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Intrinsic Geometry and the Stability of Minimum Differentiation

Aldric Labarthe, Yann Kerzreho

TL;DR

The paper develops a unified framework for horizontal differentiation by modeling the feasible product space as a smooth Riemannian manifold, enabling a joint analysis of curvature, dimension, and symmetry on equilibrium stability. Demand is specified via Multinomial Logit, with a geometric forces perspective that posits a Centrifugal force promoting differentiation and a Centripetal force pulling firms toward the geometric median; a geometric stability condition $oldsymbol{ ext Ψ}(ar{y}) gtr oldsymbol{ ext ξ} oldsymbol{ ext Φ}(ar{y})$ governs when concentration is sustainable. It proves that negative curvature and higher intrinsic dimension stabilize minimum differentiation, while symmetry can preclude concentration; it also characterizes partially concentrated equilibria in product spaces and analyzes canonical spaces (Hotelling, Salop, hypercube, cylinder). The work provides explicit results for equilibrium existence, stability thresholds, and welfare implications across spaces, showing how intrinsic geometry shapes consumer heterogeneity and substitution; it suggests empirical pathways to infer the underlying manifold from data and to distinguish equilibrium clustering from collusion. Overall, the framework offers a principled geometric lens for understanding when firms cluster versus differentiate and reveals the deep role of market geometry in shaping competitive outcomes and welfare.

Abstract

We develop a framework for horizontal differentiation in which firms compete on a product manifold representing the feasible combinations of characteristics. This approach generalizes both the Hotelling line and Salop circle to any Riemannian space, allowing for a unified analysis of product space. We show that equilibrium existence and stability are governed by intrinsic geometric properties, specifically curvature, symmetry and dimension. We prove that negative curvature and high intrinsic dimension act as stabilizers of minimum differentiation equilibria, moving the analysis beyond the symmetry-induced instabilities found in simpler, fixed domains. By characterizing curvature as a measure of consumer heterogeneity, we demonstrate that intrinsic geometry is a fundamental determinant of competitive outcomes.

Intrinsic Geometry and the Stability of Minimum Differentiation

TL;DR

The paper develops a unified framework for horizontal differentiation by modeling the feasible product space as a smooth Riemannian manifold, enabling a joint analysis of curvature, dimension, and symmetry on equilibrium stability. Demand is specified via Multinomial Logit, with a geometric forces perspective that posits a Centrifugal force promoting differentiation and a Centripetal force pulling firms toward the geometric median; a geometric stability condition governs when concentration is sustainable. It proves that negative curvature and higher intrinsic dimension stabilize minimum differentiation, while symmetry can preclude concentration; it also characterizes partially concentrated equilibria in product spaces and analyzes canonical spaces (Hotelling, Salop, hypercube, cylinder). The work provides explicit results for equilibrium existence, stability thresholds, and welfare implications across spaces, showing how intrinsic geometry shapes consumer heterogeneity and substitution; it suggests empirical pathways to infer the underlying manifold from data and to distinguish equilibrium clustering from collusion. Overall, the framework offers a principled geometric lens for understanding when firms cluster versus differentiate and reveals the deep role of market geometry in shaping competitive outcomes and welfare.

Abstract

We develop a framework for horizontal differentiation in which firms compete on a product manifold representing the feasible combinations of characteristics. This approach generalizes both the Hotelling line and Salop circle to any Riemannian space, allowing for a unified analysis of product space. We show that equilibrium existence and stability are governed by intrinsic geometric properties, specifically curvature, symmetry and dimension. We prove that negative curvature and high intrinsic dimension act as stabilizers of minimum differentiation equilibria, moving the analysis beyond the symmetry-induced instabilities found in simpler, fixed domains. By characterizing curvature as a measure of consumer heterogeneity, we demonstrate that intrinsic geometry is a fundamental determinant of competitive outcomes.

Paper Structure

This paper contains 44 sections, 23 theorems, 81 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement and game analysis
  3. General setup
  4. Objectives and optimization
  5. Interpretation and Relations to existing approaches
  6. A simultaneous game
  7. Endogenous Geometry versus Exogenous Density
  8. Theoretical Study of the concentrated equilibrium
  9. General Case
  10. Interior Nash equilibrium
  11. A Geometric Stability Condition
  12. On the consequences of symmetries and homogeneity
  13. On the consequences of curvature, volume and dimension
  14. Is the Nash equilibrium reachable?
  15. Welfare Analysis
  16. ...and 29 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be a compact, strongly geodesically convexA Riemannian manifold is strongly geodesically convex if any two points within it are connected by a unique minimizing geodesic that lies entirely within the set. Riemannian manifold. There exists a threshold $\beta_0>0$ such that for all $

Figures (8)

  • Figure 1: Curvature and Volume Measure Distortion. The top row displays: (A) a hyperbolic saddle (negative curvature), (B) a Euclidean plane (zero curvature), and (C) a Gaussian bump (positive curvature). The bottom row illustrates the density of the intrinsic volume element, projected onto the parametric domain. The grayscale heatmap indicates the local magnitude of the volume measure, where darker shading corresponds to higher density. The dark red contours represent level sets of constant geodesic distance from the origin. All manifolds have the same total volume, and all level sets represent the same distance in the three examples.
  • Figure 2: Voronoi Diagram of the market on a non-flat torus (a non-flat torus surface). As predicted by Proposition \ref{['thm:concentrated_nothomoorsym']}, firms cannot select the concentrated equilibrium. Simulated using $\beta =20, c=0.2, N=4$
  • Figure 3: Geometric condition for stability. The solid grey ellipsoid ($\mathbf{\Psi}$) must be strictly contained within the hatched circular region ($\xi \mathbf{\Phi}$). The stability margin is determined by $\|\mathbf{\Psi}\|_{op}$ (the shortest axis), which dictates the maximum stable learning rate (Corollary \ref{['cor:geo_stiffness']}).
  • Figure 4: Phase Diagram of the market structure on $\mathcal{M} = [0,1]$. The red line represents the analytical stability boundary derived in Eq. \ref{['eq:finalConcentratedOn01']}, background color amounts for the average market configuration observed in 5 simulations.
  • Figure 5: Simulation of the iterative process on $\mathcal{M} = [0,1]$. Simulations conducted with $N=3$, $c=0.2$ and linear costs. In this setting, the phase transition given by (eq. \ref{['eq:finalConcentratedOn01']}) is at $\bar{\beta} = 6$.
  • ...and 3 more figures

Theorems & Definitions (49)

  • Theorem 1
  • Definition 1: Concentrated equilibrium
  • Definition 2: Median of a manifold
  • Proposition 1
  • proof
  • Corollary 1
  • Proposition 2
  • Definition 3: Forces and Economic Scales
  • Theorem 2: The Geometric Stability Condition
  • Remark 1
  • ...and 39 more