On the Hausdorff dimension and singularities of the monopolist's free boundary curve

1. Problem Statement

The paper addresses the multidimensional monopolist's problem in mechanism design, specifically within the principal-agent framework with asymmetric information.

• Context: A monopolist seeks to price a line of products to maximize profit against a continuum of anonymous buyers with private types $x \in X \subset [0, \infty)^2$.
• Mathematical Formulation: The problem reduces to minimizing a linearly perturbed Dirichlet energy among non-negative convex functions $u$ (representing indirect utility). The objective functional is:
  $$\Pi(u) := \int_X \left[ x \cdot \nabla u(x) - u(x) - \frac{1}{2}|\nabla u(x)|^2 \right] dH^n$$
  subject to $u$ being convex.
• Key Geometric Structure: The domain $X$ is partitioned into three regions based on the dimension of the "leaves" (equivalence classes of types choosing the same product):
  1. Exclusion Region ($\Omega_0$): Buyers priced out ($u=0$).
  2. Bunching Region ($\Omega_1$): Buyers choosing identical products (rays where $u$ is affine).
  3. Customization Region ($\Omega_2$): Buyers choosing distinct products ($u$ is strictly convex).
• The Core Challenge: The boundaries between these regions are free boundaries. The paper focuses on the interface $\partial \Omega_1 \cap \partial \Omega_2$ (the boundary between bunching and customization). Previous work established that this boundary is a free boundary problem, but the regularity of the "obstacle" (the minimal convex extension of $u$ on $\Omega_1$) was only known to be $C^{1,1}$, which is insufficient for classical obstacle problem regularity theory (requiring $C^{2,\alpha}$).

2. Methodology

The authors employ a combination of geometric analysis, variational methods, and regularity theory for free boundary problems.

• Reduction to Obstacle Problem:
  The authors focus on "tame" free boundary points ($T$), where the bunching rays intersect the fixed boundary $\partial X$ smoothly and transversally. Using local coordinates $(r, t)$ adapted to the foliation of rays, they reduce the study of the free boundary near a tame point to a classical obstacle problem:
  $$\Delta v = f \mathbb{1}_{\{v>0\}}$$
  where $v = u - u_1$ ($u_1$ being the minimal convex extension of $u$ on the bunching region) and $f$ is a source term derived from the geometry.

• Regularity of the Endogenous Obstacle:
  A critical technical step is proving that the obstacle $u_1$ gains regularity. While general theory suggests $u \in C^{1,1}$, the authors prove (Proposition 2.5) that for the monopolist's problem, the obstacle $u_1$ is actually $C^2$. This is achieved by analyzing the modulus of continuity of the ray length function $R(t)$ and showing that the Laplacian of the extension depends continuously on $R(t)$.

• Low-Regular Obstacle Theory (Blank's Framework):
  Since the source term $f$ in the reduced obstacle problem is only continuous (not necessarily $C^\alpha$), the authors cannot use Caffarelli's classical $C^{1,\alpha}$ theory. Instead, they utilize the framework developed by Blank and Hao, which handles obstacles with lower regularity. This framework establishes a dichotomy between regular and singular points based on the density of the contact set.

• Blow-up Analysis:
  The authors use blow-up techniques (rescaling the solution near a boundary point) to classify points on the free boundary. They leverage the specific ray structure of the monopolist's problem to prove the uniqueness of the blow-up at singular points, a property not guaranteed in general obstacle problems.

3. Key Contributions and Results

A. Discreteness of Singular Points (Theorem 1.1 & Theorem 3.7)

The paper proves that the set of singular points ($S$) on the tame free boundary $T$ is discrete.

• Definition: A point is singular if the density of the customization region $\Omega_2$ at that point is 0 (i.e., $\lim_{r\to 0} |\Omega_2 \cap B_r| / |B_r| = 0$). Regular points have a density of $1/2$.
• Result: Singular points cannot accumulate within the interior of the tame free boundary. They can only accumulate at the fixed boundary $\partial X$ or at specific points on the boundary of the exclusion region.
• Mechanism: This is proven by showing that singular points correspond to strict local maxima of the ray length function $R(t)$. If a sequence of singular points converged, it would contradict the strict local maximality via blow-up analysis.

B. Hausdorff Dimension Bound (Theorem 1.1 & Corollary 4.4)

The authors establish a sharp bound on the size of the free boundary:

• Result: The Hausdorff dimension of the tame free boundary $T$ is at most 1 ($\dim_H(T) \le 1$).
• Significance: This confirms that the free boundary is a curve (or a collection of curves) rather than a surface with fractal dimension, despite the low regularity of the underlying obstacle.
• Regular Points: The set of regular points $R$ forms a locally $\alpha$-bi-Hölder curve for any $\alpha \in (0, 1)$.

C. Partial Regularity under Dini Conditions (Theorem 1.1(4) & Proposition 4.6)

The paper provides conditions for higher smoothness:

• If the Laplacian of the obstacle ($\Delta u_1$) satisfies the Dini continuity condition, the free boundary becomes $C^1$ (smooth) outside a closed set with empty relative interior.
• If the domain is a square (the classic Rochet-Choné example), the singularities are restricted to a discrete set of points (at most one on the diagonal and two at the corners).

D. Generalization to Polygons (Corollary 1.2)

The results extend to convex polygons and domains with finitely many non-smooth points on the boundary. The authors show that outside a specific closed set $X_1^0$ (where rays start and end on the boundary), the free boundary has Hausdorff dimension 1 and density 1/2 almost everywhere.

4. Significance and Impact

1. Resolution of a Long-Standing Challenge: The multidimensional monopolist's problem has been notoriously difficult due to the "bunching" phenomenon. This paper provides the first rigorous proof that the free boundary separating bunching from customization is essentially a 1-dimensional curve with controlled singularities.
2. Bridging Economic Theory and PDEs: The work connects economic mechanism design with advanced free boundary PDE theory. It demonstrates how specific economic structures (the ray foliation) can be exploited to overcome the lack of regularity in the obstacle, a problem that standard PDE theory could not solve alone.
3. Advancement of Obstacle Problem Theory: By applying and extending Blank's low-regularity obstacle theory to a problem with an endogenous obstacle (one generated by the solution itself), the authors push the boundaries of what is known about the regularity of free boundaries when the obstacle is only $C^2$ (or even $C^{1,1}$ in the general case, though here $C^2$ is proven).
4. Validation of Numerical Observations: The theoretical results confirm numerical simulations (e.g., by Rochet-Choné, Ekeland-Moreno-Bromberg) that suggested the free boundary is a curve, providing the necessary mathematical justification for those observations.

In summary, the paper establishes that for a broad class of domains, the monopolist's free boundary is a well-behaved curve of Hausdorff dimension 1, with singularities that are isolated and discrete, fundamentally clarifying the geometric structure of optimal pricing strategies in multidimensional markets.