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Robust Competitive Ratio for Deterministic Monopoly Pricing

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A monopolist who knows only mean, variance, and maximum value has an optimal robust price: charge low below a dispersion threshold, high above it.

desk verdict The paper solves a meaningful max-min pricing problem with an elegant decomposition, but the load-bearing monotonicity lemma has a long unverified proof and a contradictory summary statement that must be fixed before I'd trust it as a theorem. read the letter →

arxiv 2509.06619 v1 pith:VWCLQRPT submitted 2025-09-08 math.OC

Tim S.G. van Eck , Pieter Kleer , Johan S.H. van Leeuwaarden This is my paper · ORCID
classification math.OC MSC 90C4791B2490C15
keywords competitiveratiorobustmonopolypricingdistributionallyoptimizationmaximinanalysisfractionalmomentsvariancepostedpriceambiguityset
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

This paper solves a robust pricing problem: a monopolist who knows only the mean, a dispersion measure, and an upper bound on customers' willingness to pay must choose a single fixed price without knowing the valuation distribution. The criterion is the competitive ratio, revenue at the chosen price divided by the revenue a fully informed seller would obtain, and the seller maximizes this ratio against the worst-case distribution consistent with the summary statistics. The main result reduces the worst-case ratio to a simple minimum of two quantities built from tight tail-probability bounds and a conditional-expectation bound, and then determines the price that maximizes this ratio. For variance, the optimal price is closed form: a low price below a dispersion threshold and a higher price above it. The paper also shows that the adversarial worst-case distribution is the same for the competitive ratio as for expected revenue, and extends the analysis to fractional dispersion moments.

What carries the argument

The workhorse is the ratio decomposition in Theorem 3.1, which expresses the worst-case competitive ratio as the minimum of a tail-probability ratio and a price-to-conditional-expectation ratio. The proof identifies two- and three-point extremal distributions, parameterized by the price $p$, that attain tight bounds for each of the three quantities in the decomposition, and it certifies those bounds with dual solutions from semi-infinite linear programming. A second load-bearing object is the best-case revenue function $g(p)=p\,\sup_{P\in\mathcal{P}}\mathbb{P}(X\ge p)$; Lemma 4.3 proves this function is non-decreasing on $(0,\tau_2]$, which is what turns the $p^*\le p$ case into the tail-ratio lower bound. The threshold $\tau_2$ is the right support point of the two-point distribution on $\{0,\tau_2\}$ and separates the regime where the adversarial distribution is two-point from the regime where it is three-point.

What would settle it

Choose a strictly convex $\varphi$ such as $\varphi(x)=x^q$ with parameters for which $\mathcal{P}$ is nonempty and compute $g(p)$ on $(0,\tau_2]$; if $g$ decreases on any interval, evaluate $\inf_{P\in\mathcal{P}}\mathrm{CR}(p,P)$ directly at a price in that interval and compare it with the right-hand side of the decomposition, since any gap would refute Theorem 3.1. In the variance case, evaluate $R(p_l^*,\sigma)$ and $R(p_h^*,\sigma)$ around the claimed threshold $\sigma^*$; if either is beaten by a third candidate price, Theorem 5.1 fails.

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Extended reading notes

Core claim

The central discovery is a complete solution of the maximin competitive-ratio problem for the ambiguity set $\mathcal{P}(\mu,s,\beta,\varphi)$, the distributions on $[0,\beta]$ with mean $\mu$ and dispersion $\mathbb{E}_P[\varphi(X)]=s$. Theorem 3.1 states that for any fixed price $p$ the tight worst-case ratio equals $\min\left(\frac{\inf_{P\in\mathcal{P}}\mathbb{P}(X\ge p)}{\sup_{P\in\mathcal{P}}\mathbb{P}(X\ge p)}, \frac{p}{\sup_{P\in\mathcal{P}}\mathbb{E}(X\mid X\ge p)}\right)$, so the fractional ratio problem decouples into three one-sided moment problems. Theorem 3.2 adds that the same limiting extremal distribution realizes both the worst-case ratio and the worst-case expected revenue. With $\varphi(x)=x^2$, Theorem 5.1 gives the maximin price explicitly: $p_l^*$ for $\sigma\le\sigma^*$ and $p_h^*$ for $\sigma\ge\sigma^*$, a discontinuous shift from a low-price, high-conversion strategy to a high-price, selective strategy. With $\varphi(x)=x^q$ for $q>1$, the optimal price is one of four candidate prices, and introducing a unit cost only shifts the ambiguity set.

Load-bearing premise

The whole derivation hinges on the assumption that the best-case revenue function $g(p)=p\,\sup_{P\in\mathcal{P}}\mathbb{P}(X\ge p)$ never decreases while $p$ is at most $\tau_2$; if a permitted dispersion measure made it dip, the tight competitive-ratio formula and the prices derived from it would not follow.

Editorial extensions

If this is right

  • In the mean-variance-cap model, the optimal robust price is $p_l^*$ for $\sigma\le\sigma^*$ and $p_h^*$ for $\sigma\ge\sigma^*$, so a seller switches abruptly from a low-price mass-market strategy to a high-price niche strategy as dispersion crosses $\sigma^*$.
  • Without the cap $\beta$, the high-price regime vanishes and the same low-price formula $p_l^*$ remains optimal, with the worst-case ratio decaying to zero as variance grows; knowing the cap is therefore valuable.
  • The competitive-ratio optimal prices are systematically more moderate than expected-revenue optimal prices: $\pi_l^*<p_l^*$ in the low regime and $\pi_h^*>p_h^*$ in the high regime.
  • For fractional moments $\varphi(x)=x^q$ with $q>1$, the optimal price is one of four candidate prices, and larger $q$ widens the range of dispersions over which the strategy applies and enlarges the price jump.
  • A positive unit cost $c$ can be absorbed by shifting the valuation variable, so the same pricing structure applies to a seller with a production cost.

Reading between the lines

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

  • One practical consequence not developed in the paper: $\sigma^*$ is a dataable market-risk threshold, so estimating $\mu$, $\sigma$, and $\beta$ from transaction data would tell a seller directly whether to price low or high.
  • Because the same adversarial distribution governs expected revenue and competitive ratio, robust expected-revenue algorithms for other moment sets may be reusable for ratio objectives; the paper demonstrates this only for the current ambiguity set.
  • For $1<q<2$, fractional moments permit heavier tails, and the four-candidate price set suggests a monotone relationship between the tail exponent $q$ and the robust price; this is a testable extension, not a claim of the paper.
  • The paper's limiting-distribution caveat implies that exact attainment of the worst-case ratio may require an approximating sequence, and the paper does not estimate how quickly such a sequence converges; in practice a seller approaches the bound rather than hitting it.
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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 / 6 minor

Summary. The paper studies deterministic monopoly pricing when the seller knows only the mean mu, a dispersion value s = E phi(X), and an upper bound beta on the support of the valuation X, and must choose a deterministic price maximizing the worst-case competitive ratio CR(p,P) = p P(X >= p) / sup_{t>0} t P(X >= t). The main result, Theorem 3.1, expresses inf_{P in P(mu,s,beta,phi)} CR(p,P) as the minimum of the worst-to-best tail-probability ratio and the ratio p / sup_P E(X | X >= p). Propositions 4.1-4.3 give explicit formulas for the three ingredients using two- and three-point extremal distributions; Theorem 3.2 claims that the same limiting distribution is worst-case for expected revenue. Section 5 specializes to variance, with a closed-form optimal price switching between a low price and a high price at an implicit dispersion threshold sigma*, and to fractional moments, with the optimum restricted to four candidates. Proofs use semi-infinite LP duality and a novel monotonicity lemma for the best-case revenue function.

Significance. If the proofs are correct, this is a substantial contribution to distributionally robust pricing. It generalizes the mean-variance competitive-ratio analysis of Giannakopoulos et al. and Chen et al. to general convex dispersion measures and to a known maximum valuation, provides explicit extremal distributions with LP-duality certificates, and yields a closed-form pricing policy for the variance case. The observation that the worst-case distribution for the competitive ratio coincides with the worst-case distribution for expected revenue is a clean and non-obvious structural result. The paper also demonstrates a practical discontinuity in the optimal price as a function of dispersion, contrasting with the unbounded-support benchmark.

major comments (2)
  1. [Section 4.3 (proof of Theorem 3.1)] The upper-bound step evaluates the competitive ratio at P* = P*(p-) and states 'consider P* in P'. Since p- is a left-limit symbol rather than an actual support point, P*(p-) is not a distribution in P, and for any fixed distribution in P the displayed equalities REV(p,P*) = p * inf_P P(X >= p) and OPT(P*) = max{p * sup_P P(X >= p), y(p) * inf_P P(X >= p)} do not hold: the mass at p contributes to P(X >= p) in the limit. The proof needs an explicit limiting argument with a sequence P_n in P, for instance P_n = P*(p_n) with p_n increasing to p. The footnote in Section 3.2 defines asymptotic attainment, but the body proof does not use it; as written, the upper bound in (5) is not established.
  2. [Section 6 and Lemma 4.3 / Appendix E] The Conclusions state: 'This function must be non-increasing up to a certain value for the proof of Theorem 3.1 to work.' This is the opposite of what the proof in Section 4.3 requires: the case p* <= p needs sup_{t in (0,p]} g(t) = g(p), i.e., g non-decreasing, and Appendix E is written to prove non-decreasing. Because Lemma 4.3 is the only place where the maximum-valuation bound beta enters the lower-bound branch, this contradiction is not merely cosmetic. The manuscript must correct the Conclusions statement and should present the proof of Lemma 4.3 in a more verifiable form (for example, by listing the sign of each factor in (59)-(62)); as it stands, a reader cannot tell which direction is intended.
minor comments (6)
  1. [Section 5.2, Theorem 5.3(iii)] The defining equation for pbar_h uses the symbol pbar_l on the left-hand side; it should presumably be an equation in the variable that is supposed to be pbar_h. Please correct the displayed formula.
  2. [Section 4.2 (upper-bound dispersion paragraph)] This paragraph states several claims 'without proof' and says the full derivation is left to the interested reader. Since the variant with an upper bound on dispersion is not used in the main theorems, it should be marked as a remark; if it is intended as a result, the proof should be supplied.
  3. [Equations (33) and (49)] The systems displayed in (33) and (49) are missing a plus sign in the second equation ('lambda0 lambda1 alpha(p)'); please fix these typos.
  4. [Equation (8) and Section 3.1] The two-point distribution P2 in (8) uses the left-limit notation p- before it is defined, and the text does not explain that (8) is a limiting construction; please define the notation at first use.
  5. [Theorem 5.3 proof] The inference 'g1a(0+) > g1b(0) implies g1a(tau1) > g1b(tau1)' is only valid together with the already assumed inequality tau1 < pbar_l; please spell out this dependence.
  6. [Table 2] The '-' entries for beta = 1 and mu = 1 correspond to an empty ambiguity set by Lemma 4.1 (tau2 > beta); the table caption should state this.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: Theorem 3.1 is proved from independent moment-tail bounds and Lemma 4.3 rather than assumed; same-author citations are independent support.

full rationale

Walking the paper's derivation chain, I find no step in which a claimed result is equivalent to its own input by construction. Theorem 3.1 is derived, not assumed: Propositions 4.1–4.3 solve independent extremal problems for sup/inf tail probabilities and the conditional expectation, and the proof of Theorem 3.1 in Section 4.3 combines these with the explicit distribution P*(p-) for the upper bound and a case split p*≤p / p*>p for the lower bound. The p*≤p case uses Lemma 4.3, whose proof in Appendix E is a direct monotonicity argument from strict convexity of phi and the dispersion relation (22); it is not a restatement of Theorem 3.1. The pricing results in Section 5 then maximize the resulting closed-form competitive ratio, with no fitted parameter being renamed as a prediction. The same-author citations (Kleer et al., 2024; van Eck et al., 2024) supply two-point-distribution parametrization, conditional-expectation bounds, and a standard continuity fact; these are parameter-free moment results whose stated assumptions do not include the target competitive-ratio formula, so under the review rules they count as independent evidence rather than circularity. I do flag one manuscript inconsistency: Section 6 states that the best-case revenue function 'must be non-increasing up to a certain value for the proof of Theorem 3.1 to work,' while Lemma 4.3 and Appendix E prove it is non-decreasing. This is a correctness/typo issue, not a circularity. I also note the paper leaves an upper-bound-dispersion variant 'Without proof' and treats P*(p-) as a limiting left-limit object; these are rigor gaps, not circular reductions. Overall, the central derivation is self-contained against the standard moment-bound benchmarks, so the circularity score is 0.

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

The central claim rests on the moment-constrained model, convex analysis of two- and three-point distributions, and semi-infinite LP duality. No parameters are fitted to data; μ, s and β are inputs. The main external load is carried by same-author moment-space results, especially for the two-point parameterization and conditional-expectation bounds.

assumptions (5)
  • domain assumption φ is strictly convex, differentiable and nonnegative on [0,β]; dispersion constraint is exact equality E φ(X)=s.
    This is the model. Strict convexity is used to parameterize two-point distributions via equation (19), to sign dual variables in Propositions 4.1 and 4.2, and to prove monotonicity in Appendix E. The paper notes in Section 4.2 that if the dispersion constraint is an upper bound rather than equality, the piecewise solution changes for finite β and is only sketched.
  • domain assumption Ambiguity set is non-empty with μ≤τ2≤β and β>τ2.
    Lemma 4.1 shows non-emptiness is equivalent to μ≤τ2≤β; the paper then restricts to β>τ2 to avoid trivial cases. All subsequent extremal distributions require a feasible set.
  • standard math Semi-infinite linear programming strong duality and existence of extremal three-point distributions.
    Used to prove Propositions 4.1, 4.2, and Lemma 4.1 through citations to Rogosinski (1958), Shapiro (2001), and Popescu (2005).
  • domain assumption For every p in the relevant ranges, a unique two-point distribution solving (19) exists, and sup E(X|X≥p)=α(p) for p∈(0,τ1].
    Imported from (Kleer et al., 2024, Proposition 2.2, 2.3, Theorem 3.1) and (van Eekelen, 2023, Proposition 3); authors overlap with this paper. These facts support Proposition 4.3 and the construction of P2*(p).
  • domain assumption Worst-case distributions may be attained only asymptotically as left-limits p-.
    Theorem 3.2 and Propositions 4.3 use limiting distributions, and the paper explicitly defines asymptotic attainment. If true attainment by an actual distribution were required, the statement of Theorem 3.2 would not hold as written.

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Pith. "Pith review of Robust Competitive Ratio for Deterministic Monopoly Pricing." pith.science (2026). https://pith.science/paper/VWCLQRPT

@misc{pith2026250906619,
  author       = {Pith},
  title        = {Pith review of: Robust Competitive Ratio for Deterministic Monopoly Pricing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWCLQRPT}},
  note         = {Machine review of arXiv:2509.06619}
}
read the original abstract

We study deterministic monopoly pricing under partial knowledge of the market, where the seller has access only to summary statistics of the valuation distribution, such as the mean, dispersion, and maximum value. Using tools from distributionally robust optimization and max-min analysis, we evaluate pricing strategies based on their competitive ratio (CR). We characterize the worst-case market scenario consistent with the available information and provide a complete solution for minimizing the CR. Our analysis also covers optimal pricing under various measures of dispersion, including variance and fractional moments. Interestingly, we find that the worst-case market for CR coincides with that for expected revenue. Using proof techniques tailored to the CR framework, we further examine how dispersion and maximum valuation influence optimal deterministic pricing. These results offer practical guidance for setting robust prices when market information is limited.

Figures

Figures reproduced from arXiv: 2509.06619 by the authors.

Figure 1
Figure 1. Function p 7→ R(p, σ) with µ = 0.5 and β = 1. selective segment of the market. The threshold σ ∗ represents the critical level of market risk at which the seller shifts from a mass-market approach to a niche market strategy. Although this shift can be intuitively understood, the specific prices and threshold σ ∗ are influenced by subtle interactions among all the model parameters. One part of the optimal price in Th… view at source ↗
Figure 2
Figure 2. illustrates how optimal prices differ under the expected revenue and competitive ratio objectives and, specifically, how the competitive ratio gives rise to more moderate pricing strategies. While both objectives involve the same underlying trade-off between price and conversion, they differ in how they prioritize these competing factors. This difference does not stem from Nature’s behavior, as by Theorem 3.2, both … view at source ↗
Figure 3
Figure 3. Optimal price and competitive ratio for various [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

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Works this paper leans on

45 extracted references · 39 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter doi edition editor eid howpublished institution isbn issn journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1...

    work page

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in "" FUNCTION format.date year ...

  3. [3]

    Allouah, A., Bahamou, A., and Besbes, O. (2023). Optimal pricing with a single point. Management Science , 69(10):5866--5882

    work page 2023

  4. [4]

    and Baysan, M

    Averbakh, I. and Baysan, M. (2012). Semi-online two-level supply chain scheduling problems. Journal of Scheduling , 15(3):381--390

    work page 2012

  5. [5]

    and Micali, S

    Azar, P. and Micali, S. (2012). Optimal parametric auctions. Technical report, MIT-CSAIL-TR-2012-015, MIT CSAIL

    work page 2012

  6. [6]

    Azar, P. D. and Micali, S. (2013). Parametric digital auctions. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science , pages 231--232

    work page 2013

  7. [7]

    and Walsh, W

    Babaioff, M. and Walsh, W. E. (2005). Incentive-compatible, budget-balanced, yet highly efficient auctions for supply chain formation. Decision Support Systems , 39(1):123--149

    work page 2005

  8. [8]

    Bahamou, A., Besbes, O., and Mouchtaki, O. (2024). Fast revenue maximization. arXiv preprint arXiv:2407.07316

    work page Pith review arXiv 2024

Show all 45 references
  1. [9]

    and Schlag, K

    Bergemann, D. and Schlag, K. H. (2008). Pricing without priors. Journal of the European Economic Association , 6(2-3):560--569

    2008

  2. [10]

    and Schlag, K

    Bergemann, D. and Schlag, K. H. (2011). Robust monopoly pricing. Journal of Economic Theory , 146(6):2527--2543

    2011

  3. [11]

    and Elmachtoub, A

    Bergquist, J. and Elmachtoub, A. N. (2023). Static pricing guarantees for queueing systems. arXiv preprint arXiv:2305.09168

    2023

  4. [12]

    and El-Yaniv, R

    Borodin, A. and El-Yaniv, R. (2005). Online computation and competitive analysis . Cambridge University Press

    2005

  5. [13]

    F., Kos, N., Messner, M., Monteiro, P., and Moreira, H

    Carrasco, V., Luz, V. F., Kos, N., Messner, M., Monteiro, P., and Moreira, H. (2018). Optimal selling mechanisms under moment conditions. Journal of Economic Theory , 177:245--279

    2018

  6. [14]

    Carroll, G. (2019). Robustness in mechanism design and contracting. Annual Review of Economics , 11:139--166

    2019

  7. [15]

    Chen, H., Hu, M., and Perakis, G. (2022). Distribution-free pricing. Manufacturing & Service Operations Management , 24(4):1939--1958

    2022

  8. [16]

    Chen, Z., Yu, P., and Haskell, W. B. (2019). Distributionally robust optimization for sequential decision-making. Optimization , 68(12):2397--2426

    2019

  9. [17]

    Das, B., Dhara, A., and Natarajan, K. (2021). On the heavy-tail behavior of the distributionally robust newsvendor. Operations Research , 69(4):1077--1099

    2021

  10. [18]

    and Heijnen, B

    De Schepper, A. and Heijnen, B. (1995). General restrictions on tail probabilities. Journal of Computational and Applied Mathematics , 64(1-2):177--188

    1995

  11. [19]

    Elmachtoub, A. N. and Levi, R. (2016). Supply chain management with online customer selection. Operations Research , 64(2):458--473

    2016

  12. [20]

    N., Sheth, H., and Zhou, Y

    Elmachtoub, A. N., Sheth, H., and Zhou, Y. (2023). Simple policies for joint pricing and inventory management. Available at SSRN 4470538

    2023

  13. [21]

    Elmachtoub, A. N. and Shi, J. (2023). The power of static pricing for reusable resources. arXiv preprint arXiv:2302.11723

    2023 arXiv

  14. [22]

    Eren, S. S. and Maglaras, C. (2010). Monopoly pricing with limited demand information. Journal of Revenue and Pricing Management , 9(1):23--48

    2010

  15. [23]

    Giannakopoulos, Y., Po c as, D., and Tsigonias-Dimitriadis, A. (2023). Robust revenue maximization under minimal statistical information. ACM Transactions on Economics and Computation , 10(3):1--34

    2023

  16. [24]

    and Wagner, M

    Jaillet, P. and Wagner, M. R. (2008). Generalized online routing: New competitive ratios, resource augmentation, and asymptotic analyses. Operations Research , 56(3):745--757

    2008

  17. [25]

    S., and Verseveldt, B

    Kleer, P., van Leeuwaarden, J. S., and Verseveldt, B. (2024). Distribution-free expectation operators for robust pricing and stocking with heavy-tailed demand. arXiv preprint arXiv:2409.17962

    2024 arXiv

  18. [26]

    and Messner, M

    Kos, N. and Messner, M. (2015). Selling to the mean. Available at SSRN 2632014

    2015

  19. [27]

    and Muir, E

    Loertscher, S. and Muir, E. V. (2022). Monopoly pricing, optimal randomization, and resale. Journal of Political Economy , 130(3):566--635

    2022

  20. [28]

    and Muir, E

    Loertscher, S. and Muir, E. V. (2024). Market power, randomization and regulation. International Journal of Industrial Organization , 96:103081

    2024

  21. [29]

    Lucier, B. (2017). An economic view of prophet inequalities. ACM SIGecom Exchanges , 16(1):24--47

    2017

  22. [30]

    Myerson, R. B. (1981). Optimal auction design. Mathematics of Operations Research , 6(1):58--73

    1981

  23. [31]

    Popescu, I. (2005). A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. Mathematics of Operations Research , 30(3):632--657

    2005

  24. [32]

    and Mehrotra, S

    Rahimian, H. and Mehrotra, S. (2019). Distributionally robust optimization: A review. arXiv preprint arXiv:1908.05659

    2019 arXiv

  25. [33]

    and Zeckhauser, R

    Riley, J. and Zeckhauser, R. (1983). Optimal selling strategies: When to haggle, when to hold firm. The Quarterly Journal of Economics , 98(2):267--289

    1983

  26. [34]

    Rogosinski, W. W. (1958). Moments of non-negative mass. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences , 245(1240):1--27

    1958

  27. [35]

    Roos, E., Brekelmans, R., Van Eekelen, W., Den Hertog, D., and Van Leeuwaarden, J. S. (2022). Tight tail probability bounds for distribution-free decision making. European Journal of Operational Research , 299(3):931--944

    2022

  28. [36]

    Scarf, H. E. (1958). A min-max solution of an inventory problem. In Arrow, K. J., Karlin, S., and Scarf, H. E., editors, Studies in the Mathematical Theory of Inventory and Production . Stanford University Press, Palo Alto, CA

    1958

  29. [37]

    Shapiro, A. (2001). On duality theory of conic linear problems. In Goberna, M. A. and Lopez, M. A., editors, Semi-Infinite Programming: Recent Advances , pages 135--165. Kluwer Academic Publishers, Dordrecht, The Netherlands

    2001

  30. [38]

    Suzdaltsev, A. (2018). Distributionally robust pricing in auctions. Available at SSRN 3119305

    2018

  31. [39]

    Terlizzese, D. (2008). Relative minimax

    2008

  32. [40]

    S., Kleer, P., and van Leeuwaarden, J

    van Eck, T. S., Kleer, P., and van Leeuwaarden, J. S. (2024). Distributionally robust monopoly pricing: Switching from low to high prices in volatile markets. arXiv preprint arXiv:2403.19486

    2024

  33. [41]

    van Eekelen, W. J. (2023). A generalized moment approach to sharp bounds for conditional expectations. arXiv preprint arXiv:2401.00090

    2023 arXiv

  34. [42]

    Wang, S. (2024). The power of simple menus in robust selling mechanisms. Management Science

    2024

  35. [43]

    Wang, S., Liu, S., and Zhang, J. (2024). Minimax regret robust screening with moment information. Manufacturing & Service Operations Management

    2024

  36. [44]

    Weber, T. A. (2024). Monopoly pricing with unknown demand. The Scandinavian Journal of Economics

    2024

  37. [45]

    Wilson, R. (1989). Game theoretic analysis of trading. In Advances in Economic Theory: Fifth World Congress , number 12, page 33. CUP Archive

    1989

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Reviewed August 15, 2026 · model on record in the stance chip above.