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arxiv: 1907.04740 · v1 · pith:ZKJRWUWKnew · submitted 2019-07-10 · 💻 cs.GT · cs.MA

Optimal mechanisms with budget for user generated contents

Mengjing Chen , Pingzhong Tang , Zihe Wang , Shenke Xiao , Xiwang Yang This is my paper

Pith reviewed 2026-05-24 23:23 UTC · model grok-4.3

classification 💻 cs.GT cs.MA
keywords mechanism designuser-generated contentgross product maximizationoptimal mechanismslinear programmingbudget constraints
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The pith

An O(n log n) algorithm computes the optimal payment rule that maximizes total content quality under a fixed budget on user-generated sites.

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

The paper designs payment mechanisms that reward users for uploading high-quality material while respecting a platform's total budget. It first shows that the proportional-division rule, common in practice, can return arbitrarily low total quality on some inputs. The design task is then cast as maximizing the product of contributions subject to payment constraints, which reduces to a linear program whose decision variables are bounded and non-decreasing. The authors supply a sorting procedure that solves this program exactly in O(n log n) time.

Core claim

The gross-product maximization problem can be formulated as a linear program with bounded and increasing variables, and an O(n log n) algorithm recovers the optimal mechanism for any number of players.

What carries the argument

the linear program whose variables are the payments to each player and are required to be bounded and non-decreasing

If this is right

  • The proportional-division mechanism used on many platforms can produce arbitrarily poor total content quality.
  • An optimal mechanism can be computed in time linear in the number of users after a single sort.
  • The optimal payments are monotone in user type, satisfying the bounded-and-increasing condition of the linear program.

Where Pith is reading between the lines

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

  • The same linear-program reduction may extend to other crowdsourcing settings where total output is measured by a product rather than a sum.
  • Platforms could pre-compute the optimal payment schedule once per budget level and apply it in real time.

Load-bearing premise

The gross-product maximization problem can be formulated as a linear program whose variables are bounded and increasing.

What would settle it

On any small instance, enumerate all feasible payment vectors, compute the resulting gross product, and check whether the algorithm's output matches the maximum.

Figures

Figures reproduced from arXiv: 1907.04740 by Mengjing Chen, Pingzhong Tang, Shenke Xiao, Xiwang Yang, Zihe Wang.

Figure 1
Figure 1. Figure 1: avg Inequalites Lemma 8. For any i, if b(i) ≤ n, then i) the iteration where i ∗ = i comes after the iteration where i ∗ = b(i), and ii) for any i < j < b(i), the iteration where i ∗ = j comes after the iteration where i ∗ = i. Lemma 9 shows two inequality relations among avg’s related to iL, i∗ , iR in each iteration [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Before the Operation · · · iℓ iℓ + 1 · · · r r + 1 · · · x ∗ i [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
read the original abstract

In this paper, we design gross product maximization mechanisms which incentivize users to upload high-quality contents on user-generated-content (UGC) websites. We show that, the proportional division mechanism, which is widely used in practice, can perform arbitrarily bad in the worst case. The problem can be formulated using a linear program with bounded and increasing variables. We then present an $O(n\log n)$ algorithm to find the optimal mechanism, where n is the number of players.

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 / 0 minor

Summary. The paper studies mechanism design for maximizing gross product (sum of user contributions times payments or similar) on user-generated content platforms. It shows that the proportional division mechanism used in practice can perform arbitrarily poorly in the worst case. The problem is claimed to reduce to a linear program whose variables are bounded and increasing, for which the authors give an O(n log n) algorithm.

Significance. If the LP reduction and the O(n log n) algorithm are correct, the work supplies an efficient, practical method for computing optimal budget-constrained mechanisms in this setting. Explicitly showing that a widely deployed mechanism can be arbitrarily bad is a useful negative result. No machine-checked proofs or reproducible code are mentioned.

major comments (2)
  1. [Abstract] Abstract (and the formulation section): the claim that the gross-product maximization problem 'can be formulated using a linear program with bounded and increasing variables' is stated without exhibiting the LP, without deriving the bounded-and-increasing restriction from the incentive and budget constraints, and without proving that restricting the feasible region to monotone bounded variables preserves optimality. This property is load-bearing for the O(n log n) running-time claim.
  2. [Abstract] The O(n log n) algorithm is presented as solving the restricted LP, but no description of the algorithm, no argument that the number of binding constraints is O(n), and no verification that monotonicity allows a sorting-based solution appear in the supplied text. Without these steps the complexity result does not follow from the LP formulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and will revise the manuscript to improve clarity on the LP formulation and algorithm.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the formulation section): the claim that the gross-product maximization problem 'can be formulated using a linear program with bounded and increasing variables' is stated without exhibiting the LP, without deriving the bounded-and-increasing restriction from the incentive and budget constraints, and without proving that restricting the feasible region to monotone bounded variables preserves optimality. This property is load-bearing for the O(n log n) running-time claim.

    Authors: The derivation of the LP from the incentive-compatibility and budget constraints, together with the proof that restricting to bounded and monotone (increasing) variables preserves optimality, appears in Section 3 of the full manuscript. We will revise the abstract to reference Section 3 explicitly and include a short outline of the LP. revision: yes

  2. Referee: [Abstract] The O(n log n) algorithm is presented as solving the restricted LP, but no description of the algorithm, no argument that the number of binding constraints is O(n), and no verification that monotonicity allows a sorting-based solution appear in the supplied text. Without these steps the complexity result does not follow from the LP formulation.

    Authors: Section 4 contains the full description of the O(n log n) algorithm, the argument that monotonicity yields O(n) binding constraints, and the reduction to a sorting-based procedure. We will add a concise high-level description of the algorithm to the abstract and introduction. revision: yes

Circularity Check

0 steps flagged

No circularity; formulation and algorithm presented as independent steps

full rationale

The abstract states that the gross-product maximization problem 'can be formulated using a linear program with bounded and increasing variables' and then presents an O(n log n) algorithm. No equations, self-definitions, fitted parameters renamed as predictions, or self-citation chains appear in the supplied text. The bounded/increasing property is asserted as part of the modeling step rather than derived from the algorithm or vice versa, leaving the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5604 in / 913 out tokens · 12810 ms · 2026-05-24T23:23:32.184937+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    User-generated content and social media

    Michael Luca. User-generated content and social media. In Handbook of media Economics , volume 1, pages 563–592. Elsevier, 2015

    work page 2015

  2. [2]

    Differentiation with us er-generated content

    Kaifu Zhang and Miklos Sarvary. Differentiation with us er-generated content. Management Science, 61(4):898–914, 2014

    work page 2014

  3. [3]

    Exploring the user-generated content (ugc) upload ing behavior on youtube

    Jaimie Y ejean Park, Jiyeon Jang, Alejandro Jaimes, Chin -Wan Chung, and Sung-Hyon Myaeng. Exploring the user-generated content (ugc) upload ing behavior on youtube. In Pro- ceedings of the 23rd International Conference on World Wide Web, pages 529–534. ACM, 2014

    work page 2014

  4. [4]

    Learning and incentive s in user-generated content: Multi- armed bandits with endogenous arms

    Arpita Ghosh and Patrick Hummel. Learning and incentive s in user-generated content: Multi- armed bandits with endogenous arms. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science , pages 233–246. ACM, 2013

    work page 2013

  5. [5]

    Designing i ncentives for online question-and- answer forums

    Shaili Jain, Yiling Chen, and David C Parkes. Designing i ncentives for online question-and- answer forums. Games and Economic Behavior , 86:458–474, 2014

    work page 2014

  6. [6]

    All-pay contests

    Ron Siegel. All-pay contests. Econometrica, 77(1):71–92, 2009

    work page 2009

  7. [7]

    Optimal crowdsourcing con- tests

    Shuchi Chawla, Jason D Hartline, and Balasubramanian Si van. Optimal crowdsourcing con- tests. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algo- rithms, pages 856–868. SIAM, 2012

    work page 2012

  8. [8]

    Crowdsourcing and all-pay auctions

    Dominic DiPalantino and Milan V ojnovic. Crowdsourcing and all-pay auctions. In Proceed- ings of the 10th ACM conference on Electronic commerce , pages 119–128. ACM, 2009. 17

    work page 2009

  9. [9]

    Crowdsourcing with end ogenous entry

    Arpita Ghosh and Preston McAfee. Crowdsourcing with end ogenous entry. In Proceedings of the 21st international conference on World Wide Web , pages 999–1008. ACM, 2012

    work page 2012

  10. [10]

    Efficient crowdsourc ing contests

    Ruggiero Cavallo and Shaili Jain. Efficient crowdsourc ing contests. In Proceedings of the 11th International Conference on Autonomous Agents and Mul tiagent Systems-V olume 2, pages 677–686. International Foundation for Autonomous Ag ents and Multiagent Systems, 2012

    work page 2012

  11. [11]

    A game-theoretic analys is of the esp game

    Shaili Jain and David C Parkes. A game-theoretic analys is of the esp game. ACM Transac- tions on Economics and Computation (TEAC) , 1(1):3, 2013

    work page 2013

  12. [12]

    A game-theoretic anal ysis of rank-order mechanisms for user-generated content

    Arpita Ghosh and Patrick Hummel. A game-theoretic anal ysis of rank-order mechanisms for user-generated content. Journal of Economic Theory , 154:349–374, 2014

    work page 2014

  13. [13]

    Optimal prizes for all-pay contests in heterogeneous crowdsourcing

    Tie Luo, Salil S Kanhere, Sajal K Das, and Hwee-Pink Tan. Optimal prizes for all-pay contests in heterogeneous crowdsourcing. In 2014 IEEE 11th International Conference on Mobile Ad Hoc and Sensor Systems , pages 136–144. IEEE, 2014

    work page 2014

  14. [14]

    An exploration in the theory of optimu m income taxation

    James A Mirrlees. An exploration in the theory of optimu m income taxation. The review of economic studies, 38(2):175–208, 1971

    work page 1971

  15. [15]

    The optimal linear income-tax

    Eytan Sheshinski. The optimal linear income-tax. The Review of Economic Studies , 39(3): 297–302, 1972

    work page 1972

  16. [16]

    Designing an optimal contest

    Ani Dasgupta and Kofi O Nti. Designing an optimal contest . European Journal of Political Economy, 14(4):587–603, 1998

    work page 1998

  17. [17]

    Linear programming 2: theory and extensions

    George B Dantzig and Mukund N Thapa. Linear programming 2: theory and extensions . Springer Science & Business Media, 2006

    work page 2006

  18. [18]

    Fast algorithms for monotonic discounted linear programs with two variables per inequality

    Daniel Andersson and Sergei V orobyov. Fast algorithms for monotonic discounted linear programs with two variables per inequality. Preprint NI06019-LAA, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK , 2006

    work page 2006

  19. [19]

    The inverse 1-median problem on a cycle

    Rainer E Burkard, Carmen Pleschiutschnig, and Jianzho ng Zhang. The inverse 1-median problem on a cycle. Discrete Optimization, 5(2):242–253, 2008. 18

    work page 2008