Price equilibria with positive margins in loyal-strategic markets with discrete prices

Akansh Verma; Gurkirat Wadhwa; Priyank Sinha; Veeraruna Kavitha

REVIEW 3 major objections 6 minor 25 references

Price equilibria with positive margins in loyal-strategic markets with discrete prices

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read When prices are discrete and customers are loyal or strategic, manufacturer competition can settle at positive-margin Nash equilibria, yet grids that are too fine eliminate all operating equilibria.

desk verdict Worth refereeing, but the symmetric-equilibrium theorem sits on an unstated tie-break in the demand model; the paper needs a revision that names its assumptions and softens Theorem 3's claims. read the letter →

arxiv 2506.17239 v1 pith:LCMQRV7E submitted 2025-06-05 q-fin.GN

Gurkirat Wadhwa , Akansh Verma , Veeraruna Kavitha , Priyank Sinha This is my paper

classification q-fin.GN MSC 91B2691A1091A16

keywords discretepricingpriceincrementsNashequilibriumloyalcustomersstrategicmean-fieldgamesupplychaincycles

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 a supply chain with one supplier and two manufacturers who choose prices from a discrete grid, mirroring the way real currency denominations force prices to move in ticks. Its central claim is that when customers split into loyal and strategic segments and manufacturers bear fixed plus per-unit operating costs, the manufacturer pricing game has Nash equilibria at prices strictly above cost, so both firms earn positive margins. For a given supplier price, the equilibrium prices form an interval of grid points rather than a single point. The paper also claims that if the price grid is too fine, no operating equilibrium exists, so the market becomes unstable. That would overturn the textbook prediction that price competition drives margins to zero.

What carries the argument

The load-bearing machinery is the discrete price grid $P_\delta = \{0,\delta,2\delta,\dots\}$ combined with the piecewise-concave payoff functions $W_1,W_2,W_3,W_4$ that describe a manufacturer's utility depending on whether it prices below, above, at, or against a competitor who does not operate. Concavity of the relaxed continuous functions restricts best responses to a small set—the same grid point, one tick below, the duopoly optimizer, or not operating—so a symmetric equilibrium condition reduces to two quadratic inequalities in the grid price. The interval $[s_\delta(q), e_\delta(q)]$ collects the grid prices satisfying those inequalities. On the demand side, the non-loyal, product-desiring customers are allocated by a mean-field equilibrium, which at zero quality-of-service weight becomes the tie-splitting rule that all such customers buy from the cheaper manufacturer. The small-$\delta$ disappearance of equilibria follows because the interval shrinks toward a point at or below the operating cost as $\delta \to 0$.

What would settle it

Measure, in a duopoly market with fine price increments, whether every non-loyal customer buys from the strictly cheaper manufacturer whenever a price gap exists; if a substantial share buys the pricier product, the demand split in equation (3) that generates the predicted equilibrium interval is contradicted.

Watch

Extended reading notes

Core claim

At the core is Theorem 1: whenever the supplier's price $q$ is outside the complete-choking set and at or below the survival threshold, every grid price $l\delta$ in the interval $[s_\delta(q), e_\delta(q)]$ is a symmetric pure-strategy Nash equilibrium of the manufacturer game $G(q,\delta)$. At these equilibria both manufacturers charge the same grid price, split the non-loyal customers equally, keep their loyal bases, and earn equal positive profit. Theorem 2 adds asymmetric equilibria, one tick apart, in the duopoly regime; Theorem 3 shows that for every $q$ there is a small enough grid size $\delta$ below which no operating pure-strategy equilibrium survives. The paper then picks a focal equilibrium among the interval's points (the one giving both manufacturers maximum equal revenue) and, numerically, finds the supplier maximizes its own utility by choosing a price that partially chokes one manufacturer.

Load-bearing premise

The load-bearing premise is that non-loyal, product-desiring customers split between manufacturers exactly according to the mean-field equilibrium of the specified linear quality-of-service game, which at zero quality weight means every such customer buys from the strictly cheaper manufacturer and ties split evenly; this segmentation rule is postulated rather than estimated from data.

Editorial extensions

If this is right

At every symmetric equilibrium found in Theorem 1, both manufacturers earn strictly positive profit margins, contradicting the marginal-cost prediction of standard price competition.
For fixed $q$, the set of equilibrium prices is an interval of grid prices; coarser grids can support several equilibria, and sufficiently fine grids support none.
When $\delta$ is small enough there is no operating pure-strategy Nash equilibrium, so dynamic adjustment may settle into price cycles rather than a fixed price.
Asymmetric equilibria one grid step apart are possible only in the duopoly regime and only because prices are discrete.
A supplier acting as Stackelberg leader can raise its own payoff by choosing a price that partially chokes one manufacturer, pushing the downstream firms to the focal equal-price equilibrium.

Reading between the lines

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

Beyond the paper, the interval structure implies a testable comparative static: markets with coarser currency denominations should show higher and more stable margins than markets with fine denominations, all else equal.
Beyond the paper, the all-or-nothing switching rule in equation (3) is the part that real customer data would most likely violate; if even a fraction of non-loyal customers tolerates a small price gap, the predicted equilibrium interval and the no-equilibrium threshold would shift.
Beyond the paper, the same proof machinery could be extended to asymmetric manufacturers with different costs or loyal bases, where the interval endpoints would become firm-specific and the focal equilibrium would tilt toward the lower-cost firm.
Beyond the paper, one could test Theorem 3 by running best-response dynamics with an empirically estimated demand function; observing convergence to a fixed price for fine grids would falsify the no-equilibrium claim.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper has a real, checkable result—Theorem 1's interval characterization of symmetric positive-margin equilibria in a discrete-price duopoly—and the algebra around it holds up. But the demand model that feeds Theorem 1 rests on an unstated tie-break, and the paper's conclusions overclaim what follows from Theorem 3. It deserves a serious referee, but needs revision.

What's genuinely new is the combination: discrete price grids, per-unit and fixed operating costs, and a mean-field game among non-loyal "NLDP" customers, producing pure-strategy Nash equilibria at prices strictly above marginal cost, plus a no-operating-equilibrium regime for sufficiently fine grids. The paper also does useful work identifying the focal equilibrium and showing numerically how supplier price and denomination size shift the equilibrium set. The proofs are mostly transparent; Appendix A is a genuine derivation, not a hand-wave.

The main soft spot is the construction of demand at omega=0, equations (2)-(3). At omega=0 the mean-field utility is just -p_a, independent of the population split, so any distribution supported on the lowest-priced manufacturer is a mean-field equilibrium. The paper's mu_i^* = 1/|arg min| split evenly at ties is a selection, not a consequence of the mean-field equilibrium condition. That selection is doing load-bearing work: if the tie split were asymmetric, the demand functions in (7)-(9) would not be symmetric, and (l*delta, l*delta) would generally fail to be a Nash equilibrium. The paper never states the symmetric tie-break as an assumption. This is fixable, but the theorem should be restated as conditional on that rule, and the sensitivity discussed.

Two smaller issues. Theorem 3 only rules out pure-strategy operating NE for small delta; mixed strategies are untouched, so the conclusion that the model "does not have price cycles as solutions" is too strong—the paper's own Figure 5 shows Edgeworth cycles under best-response dynamics for small delta. And the best-response candidate set (23) is asserted "clearly" from concavity rather than fully proved; probably repairable but it is a gap in the proof.

Who this is for: IO and supply-chain theorists working on discrete pricing or price rigidity; the paper gives a clean example of how currency granularity can determine whether positive-margin equilibria exist. I would send it to a good referee, with the request that they look especially at the derivation of (3) and the scope of Theorem 3.

Referee Report

3 major / 6 minor

Summary. The paper studies a two-echelon supply chain with one supplier and two manufacturers. Manufacturers choose prices from a discrete grid P_δ = {0, δ, 2δ, ...} or choose not to operate; demand is determined by loyal customers plus a strategic 'NLDP' segment whose split is modeled by a mean-field equilibrium, specialized in Eq. (3) to the price-only ω=0 case with equal splitting at tied prices. The main results are Theorem 1 (an interval of symmetric operating Nash equilibria with positive margins), Theorem 2 (asymmetric operating equilibria in the duopoly regime), and Theorem 3 (no operating equilibrium for sufficiently small price denomination δ). The paper then gives a numerical study of a focal equilibrium, the supplier's optimal price, and best-response dynamics, arguing that positive-margin equilibria replace marginal-cost Bertrand outcomes and that Edgeworth cycles reappear only for small δ.

Significance. If the results are correct, the paper makes a useful contribution: it gives an explicit, checkable characterization of symmetric positive-margin equilibria in a discrete-price Bertrand-type duopoly with operating costs and loyal-strategic demand, and it shows that the equilibrium set collapses when the price grid becomes fine. The explicit interval formulas in Theorem 1 are a strength, and no fitted constants enter the theorem; the numerical experiments are consistent with the interval characterization. The paper is also honest that the supplier-level analysis is numerical and that a detailed top-layer theory is postponed. The main caveat is that the demand model, especially the tie-breaking rule at equal prices, is a modeling postulate that is load-bearing for the central theorem; the results are therefore conditional on that postulate unless the tie rule is justified or explicitly assumed.

major comments (3)

[Section II-A, Eqs. (2)-(3), and Section III-A, Eq. (9)] The tie rule μ_i^* = 1/|arg min_m p_m| at ω=0 is a selection, not a consequence of the mean-field equilibrium. At ω=0 the customer payoff π(a,μ) = p_a is independent of μ, so every split is a mean-field equilibrium, and Eq. (2) has a 0/0 expression at tied prices. This selection is load-bearing: it produces W3 in Eq. (9), and with an asymmetric tie-break the equal-price payoff for manufacturer i would be (\bar d - αp_i + 2εαp_i x)(p_i - C_M^q) - O_M, so a symmetric pair (lδ,lδ) need not be a Nash equilibrium. Please state the equal-split rule as an explicit assumption or justify it as the ω→0 limit of the non-degenerate mean-field equilibrium in Eq. (2), and discuss how Theorem 1 depends on that choice.
[Appendix A, proof of Theorem 1, Eq. (23)] The assertion that B_m(lδ) is contained in {(l−1)δ, lδ, \bar l_dδ, no} is stated as 'clearly' following from concavity, but no proof is given. The function W1 is concave, but its unconstrained maximizer can lie below lδ for parameters satisfying Assumption A.1; in that case the best grid price below lδ need not be (l−1)δ. The proof of Theorem 1 should either prove that the interval constraints (20)-(22) rule out such configurations, or enlarge the candidate set. Without this, the sufficiency part of Theorem 1 is not fully established.
[Appendix A, proof of Theorem 3] The proof of Theorem 3 is a sketch rather than a complete argument. The sentence 'for example consider the following, which' is unfinished, and the displayed limit for U_{M_i}((\bar l_d−1)δ,(\bar l_d−1)δ)−W^*_{2,δ} is not connected to the necessary conditions for asymmetric equilibria derived in the proof of Theorem 2. Since Theorem 3 is one of the paper's main claims (instability for small denomination factors), a complete proof should be supplied.

minor comments (6)

[Section II-A] There is a recurring typo: 'NDLP' should be 'NLDP' in several places (e.g., 'the NDLP customers' appears where the defined acronym is NLDP).
[Section III-A, Theorem 2] Theorem 2 says 'under certain conditions provided in Appendix A' but does not state those conditions in the theorem itself; the theorem should be self-contained by listing the inequalities from the proof.
[Eq. (15) and surrounding text] The phrase 'complete chocking regime' contains a typo ('chocking' should be 'choking'), and the notation W^*_{4,δ}(δ) is awkward because W4 also depends on q; please make the dependence on q explicit.
[Section IV, Eq. (19)] The focal equilibrium selection l^*(q) ∈ arg max W3(lδ) is used in the supplier's utility, but the paper does not specify how ties in that arg max are resolved; please state the tie-breaking rule.
[Section III, Remarks] The remark that 'one can prove this constant curve to be the equilibrium even for the dynamic game of [4] for such δ' is asserted without proof; either provide the argument or label it as a conjecture.
[Abstract and Section I] The abstract says low denomination factors 'can lead to instability' without mentioning that Theorem 3 is an asymptotic statement for sufficiently small δ; a precise statement in the introduction would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: equilibrium results are derived from explicit model assumptions, with no fitted constants or load-bearing self-citations.

full rationale

The paper's central claims are mathematical characterizations of the manufacturer game G(q,δ) under the stated demand model (1)-(3) and utility functions (4)-(5). Theorems 1-3 follow from concavity of the relaxed W1-W4 functions and the discrete grid structure; the equilibrium intervals [sδ(q), eδ(q)] and the small-δ nonexistence result are algebraic consequences of these assumptions, not rediscoveries of inputs. No parameter is calibrated to data, and the numerical values in Section IV are illustrative rather than fitted. Self-citations [1] and [7] provide background (loyal-customer demand and mean-field equilibrium definition) and Assumption A.1 is explicitly an input condition, not a conclusion imported from prior work. The ω=0 tie-break in (3) is a modeling choice — the mean-field condition is degenerate at ω=0, and the equal split is not uniquely forced by the definition — but the paper states this rule as part of its demand model rather than disguising a fitted value as a prediction. A fragile or under-derived assumption weakens external validity, but it does not make the internal derivation circular. The theorems are self-contained conditional statements, so the appropriate circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central theorem depends on a specific demand model (mean-field equilibrium split of strategic customers), a pure-strategy equilibrium restriction, Assumption A.1, symmetry, and standard concavity facts. No free parameters are fitted to data; the numerical constants are illustrative. The invented NLDP customer segment is a modeling postulate without external evidence.

free parameters (1)

Numerical scenario constants = d_bar=8, C_M=O_M=2, C_S=O_S=0.01, alpha=0.5, epsilon=0.8
Chosen for Figures 2-5 and the supplier-side numerical study. The theorems themselves are stated for general parameters under Assumption A.1, but the supplier-advantage and sensitivity claims are only demonstrated for this scenario.

assumptions (5)

domain assumption The demand model in equations (1)-(3): non-loyal, product-desiring customers split according to the mean-field equilibrium of a linear QoS game.
Formula (2) for the mean-field equilibrium is stated as easy to derive from the linear QoS utility, but the underlying utility and segmentation are not validated against data.
domain assumption Pure-strategy Nash equilibrium is the relevant solution concept; mixed strategies are not analyzed.
The paper defines operating NE as pure-strategy NE. Theorem 3 does not rule out mixed equilibria, and the abstract's 'game does not have NE' should be read as no pure operating NE.
domain assumption Assumption A.1: the market potential is sufficiently large, d_bar greater than alpha(C_S+C_M) plus twice the square root of alpha O_M.
This allows the operating equilibrium intervals to be nonempty and is taken from the authors' prior work [1].
domain assumption The two manufacturers are symmetric, with identical costs, market potential, and QoS parameters.
The paper studies a symmetric system; the positive-margin and non-uniqueness results are for this symmetric case, and asymmetric extensions are left for future work.
standard math The relaxed continuous-price utility functions W1-W4 are concave, with unique optimizers.
The proofs use concavity of quadratics to reduce best responses to a finite candidate set (23); this is a standard calculus fact.

invented entities (1)

NLDP (non-loyal, desiring-product) customer segment
purpose: Models customers who leave their own manufacturer because of price but still want the product and choose between manufacturers strategically.
No data or external evidence is provided for this segmentation; it is a modeling postulate used to derive the demand function (1)-(3).

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Cite this review

Pith. "Pith review of Price equilibria with positive margins in loyal-strategic markets with discrete prices." pith.science (2026). https://pith.science/paper/LCMQRV7E

@misc{pith2026250617239,
  author       = {Pith},
  title        = {Pith review of: Price equilibria with positive margins in loyal-strategic markets with discrete prices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LCMQRV7E}},
  note         = {Machine review of arXiv:2506.17239}
}

read the original abstract

In competitive supply chains (SCs), pricing decisions are crucial, as they directly impact market share and profitability. Traditional SC models often assume continuous pricing for mathematical convenience, overlooking the practical reality of discrete price increments driven by currency constraints. Additionally, customer behavior, influenced by loyalty and strategic considerations, plays a significant role in purchasing decisions. To address these gaps, this study examines a SC model involving one supplier and two manufacturers, incorporating realistic factors such as customer demand segmentation and discrete price setting. Our analysis shows that the Nash equilibria (NE) among manufacturers are not unique, we then discuss the focal equilibrium. Our analysis also reveals that low denomination factors can lead to instability as the corresponding game does not have NE. Numerical simulations demonstrate that even small changes in price increments significantly affect the competitive dynamics and market share distribution.

Figures

Figures reproduced from arXiv: 2506.17239 by the authors.

**Figure 2.** The variations in symmetric NE sets with supplier prices and price denominations [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 5.** Convergence to Edgeworth Cycle: for small δ with no NE, as given in Theorem 3. setting (we observe some in numerical examples)? Further, how will the overall equilibrium behavior of the SC agents change with asymmetric agents? REFERENCES [1] Wadhwa, G., Walunj, T.S. and Kavitha, V., 2024. Partition-form Cooperative Games in Two-Echelon Supply Chains. arXiv preprint arXiv:2401.04939. [2] Escrihuela-Villar, M. and Gui… view at source ↗

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Reference graph

Works this paper leans on

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" 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 gl...
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" 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 gl...
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FUNCTION id.bst "Merlin.mbs v4.21 2009-07-09. " ENTRY address annote archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass...

2009

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

11em plus .33em minus .07em @technote 4000 4000 100 4000 4000 500 `\.=1000 = #1 #1 #1 0pt [0pt][0pt] #1 * \| ** #1 \@IEEEauthorblockNstyle \@IEEEauthorblockAstyle \@IEEEauthordefaulttextstyle \@IEEEauthorblockconfadjspace -0.25em \@IEEEauthorblockNtopspace 0.0ex \@IEEEauthorblockAtopspace 0.0ex \@IEEEauthorblockNinterlinespace 2.6ex \@IEEEauthorblockAinte...

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Partition-form Cooperative Games in Two-Echelon Supply Chains

Wadhwa, G., Walunj, T.S. and Kavitha, V., 2024. Partition-form Cooperative Games in Two-Echelon Supply Chains. arXiv preprint arXiv:2401.04939

work page Pith review arXiv 2024

[3] [3]

and Guillén, J., 2025

Escrihuela-Villar, M. and Guillén, J., 2025. On Cournot and Bertrand competition in collusive mixed oligopolies. Journal of Economics, 144(3), pp.231-246

work page 2025

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Beckmann’s Bertrand–Edgeworth Duopoly Model: New Pure Strategy Equilibria

Shvedov, A.S., 2024. Beckmann’s Bertrand–Edgeworth Duopoly Model: New Pure Strategy Equilibria. Automation and Remote Control, 85(8), pp.711-717

work page 2024

[5] [5]

and Tirole, J., 1988

Maskin, E. and Tirole, J., 1988. A theory of dynamic oligopoly, II: Price competition, kinked demand curves, and Edgeworth cycles. Econometrica: Journal of the Econometric Society, pp.571-599

work page 1988

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and Lev, B., 2021

Zheng, X.X., Li, D.F., Liu, Z., Jia, F. and Lev, B., 2021. Willingness-to-cede behaviour in sustainable supply chain coordination. International Journal of Production Economics, 240, p.108207

work page 2021

[7] [7]

and Cheng, T.E., 2023

Li, C., Cao, B., Zhou, Y.W. and Cheng, T.E., 2023. Pricing, coalition stability, and profit allocation in the pull assembly supply chains under competition. OR Spectrum, 45(3), pp.977-1011

work page 2023

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Balancing rationality and social influence: Alpha-rational Nash equilibrium in games with herding

Agarwal, K., Avrachenkov, K., Kavitha, V. and Vyas, R., 2024. Balancing rationality and social influence: Alpha-rational Nash equilibrium in games with herding. arXiv preprint arXiv:2406.03928

work page Pith review arXiv 2024