured via the expected filling ratio FE-FR-beta balancing allocation, demand, and priority weight.
    The adopted fairness criterion introduced for the analysis.

pith-pipeline@v0.9.0 · 9755 in / 1279 out tokens · 68177 ms · 2026-05-07T15:40:22.469799+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 4 canonical work pages

  1. [1]

    ACM Trans

    Du, B., Huang, Z., Wu, C.: Adversarial deep learning for online resource allocation. ACM Trans. Model. Perform. Eval. Comput. Syst.6(4), 1–25 (2022)

    2022

  2. [2]

    Manshadi, V., Niazadeh, R., Rodilitz, S.: Fair dynamic rationing. Manag. Sci.69(11), 6818–6836 (2023) https://doi.org/10.1287/mnsc.2023.4700

    work page doi:10.1287/mnsc.2023.4700 2023

  3. [3]

    Science 366(6464), 447–453 (2019)

    Obermeyer, Z., Powers, B., Vogeli, C., Mullainathan, S.: Dissecting racial bias in an algorithm used to manage the health of populations. Science 366(6464), 447–453 (2019)

    2019

  4. [4]

    Samorani, M., Harris, S.L., Blount, L.G., Lu, H., Santoro, M.A.: Over- booked and overlooked: machine learning and racial bias in medical Promoting Fair Online Resource Allocation with Indivisible Units31 appointment scheduling. Manuf. Serv. Oper. Manag.24(6), 2825–2842 (2022)

    2022

  5. [5]

    Howell, J., Elliott, J.R.: Damages done: The longitudinal impacts of natural hazards on wealth inequality in the united states. Soc. Probl. 66(3), 448–467 (2019)

    2019

  6. [6]

    Health Aff.41(10), 1513–1522 (2022)

    Ne’eman, A., Bell, E., Schneider, M.C., Strolovitch, D.: Identifying and exploring bias in public opinion on scarce resource allocation during the covid-19 pandemic: Study examines bias in public opinion on scarce resource allocation during the covid-19 pandemic with an emphasis on opinions toward people with disabilities. Health Aff.41(10), 1513–1522 (2022)

    2022

  7. [7]

    Ma, W., Xu, P.: Promoting fairness among dynamic agents in online- matching markets under known stationary arrival distributions. Adv. Neural Inf. Process. Syst.37, 82205–82234 (2024)

    2024

  8. [8]

    Patriksson, M.: A survey on the continuous nonlinear resource allocation problem. Eur. J. Oper. Res.185(1), 1–46 (2008)

    2008

  9. [9]

    Informatica30(1) (2006)

    Chevaleyre, Y., Dunne, P., Endriss, U., Lang, J., Lemaître, M., Maudet, N., Padget, J., Phelps, S., Rodriguez-Aguilar, J., Sousa, P.: Issues in multiagent resource allocation. Informatica30(1) (2006)

    2006

  10. [10]

    Krylatov, A., Raevskaya, A.: Competitive resource allocation among urban congestion areas in a modern big city. J. Oper. Res. Soc. China12(1), 133–153 (2024)

    2024

  11. [11]

    In: Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21, pp

    Sankar, G.S., Louis, A., Nasre, M., Nimbhorkar, P.: Matchings with group fairness constraints: Online and offline algorithms. In: Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21, pp. 377–383 (2021)

    2021

  12. [12]

    Hosseini, H., Huang, Z., Igarashi, A., Shah, N.: Class fairness in online matching. Artif. Intell.335, 104177 (2024) https://doi.org/10.1016/j.artint. 2024.104177

    work page doi:10.1016/j.artint 2024

  13. [13]

    Zargari, F., Nekouyan, H., Sun, B., Tan, X.: Online allocation with multi- class arrivals: Group fairness vs individual welfare. Proc. ACM Meas. Anal. Comput. Syst.9(2), 1–51 (2025)

    2025

  14. [14]

    In: Proceedings of the AAAI Conference on Artificial Intelligence, vol

    Nanda, V., Xu, P., Sankararaman, K.A., Dickerson, J., Srinivasan, A.: Balancing the tradeoff between profit and fairness in rideshare platforms during high-demand hours. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, pp. 2210–2217 (2020) 32Promoting Fair Online Resource Allocation with Indivisible Units

    2020

  15. [15]

    In: Proceedings of the AAAI Conference on Artificial Intelligence, vol

    Pramanik, A., Xu, P., Xu, Y.: Equity promotion in public transportation. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, pp. 11890–11898 (2023)

    2023

  16. [16]

    Xu, E.Y., Xu, P.: Exploring the tradeoff between system profit and income equality among ride-hailing drivers. J. Artif. Intell. Res.79, 569–597 (2024)

    2024

  17. [17]

    Lien, R.W., Iravani, S.M., Smilowitz, K.R.: Sequential resource allocation for nonprofit operations. Oper. Res.62(2), 301–317 (2014)

    2014

  18. [18]

    Fadaki, M., Ansari, S., Abareshi, A., Lee, P.T.-W.: Sequential resource allocation for humanitarian operations using approximate dynamic pro- gramming. Transp. Res. Part E Logist. Transp. Rev.201, 104213 (2025)

    2025

  19. [19]

    Risk Anal.39(11), 2408–2426 (2019)

    Wang, Y., Bier, V.M., Sun, B.: Measuring and achieving equity in mul- tiperiod emergency material allocation. Risk Anal.39(11), 2408–2426 (2019)

    2019

  20. [20]

    In: Conference on Economics and Computation (EC ’24) (2024)

    Mertzanidis, M., Psomas, A., Verma, P.: Automating food drop: The power of two choices for dynamic and fair food allocation. In: Conference on Economics and Computation (EC ’24) (2024)

    2024

  21. [21]

    In: International Conference on Machine Learning, pp

    Balseiro, S., Lu, H., Mirrokni, V.: Regularized online allocation problems: Fairness and beyond. In: International Conference on Machine Learning, pp. 630–639 (2021). PMLR

    2021

  22. [22]

    Fianu, S., Davis, L.B.: A markov decision process model for equitable distribution of supplies under uncertainty. Eur. J. Oper. Res.264(3), 1101–1115 (2018)

    2018

  23. [23]

    In: Proceedings of the Fourth ACM International Conference on AI in Finance, pp

    Hassanzadeh, P., Kreacic, E., Zeng, S., Xiao, Y., Ganesh, S.: Sequential fair resource allocation under a markov decision process framework. In: Proceedings of the Fourth ACM International Conference on AI in Finance, pp. 673–680 (2023)

    2023

  24. [24]

    In: 38th Conference on Neural Information Processing Systems (NeurIPS 2024) (2024)

    Fallah, A., Jordan, M.I., Ulichney, A.: Fair allocation in dynamic mecha- nism design. In: 38th Conference on Neural Information Processing Systems (NeurIPS 2024) (2024)

    2024

  25. [25]

    ACM SIGMETRICS Perform

    Sinclair, S.R., Banerjee, S., Yu, C.L.: Sequential fair allocation: Achieving the optimal envy-efficiency tradeoff curve. ACM SIGMETRICS Perform. Eval. Rev.50(1), 95–96 (2022)

    2022

  26. [26]

    ACM SIGMETRICS Perform

    Banerjee, S., Hssaine, C., Sinclair, S.R.: Online fair allocation of perishable resources. ACM SIGMETRICS Perform. Eval. Rev.51(1), 55–56 (2023) Promoting Fair Online Resource Allocation with Indivisible Units33

    2023

  27. [27]

    arXiv preprintarXiv:2508.21753(2025)

    Onyeze, C., Sinclair, S.R., Hssaine, C., Banerjee, S.: Sequential fair allo- cation with replenishments: A little envy goes an exponentially long way. arXiv preprintarXiv:2508.21753(2025)

    work page arXiv 2025

  28. [28]

    In: International Conference on Algorithmic Decision Theory, pp

    Walsh, T.: Online cake cutting. In: International Conference on Algorithmic Decision Theory, pp. 292–305 (2011). Springer

    2011

  29. [29]

    In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) (2015)

    Aleksandrov, M., Aziz, H., Gaspers, S., Walsh, T.: Online fair division: Analysing a food bank problem. In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) (2015)

    2015

  30. [30]

    Kash, I., Procaccia, A.D., Shah, N.: No agent left behind: Dynamic fair division of multiple resources. J. Artif. Intell. Res.51, 579–603 (2014)

    2014

  31. [31]

    In: Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19) (2019)

    He, J., Procaccia, A.D., Psomas, C., Zeng, D.: Achieving a fairer future by changing the past. In: Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19) (2019)

    2019

  32. [32]

    In: Proceedings of the 21st ACM Conference on Economics and Computation, pp

    Zeng, D., Psomas, A.: Fairness-efficiency tradeoffs in dynamic fair divi- sion. In: Proceedings of the 21st ACM Conference on Economics and Computation, pp. 911–912 (2020)

    2020

  33. [33]

    Benade, G., Kazachkov, A.M., Procaccia, A.D., Psomas, A., Zeng, D.: Fair and efficient online allocations. Oper. Res.72(4), 1438–1452 (2024)

    2024

  34. [34]

    In: The 26th ACM Conference on Economics and Computation (EC ’25) (2025)

    Halpern, D., Psomas, A., Verma, P., Xie, D.: Online envy minimization and multicolor discrepancy: Equivalences and separations. In: The 26th ACM Conference on Economics and Computation (EC ’25) (2025)

    2025

  35. [35]

    In: The 26th ACM Conference on Economics and Computation (EC ’25) (2025)

    Kulkarni, P., Mehta, R., Shahkar, P.: Online fair division: Towards ex-post constant mms guarantees. In: The 26th ACM Conference on Economics and Computation (EC ’25) (2025)

    2025

  36. [36]

    In: The Thirty- Ninth AAAI Conference on Artificial Intelligence (AAAI-25) (2025)

    Cookson, B., Ebadian, S., Shah, N.: Temporal fair division. In: The Thirty- Ninth AAAI Conference on Artificial Intelligence (AAAI-25) (2025)

    2025

  37. [37]

    Gölz, P., Peters, D., Procaccia, A.D.: In this apportionment lottery, the house always wins. Oper. Res. (2025)

    2025

  38. [38]

    In: International Symposium on Algorithmic Game Theory, pp

    Elkind, E., Igarashi, A., Teh, N.: Fair division of chores with budget constraints. In: International Symposium on Algorithmic Game Theory, pp. 55–71 (2024). Springer

    2024

  39. [39]

    In: AAMAS Conference Proceedings (2022) 34Promoting Fair Online Resource Allocation with Indivisible Units

    Srinivasan, A., Xu, P.: The generalized magician problem under unknown distributions and related applications. In: AAMAS Conference Proceedings (2022) 34Promoting Fair Online Resource Allocation with Indivisible Units

    2022

  40. [40]

    In: Proceedings of the 41st International Conference on Machine Learning

    Sankararaman, K.A., Srinivasan, A., Xu, P.: Promoting external and internal equities under ex-ante/ex-post metrics in online resource allocation. In: Proceedings of the 41st International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 235, pp. 43328–43345 (2024)

    2024

  41. [41]

    ACM Trans

    Ma, W., Xu, P., Xu, Y.: Fairness maximization among offline agents in online-matching markets. ACM Trans. Econ. Comput.10(4) (2023) https://doi.org/10.1145/3569705

    work page doi:10.1145/3569705 2023

  42. [42]

    Fahrbach, M., Huang, Z., Tao, R., Zadimoghaddam, M.: Edge-weighted online bipartite matching. J. ACM69(6) (2022) https://doi.org/10.1145/ 3556971

    2022

  43. [43]

    In: Proceedings of the 13th ACM Conference on Electronic Commerce, pp

    Alaei, S., Hajiaghayi, M., Liaghat, V.: Online prophet-inequality matching with applications to ad allocation. In: Proceedings of the 13th ACM Conference on Electronic Commerce, pp. 18–35 (2012)

    2012

  44. [44]

    In: International Workshop on Approximation Algorithms for Combinatorial Optimization, pp

    Alaei, S., Hajiaghayi, M., Liaghat, V.: The online stochastic general- ized assignment problem. In: International Workshop on Approximation Algorithms for Combinatorial Optimization, pp. 11–25 (2013). Springer

    2013

  45. [45]

    Yang, L., Zeynali, A., Hajiesmaili, M.H., Sitaraman, R.K., Towsley, D.: Competitive algorithms for online multidimensional knapsack problems. Proc. ACM Meas. Anal. Comput. Syst.5(3), 1–30 (2021)

    2021