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A shared-revenue Bertrand game

T0 review · 3 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read In a shared-revenue Bertrand price game, the paper identifies cost and fee parameters under which an independent seller's optimal-price equilibrium gives consumers a lower price and gives the retailer a payoff above its own single-agent…

desk verdict A genuinely new Bertrand variant with a solid equilibrium classification, but the advertised consumer-surplus win is benchmark-dependent; the paper deserves a serious referee and a careful rewrite of its claims. read the letter →

arxiv 2502.07952 v2 pith:4HG3LQH3 submitted 2025-02-11 econ.TH

Raj Pabari , Udaya Ghai , Dominique Perrault-Joncas , Kari Torkkola , Orit Ronen , Dhruv Madeka , Aviad Rubinstein , Dean Foster
show 1 more author
Omer Gottesman
This is my paper · ORCID
classification econ.TH MSC 91A1091B2491A80
keywords BertrandcompetitionrevenuesharingreferralfeeNashequilibriumrefinementParetooptimalityoutsideoptionplatformpricing
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

The paper models a marketplace where a retailer can either sell a product itself or let an independent, lower-cost seller take the sale and pay the retailer a share $\alpha$ of revenue. The main result is that this shared-revenue setup has a middle range of referral fees in which the independent seller sells at its own optimal price, that price is no higher than the retailer's single-agent price, and the retailer's fee income can exceed the profit it would earn selling alone. The authors solve the corresponding Bertrand price game and use equilibrium refinements to predict which outcome appears for each cost pair and fee. They then let the retailer choose the fee; in the fee-optimization game the retailer always prefers a fee that keeps the seller in the market, earns at least its single-agent profit, and stays bounded away from 1. Adding an outside option in which the seller can quit and compete directly ties the maximal fee to the seller's outside payoff, so the model explains when referral programs should exist and how large the fee should be.

What carries the argument

The load-bearing device is the pair of indifference prices $p_r^{\mathrm{ind}} = c_r/(1-\alpha)$ and $p_s^{\mathrm{ind}} = c_s/(1-\alpha)$: the retailer is indifferent between selling itself and collecting the fee when the seller prices at $p_r^{\mathrm{ind}}$, while $p_s^{\mathrm{ind}}$ is the seller's breakeven price. Around these, parameter space is organized by threshold fees $\alpha_{s*}$, $\alpha_{r*}$, $\alpha_{s*,r*}$, $\alpha^\dagger_r$, $\alpha^\dagger_s$, and by the price $p^\dagger$ defined through $\pi_{r,s}(p^\dagger)=\pi_{r,r}(p_r^*)$. The mechanics are a Bertrand undercutting constraint: the seller must keep its price low enough that the retailer does not prefer to undercut and take the market, but high enough to clear its own cost; in the middle fee range the two constraints jointly admit the seller's optimal price $p_s^*$ as the refined equilibrium, which is exactly where the Pareto improvement appears.

What would settle it

For the linear demand curve $q(p)=1-p$ with costs $c_r=0.90$, $c_s=0.87$ and fee $\alpha=0.08$ (the right subplot of Figure 2), apply the lowest-price equilibrium criterion instead of relative Pareto optimality: the refined outcome flips from the seller fulfilling at $p^\dagger$ to the retailer fulfilling at $p_r^*$ with $p_r^*<p^\dagger$, so the retailer's 'always prefer the seller' conclusion fails in that region. An empirical version: measure fees and prices on a real referral platform; a fee above $\alpha_{\max}$ with sellers still participating would contradict the outside-option model.

Watch

Extended reading notes

Core claim

The paper's central claim is that revenue sharing can Pareto-improve on direct retailing: for costs $c_s \le c_s^*$ and fees $\alpha$ in the interval $[\alpha_{s*}, \alpha_{s*,r*}]$, the refined equilibrium has the independent seller fulfilling all demand at the seller's single-agent optimal price $p_s^*$, with $p_s^* \le p_r^*$, so consumers pay less than they would under the retailer's own profit-maximizing price. In that same region Lemma 6 shows that at fee $\alpha = \alpha^\dagger_r$ the retailer's fee income $\pi_{r,s}(p_s^*, \alpha^\dagger_r)$ is at least its direct-sale profit $\pi_{r,r}(p_r^*)$, so the program can raise the retailer's payoff while lowering the price to the customer. Section 5 extends this: after the retailer chooses the referral fee, every admissible, Pareto-optimal, subgame-perfect equilibrium gives the retailer a payoff at least $\pi_{r,r}(p_r^*)$, with the equilibrium fee $\alpha^*$ always bounded away from 1, and the socially suboptimal low-fee outcome never observed.

Load-bearing premise

The point predictions—that the retailer wants the seller to serve demand and sets the fee at $\alpha_{r*}$—depend on selecting among a continuum of Nash equilibria by relative Pareto optimality and by fixing a continuation profile (a rule for which price is played in the remaining interval); if players instead coordinate on the lowest-price equilibrium, the retailer may serve demand itself and the fee conclusions downgrade to interval bounds.

Editorial extensions

If this is right

  • In the middle-fee region, opening a platform to an independent seller with a cost advantage strictly below $c_s^*$ lowers the consumer price and can raise the retailer's income above its own monopoly profit.
  • A profit-maximizing platform should not charge the highest possible referral fee; the optimal fee is bounded away from 1 and is set just low enough to keep the more efficient seller active.
  • When the seller's cost advantage is small ($c_s\ge c_s^*$), the seller-optimal Pareto-improving equilibrium does not exist, so revenue sharing is only a win when the efficiency gap is large enough.
  • A credible outside option for the seller caps the referral fee at $\alpha_{\max}$ and rules out the high-fee equilibria in which the retailer ends up selling directly at $p_r^*$.
  • In equilibrium of the fee-optimization game, the region where the seller must price at the retailer's indifference point (low fee) is never chosen, and the retailer always weakly outperforms its single-agent payoff.

Reading between the lines

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

  • Editorial inference: the point prediction that the retailer always prefers the seller to fulfill demand rests on the relatively Pareto-optimal selection; if a platform coordinates on lowest-price equilibria, the outcome in the $p^\dagger \le p_s^*$ region flips to the retailer serving demand at $p_r^*$, reversing the surplus comparison there.
  • Editorial inference: the model gives a testable fee benchmark—estimate $p_r^*$, $p_s^*$, $c_r$, $c_s$ from demand data and the model predicts the Pareto-improving fee lies in $[\alpha_{s*}, \alpha_{s*,r*}]$; an observed fee outside that interval signals unobserved costs or a different selection rule.
  • Editorial inference: with heterogeneous sellers, a single aggregate fee acts as a participation tax; the model suggests low-cost sellers stay and high-cost sellers leave, so welfare gains depend on the cost distribution and seller-specific fees may dominate.
  • Editorial inference: under demand or cost uncertainty, the continuum of equilibria and the $\rho$-dependence would become a distribution over outcomes, and the paper's interval bounds (Corollary 3, Eqs. 26-27) are the natural starting objects for that extension.
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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

3 major / 3 minor

Summary. This paper analyzes a two-player Bertrand game in which a retailer and an independent seller sell an identical good; if the seller fulfills demand, it remits a share α of revenue to the retailer. The authors derive the pure-strategy Nash equilibria of the 'staying' subgame (Proposition 1 and Eqs. (18)-(19)), apply admissibility and relative Pareto optimality to select outcomes (Section 4), then extend the game to an endogenous referral fee (Section 5) and to an outside option for the seller (Section 6). The headline claim is that there are parameter regions in which the seller fulfills demand at a price below the retailer's standalone price while the retailer's payoff is at least as large as its standalone profit, so that both firms and consumers can benefit.

Significance. The paper's equilibrium characterization is technically substantial and largely transparent: the threshold fees α_r*, α_s*, α_s*,r*, and α†_r are solved in closed form from first-order and indifference conditions (Appendix C.3), no parameters are fit to data, and the limitations (no closed form for α†_s, unsimplified p_s*/p† ordering, perfect-information assumption) are acknowledged. The no-split-market result (Proposition 1) is proved under weak assumptions, and the identification of a red region where p_s* ≤ p_r* and the retailer can earn more than its standalone profit is a genuine and falsifiable prediction. However, the welfare interpretation is considerably more fragile than the abstract suggests: the consumer-surplus gain is relative to the retailer-only counterfactual, not to the seller's standalone price, and the sharp point predictions depend on a specific stack of equilibrium refinements.

major comments (3)
  1. [Section 1.2 and Section 4.4.4] The claim that the price is decreased 'relative to the price either the retailer or the independent seller would have set independently' is false as stated. In the red region the seller fulfills demand at p_s* = argmax_p ((1−α)p − c_s)q(p); because α > 0, p_s*(α) > p_s^0 = argmax_p (p − c_s)q(p) for any regular demand. For the paper's linear demand, p_s*(α) = (1 + c_s/(1−α))/2 > (1 + c_s)/2 = p_s^0. Thus Section 4.4.4's statement that customers buy at a lower price 'than without the revenue sharing program' is valid only against the retailer-alone counterfactual, not against the seller's own standalone alternative. The abstract and Section 1.2 must be revised to name the correct counterfactual, or the paper must prove a condition such as α < δ/(c_s + δ) in the outside-option extension; at α = α†_r this inequality is not guaranteed.
  2. [Sections 4.2, 5.1, and 5.3; Corollary 3] The sharp conclusions that the retailer always induces the seller to fulfill demand and sets the fee α* depend on a specific stack of refinements: rejecting weakly dominated strategies, relative Pareto optimality among equilibria, and an ex ante choice of continuation profile ρ. Section 4.2 concedes that a lowest-price criterion would instead select the retailer-fulfills-at-p_r* equilibrium, where the seller gets zero; and Corollary 3 shows that for general ρ the fee-optimization results are only interval bounds, with α*(ρ) = α_r* differing from α*(ρ-bar) = ᾱ. This is load-bearing for the paper's own summary that 'the retailer always prefers to allow the independent seller to fulfill demand in equilibrium.' The authors should either prove which headline results are invariant to the refinement/ρ choice or explicitly qualify those results as selection-dependent.
  3. [Section 6.2 and Eq. (30)] The outside-option extension does not currently deliver the advertised two-sided improvement in general. The seller's participation constraint is π_s^(eq) ≥ π_s^(ℓ)(δ), and α^(o)* is solved in closed form only in the special case α_max ≤ min{α†_r, α_s*} (Eq. (30)); for general ρ and δ it is left as the abstract optimization in Eq. (28). Moreover, the consumer-price comparison in Section 6 would require a condition such as p_s*(α^(o)*) < p_s^(ℓ)*(δ), which for linear demand is α^(o)* < δ/(c_s + δ); no such condition is proved. The Section 6 results should be presented as conditional existence results over specific parameter ranges rather than as a general 'often' statement about simultaneous improvements.
minor comments (3)
  1. [Section 6.2 header and Section 5] There are several typographical errors: 'Bertand' in the Section 6.2 header, 'revenue sharking' in the first paragraph of Section 5, and 'indepedent' in Section 7; these should be corrected.
  2. [Table 1 and Eq. (24)-(25)] The notation ρ and ρ-bar (or ρ and ρ) is easy to confuse in the text, especially because the overline is not visually distinct in equations; renaming these to something like ρ_L and ρ_H would improve readability.
  3. [Section 3.3 and Appendix C.4] The definition of α†_s is introduced informally in Section 3.3 as the fee making p_sind ≤ p† equivalent, then defined precisely in Appendix C.4 as the largest fee satisfying π_r,s(p_sind, α†_s) = π_r,r(p_r*); the main text could state this construction at the point of introduction to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all equilibrium prices and threshold fees are derived from the stated payoff functions and external Bertrand results; no fitted input is relabeled as a prediction.

full rationale

The paper's central derivation is self-contained and non-circular. The payoff functions (Eqs. 1-2) define the game, and every key price and fee — p_r*, p_s*, p_rind, p_sind, α_r*, α_s*, α_s*,r*, and α†_r — is obtained by solving the players' first-order or indifference conditions in closed form (Eqs. 32-39 and Appendix C.3); α†_s is introduced by construction as the largest fee satisfying π_r,s(p_sind,α†_s)=π_r,r(p_r*), not fitted to data. The Nash equilibrium classification (Eqs. 18-19 and 50-51) follows from best-response conditions and the concavity or unimodality of the payoff functions (Corollary 1), not from assuming the desired equilibrium. The admissibility and relative Pareto optimality refinements (Section 4) and the ex ante continuation profile ρ (Section 5.1) are additional modeling assumptions whose consequences are computed, not circular steps; the paper explicitly concedes in Section 4.2 that a lowest-price criterion would select the retailer-fulfills equilibrium, so the refinement choice is transparent rather than smuggled in. The leaving subgame uses the standard asymmetric-cost Bertrand equilibrium from external sources (Blume 2003; Kartik 2011), and the paper explicitly notes the leaving subgame is independent of α. No parameter is fit to a subset of outcomes and then predicted; no load-bearing uniqueness theorem is imported from the authors' prior work; and the paper does not justify its core construction by citing earlier work by this author team. The Section 1.2 sentence claiming the customer price is decreased relative to what either player would set independently is factually questionable for the seller, since p_s*(α) exceeds the seller's standalone price for α>0, but this is a correctness or overstatement issue, not circularity: the derivation of p_s*(α) does not assume that comparison. For these reasons the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The paper's central claims rest on eight stated assumptions: cost asymmetry (c_s < c_r), strict concavity of demand, pure strategies, revenue-share (not profit-share) fee structure, an ad hoc outside-option cost δ, a stay-if-indifferent tie-break, an a priori continuation profile ρ, and perfect information. Two parameters (δ, β) enter by hand; β is proven irrelevant by Proposition 1, and δ is a model input, not a fitted quantity. No physical or economic entity is invented.

free parameters (2)
  • δ (outside-option cost differential) = ∈ (0, c_r - c_s)
    Introduced ad hoc in Section 6.1 to make leaving costly enough that staying can be an equilibrium ("we will assume that there exists a δ > 0 such that the independent seller does not always prefer to leave"). It is a model input, not fitted to any data; the central outside-option results (28)-(30) depend on it.
  • β (market-split share at equal prices) = any β ∈ [0,1]
    Chosen by hand in Section 2 "for mathematical completeness"; Proposition 1 proves no split-market equilibrium survives, so β is shown irrelevant. Listed for exhaustiveness.
assumptions (8)
  • domain assumption c_s < c_r, the independent seller has strictly lower cost than the retailer
    Assumption (II) of Section 2.1; Lemma 2 (Appendix B.2) shows that without it no equilibrium has the seller fulfilling demand, so the game becomes uninteresting. This asymmetry is the engine of the revenue-sharing result.
  • domain assumption Demand q(p) is strictly concave, monotonically decreasing, twice differentiable, nonnegative, and compactly supported
    Assumed in Section 2 and footnote 2; needed for strict concavity and unimodality of the payoff functions (Lemma 3, Corollary 1), which underpins the unique prices p_r*, p_s*, p† and the threshold-fee ordering. The paper notes relaxations to subderivatives.
  • domain assumption Players restrict attention to pure strategies
    Section 3.1 ("we will only consider pure strategies"); mixed-strategy equilibria are excluded, which matters because the game has continua of equilibria and standard mixed-equilibrium existence arguments do not apply.
  • domain assumption The referral fee is a share of revenue, not of profit, and no fixed fee is paid
    Payoffs in Eqs. (1)-(2) define the game; the entire equilibrium structure (indifference price p_r^ind = c_r/(1-α), etc.) follows from this functional form. A profit share or two-part tariff would change all thresholds; the paper cites the two-part tariff literature but does not analyze it here.
  • ad hoc to paper There exists an outside-option cost differential δ ∈ (0, c_r - c_s) so that the seller's cost if it leaves is c_s + δ
    Section 6.1: "we will posit that if they choose to sell on their own, the effective cost to the independent seller will be c_s + δ" and "we will assume that there exists a δ > 0 such that the independent seller does not always prefer to leave." The sign restriction makes the threat credible; the outside-option analysis is built on it.
  • ad hoc to paper If the seller is indifferent between staying and leaving, it stays
    Section 6.2: "Pareto optimality also tells us that if the independent seller is indifferent between staying and leaving, they will choose to stay." This tie-break pins down the equilibrium of the full game and lets the retailer choose any fee up to α_max.
  • ad hoc to paper The continuation profile ρ is an a priori chosen admissible, Pareto optimal equilibrium of the staying subgame for every (α, c_r, c_s)
    Section 5.1: subgame perfection requires selecting one of the continuum of admissible, Pareto optimal equilibria; the paper's point predictions α*(ρ) = α_r* and α*(ρ-bar) = ᾱ are specific to the chosen ρ, and Corollary 3 only bounds the general case (Eqs. 26-27).
  • domain assumption Perfect and complete information with simultaneous price setting
    Section 1.2 game tree and Section 7 ("we have assumed throughout that this is a game of perfect, complete information"). Real marketplaces have private costs and demand uncertainty, which the paper lists as future work.

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Pith. "Pith review of A shared-revenue Bertrand game." pith.science (2026). https://pith.science/paper/4HG3LQH3

@misc{pith2026250207952,
  author       = {Pith},
  title        = {Pith review of: A shared-revenue Bertrand game},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4HG3LQH3}},
  note         = {Machine review of arXiv:2502.07952}
}
read the original abstract

We introduce and analyze a variation of the Bertrand game in which the revenue is shared between two players. This game models situations in which one economic agent can provide goods/services to consumers either directly or through an independent seller/contractor in return for a share of the revenue. We analyze the equilibria of this game, and show how they can predict different business outcomes as a function of the players' costs and the transferred revenue shares. Importantly, we identify game parameters for which independent sellers can simultaneously increase the original player's payoff while increasing consumer surplus. We then extend the shared-revenue Bertrand game by considering the shared revenue proportion as an action and giving the independent seller an outside option to sell elsewhere. This work constitutes a first step towards a general theory for how partnership and sharing of resources between economic agents can lead to more efficient markets and improve the outcomes of both agents as well as consumers.

Figures

Figures reproduced from arXiv: 2502.07952 by the authors.

Figure 1
Figure 1. Sample payoff curves and important price points for linear demand curve [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Shared-price Nash equilibria where the independent seller fulfills demand [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Equilibrium outcomes as a function of 𝛼 vs 𝑐𝑟 (left) and 𝛼 vs 𝑐𝑠 (right). Case (ii), 𝑐𝑠 ≥ 𝑐𝑠 ∗ : (𝑃, 𝑥) =    (𝑝𝑟ind , 𝑠) 𝛼 ≤ 𝛼 † 𝑟 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Equilibrium payoff curves (2) The retailer and independent seller set the equilibrium price configuration 𝜌 (𝛼∗, 𝑐𝑟 , 𝑐𝑠 ) 5.2 Interpreting equilibria To complete our analysis, our natural next step is to find 𝛼∗. We’ll denote as 𝜋 (eq) 𝑟 (𝜌 (𝛼), 𝛼, 𝑐𝑟 , 𝑐𝑠 ) the payof…
Figure 5
Figure 5. Figure 5: Equilibrium payoff curves with an outside option [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

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Works this paper leans on

17 extracted references · 15 canonical work pages

  1. [2]

    https://doi.org/10.1016/S1573-4420(87)80010- 1 Steven Tadelis

    Elsevier, 991–1042. https://doi.org/10.1016/S1573-4420(87)80010- 1 Steven Tadelis

    work page doi:10.1016/s1573-4420(87)80010-

  2. [12]

    https://doi.org/10.1162/154247603322493212 arXiv:https://academic.oup.com/jeea/article-pdf/1/4/990/10312916/jeea0990.pdf M

    Platform Competition in Two-Sided Markets.Journal of the European Economic Association1, 4 (06 2003), 990–1029. https://doi.org/10.1162/154247603322493212 arXiv:https://academic.oup.com/jeea/article-pdf/1/4/990/10312916/jeea0990.pdf M. Shubik and R. Levitan. 1980.Market Structure and Behavior. Harvard University Press. https://books.google.com/books? id=j...

    doi:10.1162/154247603322493212 2003

  3. [15]

    http://www.jstor.org/stable/3083446 J

    Complexity, Flexibility, and the Make-or-Buy Decision.The American Economic Review92, 2 (2002), 433–437. http://www.jstor.org/stable/3083446 J. Tirole. 1988.The Theory of Industrial Organization. MIT Press. https://books.google.com/books?id=HIjsF0XONF8C Joel Watson

    work page arXiv 2002

  4. [17]

    Two-part tariff competition in duopoly.International Journal of Industrial Organization22, 6 (2004), 799–820. https://doi.org/10.1016/j.ijindorg.2004.03.005 Pabari et al.20 A NOTATION TABLE Notation Value(s) Description 𝑐𝑥 ∈(0,1)Marginal cost for player𝑥 𝑝𝑥 ∈(0,1)Price set by player𝑥 𝛼∈(0,1)Referral fee 𝛽∈[0,1]Market split proportion if prices are equal 𝑞...

    work page doi:10.1016/j.ijindorg.2004.03.005 2004

  5. [1937]

    https://doi.org/10.1111/j.1468-0335.1937.tb00002.x arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1468-0335.1937.tb00002.x Jennifer C

    The Nature of the Firm.Economica4, 16 (1937), 386–405. https://doi.org/10.1111/j.1468-0335.1937.tb00002.x arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1468-0335.1937.tb00002.x Jennifer C. Coats

    work page arXiv 1937

  6. [1976]

    https://doi.org/10.1016/0047-2727(76)90044-X Joseph E

    On the specification of models of optimum income taxation.Journal of Public Economics6, 1 (1976), 123–162. https://doi.org/10.1016/0047-2727(76)90044-X Joseph E. Stiglitz

    work page doi:10.1016/0047-2727(76)90044-x 1976

  7. [1989]

    http://www.jstor.org/stable/2937866 Andreas Blume

    An Outside Option Experiment.The Quarterly Journal of Economics 104, 4 (1989), 753–770. http://www.jstor.org/stable/2937866 Andreas Blume

    work page arXiv 1989

  8. [2002]

    https://doi.org/10.1081/PAD-120013250 arXiv:https://doi.org/10.1081/PAD-120013250 Thomas Demuynck

    Applications of Principal-Agent Models to Government Contracting and Accountability Decision Making.International Journal of Public Administration25, 4 (2002), 441–461. https://doi.org/10.1081/PAD-120013250 arXiv:https://doi.org/10.1081/PAD-120013250 Thomas Demuynck

    work page doi:10.1081/pad-120013250 2002

Show all 17 references
  1. [2003]

    https://doi.org/10.1016/S0165- 1765(02)00217-3 Chen-Yu Chang

    Bertrand without fudge.Economics Letters78, 2 (2003), 167–168. https://doi.org/10.1016/S0165- 1765(02)00217-3 Chen-Yu Chang

    2003 doi

  2. [2004]

    https://www.heritage

    The Laffer Curve: Past, Present, and Future.Heritage Foundation1765 (2004). https://www.heritage. org/taxes/report/the-laffer-curve-past-present-and-future M.J. Osborne and A. Rubinstein. 1990.Bargaining and Markets. Academic Press. https://books.google.com/books?id= JaoUAQAAM...

    2004

  3. [2006]

    https://doi

    Competition in two-sided markets.The RAND Journal of Economics37, 3 (2006), 668–691. https://doi. org/10.1111/j.1756-2171.2006.tb00037.x arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1756-2171.2006.tb00037.x Joseph Bertrand

    2006

  4. [2011]

    https://doi.org/ 10.1016/j.econlet.2011.01.013 Arthur B

    A note on undominated Bertrand equilibria.Economics Letters111, 2 (2011), 125–126. https://doi.org/ 10.1016/j.econlet.2011.01.013 Arthur B. Laffer

    2011 doi

  5. [2014]

    2014), 9 pages

    Principal-Agent Model of Risk Allocation in Construction Contracts and Its Critique.Journal of Construction Engineering and Management140, 2, Article 1 (Aug. 2014), 9 pages. https://doi.org/10.1061/(ASCE)CO.1943- 7862.0000779 R. H. Coase

    2014

  6. [2015]

    https://doi.org/10.1016/j.ijindorg.2015.05.002 Navin Kartik

    On two-part tariff competition in a homogeneous product duopoly.International Journal of Industrial Organization41 (2015), 30–41. https://doi.org/10.1016/j.ijindorg.2015.05.002 Navin Kartik

    2015 doi

  7. [2019]

    https://doi.org/10.1007/s11238-019-09698-4 Srihari Govindan and Robert B

    Bertrand competition with asymmetric costs: a solution in pure strategies.Theory and Decision 87, 1 (April 2019), 147–154. https://doi.org/10.1007/s11238-019-09698-4 Srihari Govindan and Robert B. Wilson. 2016.Nash Equilibrium, Refinements of. Palgrave Macmillan UK, London, 1–...

    2019 doi

  8. [2020]

    https: //doi.org/10.1016/j.econlet.2020.109110 O.E

    On the outside-option principle with one-sided options.Economics Letters191 (2020), 109110. https: //doi.org/10.1016/j.econlet.2020.109110 O.E. Williamson. 1975.Markets and Hierarchies, Analysis and Antitrust Implications: A Study in the Economics of Internal Organization. Fre...

    2020

  9. [2024]

    https://doi.org/10.1007/s00712-024-00874-x Mark Armstrong

    Platform investment and seller competition in two-sided markets.Journal of Economics144 (06 2024), 1–29. https://doi.org/10.1007/s00712-024-00874-x Mark Armstrong

    2024 doi

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