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Approximately Optimal Mechanism Design for Competing Sellers

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Pith's one-line read A first mover with one lottery secures a quarter of monopoly revenue

desk verdict A new and mostly sound result: constant-factor Stackelberg approximation with lotteries, with one non-central theorem (3.11) that overclaims its domain. read the letter →

arxiv 2505.19453 v1 pith:PWJ5MQ6B submitted 2025-05-26 cs.GT

Brendan Lucier , Raghuvansh R. Saxena This is my paper · ORCID
classification cs.GT MSC 91B2691A65
keywords mechanismdesigncompetingsellersStackelbergequilibriumsingle-lotterymechanismsposted-priceduopolyrevenuevirtualvaluesapproximatemaximization
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 studies two sellers who each choose an arbitrary sales mechanism—possibly including lotteries—to sell identical goods to one buyer who can visit both sellers in sequence. It tries to show that in the Stackelberg version of this game, the first-moving seller can commit to a single-lottery menu and guarantee herself at least one quarter of the revenue a monopolist would earn, for any value distribution that is regular or has decreasing marginal revenue (DMR). The argument rests on a structural lemma: against any single-lottery mechanism, the second seller has a best response that is a posted take-it-or-leave-it price, which turns Alice's design problem into a comparison of fixed prices. The paper also shows the guarantee cannot be pushed above a $1/e$ fraction of monopoly revenue at any Stackelberg equilibrium, and that in simultaneous-move Nash equilibrium both sellers may earn zero even when a monopolist could extract full surplus. If correct, this establishes that randomization together with commitment can overcome Bertrand-style competition, while neither ingredient alone suffices.

What carries the argument

The load-bearing object is the auxiliary distribution $D_s$ constructed for a fixed threshold $s$, the buyer type that separates those who visit Bob first from those who visit Alice first. When Alice uses a single lottery with price $p$ and allocation probability $z$, the paper shifts the density of types in $[p,s]$ to modified values $a+(1-z)v$, scales down the density above $s$ by $1-z$, and adds a point mass at zero; for any Bob mechanism with threshold $s$, Bob's duopoly revenue equals the revenue a monopolist would earn from $D_s$. This reduction, combined with the standard revenue-as-virtual-welfare identity, shows that Bob's optimal mechanism has allocations concentrated on $\{0,x\}$ for types below the threshold and that its revenue is bounded by a convex combination of posted-price revenues, forcing a posted price to be a best response. The proof also introduces 'bottom proper' mechanisms, where every Bob-first type takes the same lottery or nothing, and shows any Bob mechanism can be replaced by a bottom proper one that earns at least as much. The regularity or DMR assumption is exactly what controls the virtual values of the distorted distribution.

What would settle it

Find a regular or DMR distribution and a single-lottery mechanism for Alice such that Bob's revenue-maximizing response beats every posted price; since Lemma 1.2 claims a posted price is always a best response, such an instance would disprove the key structural result. Concretely, on a finely discretized truncated exponential distribution, one could solve the linear program for Bob's optimal mechanism against the half-price lottery and compare its revenue with the best posted price.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for any buyer-value distribution that is regular or has decreasing marginal revenue, there is a Stackelberg equilibrium in which the principal seller, Alice, earns at least $\mathrm{Rev}(D)/4$, where $\mathrm{Rev}(D)$ is the maximum expected revenue of a monopolist. The equilibrium is built from a single lottery: Alice charges price $p$ for a probability $1/2$ of the good, where $p$ is the smallest price at which the monopoly revenue curve reaches half its maximum. Against that menu, the paper proves Bob's best response is the revenue-maximizing posted price, and Alice's revenue is at least $\Gamma(v)/4$ while Bob's is at least $\Gamma(v)/2$. The companion negative results are Theorem 1.3, which caps Alice's Stackelberg revenue at $\mathrm{Rev}(D)/e$ even with arbitrary mechanisms, and Theorem 1.4, which shows that with a point-mass buyer type every pure Nash equilibrium gives both sellers zero revenue.

Load-bearing premise

The buyer must finish the first seller's mechanism completely before choosing whether to approach the second; if a buyer could switch mid-mechanism or run the two mechanisms simultaneously, the paper's reduction and its $1/4$ revenue guarantee are not established.

Editorial extensions

If this is right

  • Alice's quarter-of-monopoly guarantee is achieved by a menu with a single lottery at price $p$ and allocation probability $1/2$, so the optimal first-mover strategy has no menu complexity.
  • Bob can always be assumed to answer Alice's single-lottery menu with a take-it-or-leave-it price, which reduces a mechanism-design competition to comparison of two prices.
  • The factor 4 is tight among single-lottery menus for Alice, and the factor $e$ is tight when Bob is restricted to posted prices; the true Stackelberg approximation factor for arbitrary mechanisms lies between $e$ and 4.
  • Without first-mover commitment, the approximation fails: with a fixed buyer value, every pure Nash equilibrium leaves both sellers with zero revenue.
  • The $1/4$ guarantee holds for every regular or DMR distribution, and the $1/e$ posted-price result holds for every distribution, even non-regular ones.

Reading between the lines

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

  • Inference: The posted-price best-response lemma is proved only for single-lottery Alice menus; if a similar collapse to posted prices holds for larger menus, the upper bound of $1/e$ might be reachable by a simple first-mover menu, which would close the gap to 4. The paper does not claim this.
  • Inference: The auxiliary-distribution construction depends on the buyer resolving one mechanism fully before the next; a model that allows interleaving would require a different distortion argument, and the $1/4$ bound may fail there.
  • Inference: The construction behind the $1/e$ guarantee—a menu that makes Bob indifferent across a range of posted prices—suggests a general template for first movers facing price-undercutting rivals: flatten the rival's best-response revenue over the monopoly price interval.
  • Inference: Because the positive result needs only regular or DMR priors while the posted-price $1/e$ result needs no regularity, the technical bottleneck is the virtual-value control of the distorted distribution, not competition per se.
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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

0 major / 5 minor

Summary. The paper studies a sequential-move duopoly in which two sellers each choose an arbitrary mechanism (equivalently, a pricing function over lotteries) to sell an identical good to a single buyer with a known value distribution. The main result, Theorem 1.1, asserts that in a Stackelberg equilibrium the committed leader (Alice) can guarantee at least Rev(D)/4, the quarter of the optimal monopoly revenue, whenever the distribution is regular or DMR; moreover this is achieved by a single-lottery mechanism. The central structural lemma, Lemma 3.2, states that any best response to such a lottery is a posted price. The paper also gives a 1/e upper bound for point-mass distributions, shows that the factor 4 is tight within the single-lottery class, proves a 1/e approximation when Bob is restricted to posted prices, and demonstrates that the positive result fails for pure Nash equilibria. The technical development is extensive, with the Stackelberg proof built on an auxiliary-distribution reduction (Section 3.2) and a revenue inequality for a single seller (Lemma B.15).

Significance. If the central claims hold, this is an interesting and nontrivial contribution: it shows that simple randomized mechanisms can restore a constant-fraction of monopoly revenue in a natural model of seller competition with commitment, and it identifies a clean structural property of best responses to lotteries. The paper is careful about its sequential timing assumption and acknowledges that simultaneous or partial-participation models are not covered. The proof machinery is detailed, with omitted steps supplied in appendices; the auxiliary-distribution construction and the derivation of the 1/4 bound are particularly elegant. The 1/e upper bound and the tightness results give a fairly complete picture of the Stackelberg payoff, modulo the gap in the stated generality of Theorem 3.11 discussed below.

minor comments (5)
  1. [Section 3.6, Theorem 3.11 and Eq. (7)] Theorem 3.11 is stated for any distribution D, but the proposed mechanism A in Eq. (7) uses the inverse revenue curve Γ_D^{-1}, and the proof of Lemma B.18 explicitly asserts that Γ_D^{-1}(·) is monotone increasing. This monotonicity is not guaranteed for an arbitrary distribution, and without it the pricing function A may not be proper (convex) or even well defined on the required range. The theorem should either be restricted to distributions for which Γ_D is monotone on the relevant interval (for example, increasing hazard-rate distributions), or the proof must be modified to avoid relying on this assumption. This is not load-bearing for Theorem 1.1, but it is a correctness issue in a stated theorem.
  2. [Section 3.5, proof of Theorem 3.9] In the case where the buyer buys from Alice first, the proof contains the sentence "we must have B(xB) ≤ xB = 1", but xB is the buyer's allocation probability from Bob and need not equal 1. This appears to be a typo (likely "xB ≤ 1") and should be corrected for clarity.
  3. [Section 3.5, line after Eq. (3)] The sentence "The buyer breaks ties in favor of buying from Bob first, then in favor of larger values of x" introduces a tie-breaking rule on the allocation quantity that is not stated in Section 2. The relation between this rule and the earlier tie-breaking convention ("first in favor of maximizing expected allocation from Bob, then from Alice") should be made explicit.
  4. [Section 2, Timing] The model requires the buyer to fully resolve one seller's mechanism before interacting with the other. The paper correctly notes in Section 5 that this is a modeling assumption and that other timing protocols are outside the scope, but it would be helpful to state this limitation near the definition of the timing in Section 2, so that the scope of the main theorems is unambiguous from the outset.
  5. [Throughout] There are a few typographical errors, e.g., "equilibirum" in Section 3.5 and "whehter" in the proof of Theorem 3.9. These should be fixed in a final revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1's derivation is self-contained and independent of its conclusions.

full rationale

The paper's main claim, Theorem 1.1, is derived by constructing an explicit strategy for Alice (a single lottery with probability z=1/2 and price p satisfying Gamma_D(p)=Rev(D)/2) and then proving that Bob has a best response that is a posted price (Lemma 3.2). The proof of Lemma 3.2 rests on the auxiliary distribution D_s defined in Equation (4), which is a transformation of the original distribution D with a point mass at 0. This transformation is not defined in terms of the target revenue bound; it is a device to map Bob's distorted best-response problem onto a monopolist problem. The inequalities in Lemmas 3.7 and 3.8 use standard Myerson virtual-value machinery and monotonicity arguments; the virtual-value decomposition is quoted from the classic [Mye81] result, which is external and not self-citational. No parameter is fitted to the conclusion: p is chosen from the revenue curve of D, and the 1/4 bound follows algebraically from the construction, not by assuming the bound. The upper bound (Theorem 3.9) uses Lemma 3.10, which is proven directly from the lower convex envelope of Alice's mechanism and Bob's best-response behavior; it does not import the result being proved. Theorem 3.11's construction A(x) is defined via Gamma_D^{-1}, but the paper explicitly proves the resulting revenue equality in Lemma B.20; no step reduces an equation to itself by construction. The model's sequential timing is stated as an explicit assumption and its limitation is acknowledged in Section 5; this affects external applicability, not circularity. The only identified gap, an unproved monotonicity of Gamma_D^{-1} used parenthetically in Lemma B.18, is a correctness concern, not a circularity: it does not make any theorem equivalent to its own premise. Therefore, the derivation chain is self-contained, and any self-citations (e.g., [CBL24] in related work) are contextual and not load-bearing.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No empirical data or fitted constants appear; the only distribution-dependent quantities, such as the lottery price p defined by Gamma_D(p)=Gamma_D(v)/2 in Theorem 3.1, are explicit construction parameters rather than free parameters. The main premises are the standard Myerson and taxation results, the sequential-resolution timing, the tie-breaking convention, and the regularity or DMR conditions. The unproved monotonicity of Gamma_D^{-1} in Section 3.6 is the notable load-bearing assumption for Theorem 3.11.

assumptions (6)
  • standard math Myerson's revenue equals expected virtual welfare (Proposition 2.3).
    Used throughout to convert revenue to virtual welfare integrals; standard in mechanism design, cited to Myerson 1981.
  • standard math Taxation principle: any mechanism is strategically equivalent to a proper pricing function (Observation 2.1).
    Proven in Appendix A; reduces general mechanisms to menus of (allocation probability, price).
  • domain assumption The value distribution D is atomless with well-defined density and admits a finite revenue-maximizing price; the main theorem is stated for regular or DMR distributions.
    Section 2 assumptions; regularity or DMR is used in Lemma 3.8 to control the virtual values of the auxiliary distribution.
  • domain assumption The buyer has quasi-linear, risk-neutral utility, can visit the two sellers in either order, and must resolve one mechanism completely before engaging the other.
    Section 2 Timing; this sequential structure is load-bearing for the derived revenue expressions and is acknowledged as a limitation in Section 5.
  • ad hoc to paper Tie-breaking rule: the buyer first maximizes expected allocation from Bob, then from Alice.
    Section 2; makes the revenue functions single-valued and defines AB-start(B). Changing this convention could change equilibrium revenue.
  • ad hoc to paper For Theorem 3.11, the inverse revenue curve Gamma_D^{-1} is monotone increasing on the relevant range.
    Section 3.6 Eq. (7) and Lemma B.18; asserted without proof for an arbitrary distribution D, while the theorem claims all distributions. This is the flagged gap.

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Pith. "Pith review of Approximately Optimal Mechanism Design for Competing Sellers." pith.science (2026). https://pith.science/paper/PWJ5MQ6B

@misc{pith2026250519453,
  author       = {Pith},
  title        = {Pith review of: Approximately Optimal Mechanism Design for Competing Sellers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWJ5MQ6B}},
  note         = {Machine review of arXiv:2505.19453}
}
read the original abstract

Two sellers compete to sell identical products to a single buyer. Each seller chooses an arbitrary mechanism, possibly involving lotteries, to sell their product. The utility-maximizing buyer can choose to participate in one or both mechanisms, resolving them in either order. Given a common prior over buyer values, how should the sellers design their mechanisms to maximize their respective revenues? We first consider a Stackelberg setting where one seller (Alice) commits to her mechanism and the other seller (Bob) best-responds. We show how to construct a simple and approximately-optimal single-lottery mechanism for Alice that guarantees her a quarter of the optimal monopolist's revenue, for any regular distribution. Along the way we prove a structural result: for any single-lottery mechanism of Alice, there will always be a best response mechanism for Bob consisting of a single take-it-or-leave-it price. We also show that no mechanism (single-lottery or otherwise) can guarantee Alice more than a 1/e fraction of the monopolist revenue. Finally, we show that our approximation result does not extend to Nash equilibrium: there exist instances in which a monopolist could extract full surplus, but neither competing seller obtains positive revenue at any equilibrium choice of mechanisms.

Figures

Figures reproduced from arXiv: 2505.19453 by the authors.

Figure 1
Figure 1. Constructing the distribution Ds (red) from the distribution D (black). The density below p is unchanged, the density in the interval [p, s] is squished into a smaller interval [p, a + (1 − z) · s], and the density above s is scaled down by a factor of 1 − z with the remaining probability moved to an atom at 0. 3.3 Special Case of Lemma 3.2: Bottom Proper Mechanisms We now define a special class of mechanisms that w… view at source ↗
Figure 2
Figure 2. Allocation rule of a general mechanism (black) and the corresponding bottom [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

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Reference graph

Works this paper leans on

38 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [1]

    Approximating gains from trade in two-sided markets via simple mechanisms

    Johannes Brustle, Yang Cai, Fa Wu, and Mingfei Zhao. Approximating gains from trade in two-sided markets via simple mechanisms. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017 , pages 589--590, 2017

    work page 2017

  2. [2]

    Bertrand Networks

    Moshe Babaioff, Brendan Lucier, and Noam Nisan. Bertrand networks. arXiv preprint arXiv:1304.6806 , 2013

    work page Pith review arXiv 2013

  3. [3]

    Price competition in online combinatorial markets

    Moshe Babaioff, Noam Nisan, and Renato Paes Leme. Price competition in online combinatorial markets. In Proceedings of the 23rd international conference on World wide web , pages 711--722, 2014

    work page 2014

  4. [4]

    Imperfect competition in auction designs

    Roberto Burguet and J \'o zsef S \'a kovics. Imperfect competition in auction designs. International Economic Review , 40(1):231--247, 1999

    work page 1999

  5. [5]

    Matthew Weinberg

    Hedyeh Beyhaghi and S. Matthew Weinberg. Optimal (and benchmark-optimal) competition complexity for additive buyers over independent items. In Proceedings of the 51st ACM Symposium on Theory of Computing Conference (STOC) , 2019

    work page 2019

  6. [6]

    Bundling in oligopoly: Revenue maximization with single-item competitors

    Linda Cai, Moshe Babaioff, and Brendan Lucier. Bundling in oligopoly: Revenue maximization with single-item competitors. In Proceedings of the 25th ACM Conference on Economics and Computation , pages 465--465, 2024

    work page 2024

  7. [7]

    Matthew Weinberg

    Yang Cai, Nikhil Devanur, and S. Matthew Weinberg. A duality based unified approach to bayesian mechanism design. In Proceedings of the 48th ACM Conference on Theory of Computation(STOC) , 2016

    work page 2016

  8. [8]

    Researches into the mathematical principles of the theory of wealth

    Augustin Cournot. Researches into the mathematical principles of the theory of wealth. In Forerunners of Realizable Values Accounting in Financial Reporting , pages 3--13. Routledge, 2020

    work page 2020

Show all 38 references
  1. [9]

    Multiproduct duopolists

    Paul Champsaur and Jean-Charles Rochet. Multiproduct duopolists. Econometrica: Journal of the Econometric Society , pages 533--557, 1989

    1989

  2. [10]

    Bertrand competition in networks

    Shuchi Chawla and Tim Roughgarden. Bertrand competition in networks. In International Symposium on Algorithmic Game Theory , pages 70--82. Springer, 2008

    2008

  3. [11]

    Linda Cai and Raghuvansh R. Saxena. 99 \ In P \' e ter Bir \' o , Shuchi Chawla, and Federico Echenique, editors, EC '21: The 22nd ACM Conference on Economics and Computation, Budapest, Hungary, July 18-23, 2021 , pages 224--241. ACM , 2021

    2021

  4. [12]

    Optimal auctions with simultaneous and costly participation

    Gorkem Celik and Okan Yilankaya. Optimal auctions with simultaneous and costly participation. The BE Journal of Theoretical Economics , 9(1), 2009

    2009

  5. [13]

    Simple mechanisms for subadditive buyers via duality

    Yang Cai and Mingfei Zhao. Simple mechanisms for subadditive buyers via duality. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017 , pages 170--183, 2017

    2017

  6. [14]

    The Complexity of Optimal Mechanism Design

    Constantinos Daskalakis, Alan Deckelbaum, and Christos Tzamos. The Complexity of Optimal Mechanism Design . In the 25th ACM-SIAM Symposium on Discrete Algorithms (SODA) , 2014

    2014

  7. [15]

    Strong duality for a multiple-good monopolist

    Constantinos Daskalakis, Alan Deckelbaum, and Christos Tzamos. Strong duality for a multiple-good monopolist. Econometrica , 85(3):735--767, 2017

    2017

  8. [16]

    Devanur and S

    Nikhil R. Devanur and S. Matthew Weinberg. The optimal mechanism for selling to a budget constrained buyer: The general case. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017 , pages 39--40, 2017

    2017

  9. [17]

    Matthew Weinberg

    Alon Eden, Michal Feldman, Ophir Friedler, Inbal Talgam - Cohen, and S. Matthew Weinberg. The competition complexity of auctions: A bulow-klemperer result for multi-dimensional bidders. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, ...

    2017

  10. [18]

    Matthew Weinberg

    Alon Eden, Michal Feldman, Ophir Friedler, Inbal Talgam - Cohen, and S. Matthew Weinberg. A simple and approximately optimal mechanism for a buyer with complements: Abstract. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, Ju...

    2017

  11. [19]

    Designing and learning optimal finite support auctions

    Edith Elkind. Designing and learning optimal finite support auctions. In Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms , pages 736--745, 2007

    2007

  12. [20]

    The fedex problem

    Amos Fiat, Kira Goldner, Anna R Karlin, and Elias Koutsoupias. The fedex problem. In Proceedings of the 2016 ACM Conference on Economics and Computation , pages 21--22, 2016

    2016

  13. [21]

    The value of information concealment

    Hu Fu, Christopher Liaw, Pinyan Lu, and Zhihao Gavin Tang. The value of information concealment. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018 , pages 2533--2544, 2018

    2018

  14. [22]

    Revenue maximization for buyers with costly participation

    Yannai A Gonczarowski, Nicole Immorlica, Yingkai Li, and Brendan Lucier. Revenue maximization for buyers with costly participation. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 41--73. SIAM, 2024

    2024

  15. [23]

    On taxation and incentives: Further reflections on the limits to redistribution

    Roger Guesnerie. On taxation and incentives: Further reflections on the limits to redistribution . Inst. f \"u r Ges.-und Wirtschaftswiss., Wirtschaftstheoretische Abt., Univ., 1981

    1981

  16. [24]

    Straightforward individual incentive compatibility in large economies

    Peter J Hammond. Straightforward individual incentive compatibility in large economies. The Review of Economic Studies , 46(2):263--282, 1979

    1979

  17. [25]

    Annual survey of economic theory: the theory of monopoly

    John R Hicks. Annual survey of economic theory: the theory of monopoly. Econometrica: Journal of the Econometric Society , pages 1--20, 1935

    1935

  18. [26]

    The menu-size complexity of auctions

    Sergiu Hart and Noam Nisan. The menu-size complexity of auctions. In the 14th ACM Conference on Electronic Commerce (EC) , 2013

    2013

  19. [27]

    Sergiu Hart and Philip J. Reny. Maximizing Revenue with Multiple Goods: Nonmonotonicity and Other Observations . Theoretical Economics , 10(3):893--922, 2015

    2015

  20. [28]

    Advanced microeconomic theory

    Geoffrey Alexander Jehle. Advanced microeconomic theory . Pearson Education India, 2001

    2001

  21. [29]

    On the competition complexity of dynamic mechanism design

    Siqi Liu and Christos - Alexandros Psomas. On the competition complexity of dynamic mechanism design. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018 , pages 2008--2025, 2018

    2018

  22. [30]

    Mechanism design by competing sellers

    R Preston McAfee. Mechanism design by competing sellers. Econometrica: Journal of the econometric society , pages 1281--1312, 1993

    1993

  23. [31]

    Monopoly and product quality

    Michael Mussa and Sherwin Rosen. Monopoly and product quality. Journal of Economic theory , 18(2):301--317, 1978

    1978

  24. [32]

    A. M. Manelli and D. R. Vincent. Multidimensional Mechanism Design: Revenue Maximization and the Multiple-Good Monopoly . Journal of Economic Theory , 137(1):153--185, 2007

    2007

  25. [33]

    Optimal auction design

    Roger B Myerson. Optimal auction design. Mathematics of operations research , 1981

    1981

  26. [34]

    Optimal mechanism for selling two goods

    Gregory Pavlov. Optimal mechanism for selling two goods. The B.E. Journal of Theoretical Economics , 11(3), 2011

    2011

  27. [35]

    Competition among sellers who offer auctions instead of prices

    Michael Peters and Sergei Severinov. Competition among sellers who offer auctions instead of prices. Journal of Economic Theory , 75(1):141--179, 1997

    1997

  28. [36]

    Haggling over substitutes

    John Thanassoulis. Haggling over substitutes. Journal of Economic Theory , 117:217--245, 2004

    2004

  29. [37]

    Oligopoly Pricing: Old Ideas and New Tools

    X Vives. Oligopoly Pricing: Old Ideas and New Tools . MIT Press, 1999

    1999

  30. [38]

    The theory of the market economy

    Heinrich Von Stackelberg, Alan T Peacock, Erich Schneider, and TW Hutchison. The theory of the market economy. Economica , 20(80):384, 1953

    1953

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