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arxiv: 2602.14120 · v2 · submitted 2026-02-15 · 💻 cs.GT

Recognition: no theorem link

Evaluating the Performance of Approximation Mechanisms under Budget Constraints

Juan Carlos Carbajal , Ahuva Mualem
Authors on Pith no claims yet

Pith reviewed 2026-05-15 22:14 UTC · model grok-4.3

classification 💻 cs.GT
keywords revenue maximizationprivate budgetsapproximation mechanismsmenu sizemechanism designsingle-item auctioncorrelationbounded support
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The pith

No mechanism with finite or sublinear menu size can guarantee any positive fraction of optimal revenue when buyers have private budgets, even for many bounded distributions.

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

The paper evaluates how well simple approximation mechanisms perform against optimal revenue-maximizing mechanisms in single-object sales where the buyer has both a private value and a private budget. For distributions with bounded support, mechanisms whose menu size grows only polylogarithmically can approximate the optimum arbitrarily closely, even when values and budgets are correlated. For unbounded support and for bounded distributions concentrated inside the unit square, however, no mechanism whose menu is finite or sublinear can secure any constant fraction of the optimum, and this negative result holds even when values and budgets are independent. The authors also exhibit cases in which relaxing the mechanism produces unbounded revenue gains under negative correlation and identify instances of revenue non-monotonicity.

Core claim

In the single-item setting with a buyer whose valuation and budget are both private, simple mechanisms (those with finite or sublinear menu size) cannot guarantee any positive fraction of the revenue of an optimal mechanism once the joint distribution of value and budget has unbounded support or is concentrated in the unit square. Polylogarithmic-menu mechanisms succeed in approximating the optimum arbitrarily well when support is bounded. These limits persist under independence and are accompanied by unbounded gains from certain relaxations under negative correlation and by revenue non-monotonicity in some instances.

What carries the argument

Menu size of the mechanism (finite, sublinear, or polylogarithmic) as the measure of simplicity when approximating optimal revenue in the presence of private budgets.

If this is right

  • Polylogarithmic-menu mechanisms achieve arbitrary approximation for any bounded-support distribution, including correlated value-budget pairs.
  • No finite-menu mechanism secures a positive revenue fraction for unbounded-support distributions or for unit-square-concentrated distributions, even under independence.
  • Certain relaxations of the mechanism produce unbounded revenue improvement under negative correlation.
  • Revenue can be non-monotonic in the reported budget for some distributions.

Where Pith is reading between the lines

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

  • The fragility result suggests that mechanism design with private budgets may require either highly complex menus or entirely new notions of approximation that tolerate menu size explicitly.
  • Similar menu-size barriers could appear in multi-item settings once budgets are private.
  • Concrete distributions with exponential tails would be a natural next test case for the unbounded-support impossibility.

Load-bearing premise

The impossibility results depend on the three specific performance measures chosen and on the partition of distributions into bounded versus unbounded support classes.

What would settle it

A concrete finite-menu mechanism that obtains a positive constant fraction of optimal revenue for some distribution with unbounded support and independent valuation and budget.

read the original abstract

We study revenue maximization in a buyer-seller setting where the seller has a single object and the buyer has both a private valuation and a private budget. Private budgets complicate the classic single-product monopoly problem, making optimal mechanisms difficult to characterize. To address this, we evaluate the robust performance of approximation mechanisms relative to optimal mechanisms using three performance measures: the guaranteed fraction of optimal revenue, the maximal value of relaxation, and a revenue non-monotonicity gap. Our analysis reveals sharp contrasts. For distributions with bounded support, simple mechanisms with polylogarithmic menu size can approximate optimal revenue arbitrarily well, even when valuations and budgets are correlated. By contrast, for distributions with unbounded support, and even for bounded distributions concentrated in the unit square, no simple mechanism -- or any mechanism with a finite or sublinear menu -- can guarantee a positive fraction of optimal revenue. In particular, no finite-menu mechanism guarantees any positive fraction even under independence. We also show unbounded revenue gains from certain relaxations under negative correlation and identify cases of revenue non-monotonicity. Overall, our results show that approximation guarantees with private budgets are fragile, revealing fundamental limits to simplicity and robustness in mechanism design.

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

2 major / 3 minor

Summary. The manuscript evaluates approximation mechanisms for single-item revenue maximization when the buyer has both a private valuation and a private budget. Using three performance measures—the guaranteed fraction of optimal revenue (worst-case infimum over distributions), the maximal value of relaxation, and a revenue non-monotonicity gap—the paper claims that polylogarithmic-menu mechanisms achieve arbitrarily close approximation to the optimum for any bounded-support distribution (even under correlation), while no finite or sublinear-menu mechanism can guarantee a positive fraction for unbounded-support distributions or for certain bounded distributions concentrated in the unit square, even under independence. Additional results establish unbounded revenue gains from relaxations under negative correlation and identify cases of revenue non-monotonicity.

Significance. If the central separations hold, the work demonstrates the fragility of simple approximation guarantees once budgets are private, sharply distinguishing bounded-support regimes (where polylog menus suffice) from unbounded and concentrated cases (where finite menus fail entirely). The use of explicit worst-case infima over distribution classes and the three complementary performance measures provides a robust framework for assessing simplicity in budget-constrained mechanism design.

major comments (2)
  1. [Main results section (around the unbounded-support theorem)] The impossibility claim that no finite-menu mechanism guarantees a positive fraction even under independence (stated in the abstract and presumably proved in the main results section) is load-bearing; the explicit family of distributions and the calculation showing that the revenue ratio tends to zero for any fixed menu size must be presented with sufficient detail to allow verification of the infimum.
  2. [Bounded-support approximation theorem] The polylogarithmic menu-size bound for bounded-support distributions (allowing approximation arbitrarily close to 1) relies on a specific construction; the dependence of the menu size on the support parameters and the correlation structure should be stated explicitly, together with the proof that the guaranteed fraction can be driven to 1.
minor comments (3)
  1. [Model and definitions] The three performance measures are introduced in the abstract without symbols; defining them with consistent notation (e.g., α for guaranteed fraction) at the beginning of the model section would improve readability.
  2. [Numerical illustrations] Figure captions and axis labels for any plots comparing menu sizes versus guaranteed fractions should explicitly state the distribution class (bounded vs. unbounded) and the correlation regime used.
  3. [Introduction or related work] A short discussion of how the results relate to prior menu-complexity bounds in the independent private-values setting (without budgets) would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight opportunities to improve clarity in the presentation of our main results. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Main results section (around the unbounded-support theorem)] The impossibility claim that no finite-menu mechanism guarantees a positive fraction even under independence (stated in the abstract and presumably proved in the main results section) is load-bearing; the explicit family of distributions and the calculation showing that the revenue ratio tends to zero for any fixed menu size must be presented with sufficient detail to allow verification of the infimum.

    Authors: We agree that additional detail will strengthen verifiability. In the revised manuscript we will expand the proof of the unbounded-support impossibility result to explicitly define the family of distributions (including the precise parameterization that forces the ratio to zero) and provide the full step-by-step calculation showing that, for any fixed menu size k, the revenue ratio infimum over the family tends to zero. This will be placed in the main text rather than deferred to an appendix. revision: yes

  2. Referee: [Bounded-support approximation theorem] The polylogarithmic menu-size bound for bounded-support distributions (allowing approximation arbitrarily close to 1) relies on a specific construction; the dependence of the menu size on the support parameters and the correlation structure should be stated explicitly, together with the proof that the guaranteed fraction can be driven to 1.

    Authors: We will revise the bounded-support section to state explicitly the functional dependence of the menu size on the support bounds (valuation upper bound V and budget upper bound B) and on the correlation structure. We will also include a concise but self-contained outline of the construction together with the argument showing that the guaranteed fraction can be driven arbitrarily close to 1 by choosing sufficiently large (but still polylogarithmic) menu size. The dependence will be summarized in a theorem statement for easy reference. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes impossibility and approximation results via direct worst-case analysis over distribution classes, comparing explicit mechanism menus against optimal revenue benchmarks using three explicitly defined performance measures. No step reduces a claimed prediction or guarantee to a fitted parameter, self-definition, or load-bearing self-citation; the bounded-versus-unbounded contrast and finite-menu fragility follow from infima over independent distribution families without internal collapse. The argument remains self-contained through explicit constructions and relaxations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on standard mechanism-design modeling assumptions and on the chosen performance measures; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Buyers have quasi-linear utilities and are risk-neutral.
    Standard background assumption in mechanism design invoked for the revenue-maximization setting.

pith-pipeline@v0.9.0 · 5501 in / 1222 out tokens · 31520 ms · 2026-05-15T22:14:45.236778+00:00 · methodology

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Optimal deterministic mechanisms for an additive buyer

    Moshe Babaioff, Noam Nisan, and Aviad Rubinstein. Optimal deterministic mechanisms for an additive buyer. InProceedings of the 2018 ACM Conference on economics and compu- tation, EC ’18, pages 429–429. ACM, 2018. ISBN 1450358292

    work page 2018

  2. [2]

    Matthew Weinberg

    Moshe Babaioff, Nicole Immorlica, Brendan Lucier, and S. Matthew Weinberg. A simple and approximately optimal mechanism for an additive buyer.J. ACM, 67(4), June 2020. ISSN 0004-5411. doi: 10.1145/3398745. URLhttps://doi.org/10.1145/3398745

    work page doi:10.1145/3398745 2020

  3. [3]

    Gonczarowski, and Noam Nisan

    Moshe Babaioff, Yannai A. Gonczarowski, and Noam Nisan. The Menu-Size Complexity of Revenue Approximation.Games and Economic Behavior, 134:281–307, 2022. ISSN 0899-

    work page 2022

  4. [4]

    doi: 10.1016/j.geb.2021.03.001

    work page doi:10.1016/j.geb.2021.03.001 2021

  5. [5]

    Monotonic mechanisms for selling mul- tiple goods

    Ran Ben-Moshe, Sergiu Hart, and Noam Nisan. Monotonic mechanisms for selling mul- tiple goods. arXiv, 2024

    work page 2024

  6. [6]

    Incentive Compatible Budget Elicitation in Multi-Unit Auctions

    Sayan Bhattacharya, Vincent Conitzer, Kamesh Munagala, and Lirong Xia. Incentive Compatible Budget Elicitation in Multi-Unit Auctions. InProceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 554–572, 2010. 28

    work page 2010

  7. [7]

    Optimal and Efficient Mechanisms with Asym- metrically Budget Constrained Buyers.Games and Economic Behavior, 127:155–178, 2021

    Alexei Boulatov and Sergei Severinov. Optimal and Efficient Mechanisms with Asym- metrically Budget Constrained Buyers.Games and Economic Behavior, 127:155–178, 2021. ISSN 0899-8256. doi: 10.1016/j.geb.2021.02.001

    work page doi:10.1016/j.geb.2021.02.001 2021

  8. [8]

    Matthew Weinberg

    Patrick Briest, Shuchi Chawla, Robert Kleinberg, and S. Matthew Weinberg. Pricing Lotteries.Journal of Economic Theory, 156:144–174, 2015. ISSN 0022-0531. doi: 10.1016/j. jet.2014.04.011

    work page doi:10.1016/j 2015

  9. [9]

    Hartline, and Robert Kleinberg

    Shuchi Chawla, Jason D. Hartline, and Robert Kleinberg. Algorithmic pricing via vir- tual valuations. InProceedings of the 8th ACM Conference on Electronic Commerce, EC ’07, pages 243–251, New York, NY, USA, 2007. Association for Computing Machinery. ISBN 9781595936530. doi: 10.1145/1250910.1250946. URLhttps://doi.org/10.1145/ 1250910.1250946

    work page doi:10.1145/1250910.1250946 2007

  10. [10]

    Bayesian mechanism design for budget-constrained agents

    Shuchi Chawla, David Malec, and Azarakhsh Malekian. Bayesian mechanism design for budget-constrained agents. InProceedings of the 12th ACM Conference on Electronic Commerce, EC’11, pages 253–262. ACM, 2011. ISBN 9781450302616

    work page 2011

  11. [11]

    Standard Auctions with Financially Constrained Bidders

    Yeon-Koo Che and Ian Gale. Standard Auctions with Financially Constrained Bidders. Review of Economic Studies, 65(1):1–21, January 1998

    work page 1998

  12. [12]

    The Optimal Mechanism for Selling to a Budget- Constrained Buyer.Journal of Economic Theory, 92:198–233, 2000

    Yeon-Koo Che and Ian Gale. The Optimal Mechanism for Selling to a Budget- Constrained Buyer.Journal of Economic Theory, 92:198–233, 2000

    work page 2000

  13. [13]

    Assigning Resources to Budget-Constrained Agents.Review of Economic Studies, 80(1):73–107, February 2013

    Yeon-Koo Che, Ian Gale, and Jinwoo Kim. Assigning Resources to Budget-Constrained Agents.Review of Economic Studies, 80(1):73–107, February 2013

    work page 2013

  14. [14]

    A simple mechanism for a budget-constrained buyer.ACM Trans

    Yu Cheng, Nick Gravin, Kamesh Munagala, and Kangning Wang. A simple mechanism for a budget-constrained buyer.ACM Trans. Econ. Comput., 9(2), January 2021. ISSN 2167-8375. doi: 10.1145/3434419. URLhttps://doi.org/10.1145/3434419

    work page doi:10.1145/3434419 2021

  15. [15]

    How to best auction natural resources

    Peter Cramton. How to best auction natural resources. In Philip Daniel, Michael Keen, and Charles McPherson, editors,The Taxation of Petroleum and Minerals: Principles, Prob- lems and Practice, chapter 10. Routledge, 2010

    work page 2010

  16. [16]

    Devanur, and S

    Constantinos Daskalakis, Nikhil R. Devanur, and S. Matthew Weinberg. Revenue maxi- mization and ex-post budget constraints.ACM Transactions on Economics and Computation, 6(3-4):1–19, 2018

    work page 2018

  17. [17]

    The Optimal Mechanism for Selling to a Budget-Constrained Buyer: the General Case

    Nikhil R Devanur and S Matthew Weinberg. The Optimal Mechanism for Selling to a Budget-Constrained Buyer: the General Case. InEC 2017 - Proceedings of the 2017 ACM Conference on Economics and Computation, pages 39–40, New York, New York, USA, June

    work page 2017

  18. [18]

    Microsoft Research, ACM Press

    work page

  19. [19]

    Hartline, and Yingkai Li

    Yiding Feng, Jason D. Hartline, and Yingkai Li. Simple mechanisms for agents with non-linear utilities. InProceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3802–3816. 2023

    work page 2023

  20. [20]

    Approximate Revenue Maximization with Multiple Items

    Sergiu Hart and Noam Nisan. Approximate Revenue Maximization with Multiple Items. Journal of Economic Theory, 172:313–347, 2017. ISSN 0022-0531. doi: 10.1016/j.jet.2017.09. 001

    work page doi:10.1016/j.jet.2017.09 2017

  21. [21]

    Selling multiple correlated goods: Revenue maximization and menu-size complexity.Journal of Economic Theory, 183:991–1029, 2019

    Sergiu Hart and Noam Nisan. Selling multiple correlated goods: Revenue maximization and menu-size complexity.Journal of Economic Theory, 183:991–1029, 2019. 29

    work page 2019

  22. [22]

    Sergiu Hart and Philip J. Reny. Maximal Revenue with Multiple Goods: Nonmonotonic- ity and Other Observations.Theoretical Economics, 10(3):893–922, 2015. ISSN 1933-6837. doi: 10.3982/te1517

    work page doi:10.3982/te1517 2015

  23. [23]

    Sergiu Hart and Philip J. Reny. The Better Half of Selling Separately.ACM Transactions on Economics and Computation, 7(4):18, 2019. ISSN 2167-8375. doi: 10.1145/3369927

    work page doi:10.1145/3369927 2019

  24. [24]

    Kotowski

    Maciej H. Kotowski. First-Price Auctions with Budget Constraints.Theoretical Economics, 15(1):199–237, 2020. ISSN 1933-6837. doi: 10.3982/te2982

    work page doi:10.3982/te2982 2020

  25. [25]

    Optimal Auction with Financially Constrained Buyers.Economics Letters, 52(2):181–186, 1996

    J J Laffont and J Robert. Optimal Auction with Financially Constrained Buyers.Economics Letters, 52(2):181–186, 1996

    work page 1996

  26. [26]

    Xinye Li and Andrew Chi-Chih Yao. On revenue maximization for selling multiple independently distributed items.Proceedings of the National Academy of Sciences of the United States of America, 110(28):11232–11237, 2013. ISSN 00278424, 10916490. URLhttp: //www.jstor.org/stable/42712717

    work page arXiv 2013

  27. [27]

    Sequentially Rationalizable Choice.American Eco- nomic Review, 97(5):1824–1839, 2007

    Paola Manzini and Marco Mariotti. Sequentially Rationalizable Choice.American Eco- nomic Review, 97(5):1824–1839, 2007

    work page 2007

  28. [28]

    Multidimensional Incentive Compatibility and Mechanism Design.Journal of Economic Theory, 46(2):335–354, 1988

    R.Preston McAfee and John McMillan. Multidimensional Incentive Compatibility and Mechanism Design.Journal of Economic Theory, 46(2):335–354, 1988. ISSN 0022-0531. doi: 10.1016/0022-0531(88)90135-4

    work page doi:10.1016/0022-0531(88)90135-4 1988

  29. [29]

    Optimal Auction Design.Mathematics of Operations Research, 6(1): 58–73, 1981

    Roger B Myerson. Optimal Auction Design.Mathematics of Operations Research, 6(1): 58–73, 1981

    work page 1981

  30. [30]

    Optimal Auctions with Financially Constrained Buyers.Journal of Economic Theory, 150:383–425, March 2014

    Mallesh M Pai and Rakesh V Vohra. Optimal Auctions with Financially Constrained Buyers.Journal of Economic Theory, 150:383–425, March 2014

    work page 2014

  31. [31]

    Matthew Weinberg

    Alexandros Psomas, Ariel Schvartzman, and S. Matthew Weinberg. On infinite separa- tions between simple and optimal mechanisms. InProceedings of the 36th International Conference on Neural Information Processing Systems, NIPS ’22, Red Hook, NY, USA, 2022. Curran Associates Inc. ISBN 9781713871088

    work page 2022

  32. [32]

    Mechanism design with budget constraints and a population of agents

    Michael Richter. Mechanism design with budget constraints and a population of agents. Games and Economic Behavior, 115:30–47, 2019

    work page 2019

  33. [33]

    Ironing, Sweeping, and Multidimensional Screening.Econometrica, 66(4):783–826, 1998

    Jean-Charles Rochet and Philippe Chone. Ironing, Sweeping, and Multidimensional Screening.Econometrica, 66(4):783–826, 1998. ISSN 0012-9682. doi: 10.2307/2999574

    work page doi:10.2307/2999574 1998

  34. [34]

    Approximately Optimal Mechanism De- sign.Annual Review of Economics, 11:355—381, 2019

    Tim Roughgarden and Inbal Talgam-Cohen. Approximately Optimal Mechanism De- sign.Annual Review of Economics, 11:355—381, 2019. ISSN 1941-1383. doi: 10.1146/ annurev-economics-080218-025607

    work page 2019

  35. [35]

    Matthew Weinberg

    Aviad Rubinstein and S. Matthew Weinberg. Simple mechanisms for a subadditive buyer and applications to revenue monotonicity.ACM Trans. Econ. Comput., 6(3–4), October

    work page

  36. [36]

    doi: 10.1145/3105448

    ISSN 2167-8375. doi: 10.1145/3105448. URLhttps://doi.org/10.1145/3105448

    work page doi:10.1145/3105448

  37. [37]

    Up in the Air: GTE’s Experience in the MTA Auction for Personal Communication Services Licenses.Journal of Economics & Management Strategy, 6(3):549– 572, 1997

    David J Salant. Up in the Air: GTE’s Experience in the MTA Auction for Personal Communication Services Licenses.Journal of Economics & Management Strategy, 6(3):549– 572, 1997

    work page 1997

  38. [38]

    George Shanthikumar.Stochastic Orders

    Moshe Shaked and J. George Shanthikumar.Stochastic Orders. Springer, New York, NY, 2007. 30 APPENDIX A. Extension Lemma In this appendix we show that theExtension Lemmaof Hart and Reny [20] applies to a setting with private valuations and private budgets. LetD⊆R 2 + be the type domain, i.e., the support of the random vector(V,W). Letµ= (x,s)be a direct me...

    work page 2007