arxiv: 2604.26487 · v1 · submitted 2026-04-29 · 🧮 math.FA

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Equilibrium in the Canonical Stackelberg Triopoly via Response Functions and Fixed Point Theory

Anton Badev, Boyan Zlatanov, Martin Pavlov

Pith reviewed 2026-05-07 12:42 UTC · model grok-4.3

classification 🧮 math.FA

keywords stackelberg triopolycoupled fixed pointsbest response functionssequential oligopolymarket equilibriummyopic dynamicsrecursive formulation

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The pith

In the canonical Stackelberg triopoly, equilibrium exists and is unique when the best-response conditions are recast as a coupled fixed-point problem.

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

The authors extend the Stackelberg duopoly model to three firms that move in sequence, with each anticipating how later firms will react to earlier choices. They reformulate the equilibrium conditions using coupled fixed-point theory to prove that a unique market equilibrium exists under standard assumptions on demand and costs. This holds even though simple myopic best-response adjustment processes are not guaranteed to converge to that equilibrium when demand is linear. The paper also supplies a recursive formulation of the equilibrium that supports analysis of the case with many firms.

Core claim

In the canonical Stackelberg triopoly each firm anticipates the reactions of all subsequent players, and the equilibrium is the unique solution to the system of coupled fixed-point equations obtained from the firms' best-response functions under the given demand and cost structure.

What carries the argument

The coupled fixed-point reformulation of the best-response functions, which permits direct application of fixed-point theorems to establish existence and uniqueness.

If this is right

The equilibrium quantities can be found by solving the fixed-point system instead of simulating adjustment dynamics.
Myopic best-response iteration may cycle or diverge, requiring more advanced solution methods.
A recursive description allows the equilibrium to be tracked as the number of sequential firms increases toward the competitive limit.
The approach applies directly to any finite number of firms in this sequential quantity-setting game.

Where Pith is reading between the lines

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

The failure of myopic dynamics even in the linear case suggests that convergence properties depend on the order of moves rather than just functional form.
This fixed-point method could be tested in other sequential games where players have perfect foresight of later stages.
As the number of firms grows, the recursive formulation might reveal convergence to the standard Cournot outcome or a different limit.
Computational algorithms based on fixed-point iteration may be more stable than best-response dynamics for finding the equilibrium.

Load-bearing premise

The best-response functions of the firms must satisfy the technical conditions required by the coupled fixed-point theorem, which the model assumes hold for its particular demand and cost specifications.

What would settle it

An explicit numerical example with linear demand and quadratic costs where the fixed-point equations have no solution or multiple solutions, or where the computed equilibrium fails to satisfy the original best-response conditions.

Figures

Figures reproduced from arXiv: 2604.26487 by Anton Badev, Boyan Zlatanov, Martin Pavlov.

**Figure 1.** Figure 1: Consumer surplus under the Cournot and Stackelberg eq view at source ↗

**Figure 2.** Figure 2: Aggregate quantities across models. Green: Cournot wit view at source ↗

**Figure 3.** Figure 3: Individual quantities under linear costs as the number of p view at source ↗

**Figure 4.** Figure 4: Individual quantities under quadratic costs as the numbe view at source ↗

**Figure 5.** Figure 5: Convergence of aggregate quantities to their large-mar view at source ↗

**Figure 6.** Figure 6: Equilibrium prices across models. Green: Cournot with linear view at source ↗

**Figure 7.** Figure 7: Convergence of prices to their large-market limits. Green view at source ↗

read the original abstract

We analyze a canonical extension of the Stackelberg duopoly to a sequential framework, where each firm strategically anticipates the reactions of all subsequent players. In a triopoly (three-firm) settings, we obtain existence and uniqueness of market equilibrium via a reformulation of the equilibrium conditions that draws on coupled fixed-point theory. Even with linear demand, convergence of myopic best-response dynamics is not guaranteed. A recursive equilibrium formulation enables the analysis of the limiting case as the number of participants grow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper recasts Stackelberg triopoly equilibrium as a coupled fixed-point problem to claim existence and uniqueness, plus a recursive extension for more firms, but the uniqueness step looks under-supported given the noted non-convergence of best-response dynamics.

read the letter

The main takeaway is that this work reformulates the three-firm sequential Stackelberg game by treating the best-response functions as a coupled fixed-point map. That move lets them invoke standard existence results from fixed-point theory and then sketch a recursive version that could handle larger numbers of firms without re-solving the full system each time. The abstract also flags that myopic best-response iteration fails to converge even under linear demand, which is a useful negative result to record.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the canonical Stackelberg duopoly to a sequential triopoly in which each firm anticipates the reactions of all subsequent players. It reformulates the equilibrium conditions as a coupled fixed point of the best-response maps and asserts existence and uniqueness via fixed-point theory. The paper observes that myopic best-response dynamics fail to converge even under linear inverse demand and introduces a recursive formulation to analyze the limiting behavior as the number of participants grows.

Significance. If the existence and uniqueness results are placed on a fully rigorous footing, the work would supply a fixed-point-theoretic treatment of sequential oligopoly that separates static equilibrium from the stability of iterative best-response dynamics. The recursive formulation offers a potential route to large-market limits. These elements are standard in the field but would be strengthened by explicit verification of the mapping properties under the model's demand and cost assumptions.

major comments (2)

[Abstract and main theorem] Abstract and main result: The uniqueness claim rests on a coupled fixed-point reformulation, yet the abstract states that myopic best-response iteration does not converge under linear demand. This implies that the composite map is not contractive (spectral radius of its Jacobian at least 1). Uniqueness therefore requires an alternative argument, such as strict monotonicity of the profit functions or negative-definiteness of the relevant Hessian. The manuscript must exhibit the explicit best-response functions for the triopoly, the Jacobian of the coupled map, and the uniqueness proof (likely in the section presenting the fixed-point theorem).
[Linear demand analysis] Linear-demand case: The non-convergence statement is load-bearing for distinguishing equilibrium existence from dynamic stability, but without the concrete affine best-response expressions or eigenvalue calculation it is impossible to confirm that the spectral-radius observation does not also undermine uniqueness. Please supply the explicit map and its spectral properties under linear inverse demand.

minor comments (2)

The notation for the individual and coupled response functions should be introduced with explicit functional forms at the beginning of the technical development.
Standard references to Brouwer's fixed-point theorem, Schauder's theorem, and Banach's contraction principle should be cited when the respective conditions are invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments highlight important points about distinguishing static equilibrium from dynamic stability and the need for explicit calculations under linear demand. We address each major comment below and will revise the manuscript accordingly to provide the requested details and strengthen the presentation.

read point-by-point responses

Referee: [Abstract and main theorem] Abstract and main result: The uniqueness claim rests on a coupled fixed-point reformulation, yet the abstract states that myopic best-response iteration does not converge under linear demand. This implies that the composite map is not contractive (spectral radius of its Jacobian at least 1). Uniqueness therefore requires an alternative argument, such as strict monotonicity of the profit functions or negative-definiteness of the relevant Hessian. The manuscript must exhibit the explicit best-response functions for the triopoly, the Jacobian of the coupled map, and the uniqueness proof (likely in the section presenting the fixed-point theorem).

Authors: We agree that non-convergence of myopic best-response dynamics implies the composite map is not contractive, so uniqueness cannot rely on contraction mapping arguments. Our proof instead uses the sequential structure and a coupled fixed-point theorem that exploits the strict concavity of each firm's profit function (negative-definite Hessian with respect to its own output) together with the monotonicity properties induced by the Stackelberg anticipation. To make this fully transparent, we will add the explicit best-response functions for the three firms, derive the Jacobian of the coupled map, and expand the uniqueness argument in the fixed-point section, explicitly noting that it does not require spectral radius less than one. revision: yes
Referee: [Linear demand analysis] Linear-demand case: The non-convergence statement is load-bearing for distinguishing equilibrium existence from dynamic stability, but without the concrete affine best-response expressions or eigenvalue calculation it is impossible to confirm that the spectral-radius observation does not also undermine uniqueness. Please supply the explicit map and its spectral properties under linear inverse demand.

Authors: We acknowledge that the current manuscript does not include the explicit affine best-response expressions or the eigenvalue analysis for the linear-demand case. In the revision we will derive the closed-form best-response functions under linear inverse demand and standard quadratic costs, construct the composite map, and compute its Jacobian at the equilibrium. We will show that the largest eigenvalue has magnitude at least 1, confirming non-convergence of iterations, while uniqueness continues to hold via the monotonicity argument of the coupled fixed-point theorem. This addition will be placed in a new subsection on the linear-demand specialization. revision: yes

Circularity Check

0 steps flagged

No circularity: direct application of standard fixed-point theorems to response functions

full rationale

The derivation recasts Stackelberg equilibrium conditions as a coupled fixed point of best-response maps and invokes Brouwer/Schauder-type theorems for existence together with an (unspecified) uniqueness argument. No equation or claim reduces the equilibrium to a fitted parameter, a self-defined quantity, or a prior result by the same authors; the abstract explicitly distinguishes the fixed-point existence claim from the separate observation that myopic iteration need not converge. The result therefore rests on external mathematical theorems applied to the model's primitives rather than on any self-referential reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of coupled fixed-point theorems to the best-response functions of the sequential triopoly; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Coupled fixed-point theorems from functional analysis apply to the system of best-response functions under the model's demand and cost assumptions.
The existence-uniqueness result is obtained by reformulating equilibrium conditions to invoke these theorems.

pith-pipeline@v0.9.0 · 5375 in / 1310 out tokens · 38128 ms · 2026-05-07T12:42:11.578748+00:00 · methodology

discussion (0)

Reference graph

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converges to the ﬁxed point ξ

there is a unique ﬁxed point ξ ∈ X of T and, moreover, for any initial guess x0 ∈ X, the iterated sequence xn = T xn− 1 for n = 1 , 2, . . . converges to the ﬁxed point ξ
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there holds a priori error estimate: ρ(ξ, x n) ≤ kn 1− k ρ(x0, x 1)
[54]

there holds a posteriori error estimate: ρ(ξ, x n) ≤ k 1− k ρ(xn− 1, x n)
[55]

the rate of convergence is: ρ(ξ, x n) ≤ kρ(ξ, x n− 1), where k = k1+k2+k3 1− k2− k3 and ki, i = 1, 2, 3 are the constants from (17). A.3 N -tupled Fixed Points Following a sequence of articles, dealing with coupled ﬁxed points, sta rting with [10, 22], tripled ﬁxed points, starting with [9], the ideas were generalized to N tuples of ﬁxed points [41]. The ...