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Ironing Without Concavification

Filip Tokarski

TL;DR

This note addresses standard single-agent screening with a monotonicity constraint by solving a relaxed problem without monotonicity and then optimally truncating the resulting allocation to enforce monotone behavior, with the objective $F[x] = ∫_0^1 J(x(θ), θ) dθ$. It shows that, when a solution exists, the optimum can be obtained from a truncated, piecewise-monotone family $x^*_v$ determined by the monotonicity-change points of the relaxed solution $x_R$, and it reduces the search to a finite-dimensional problem over $v$. The paper provides a constructive algorithm under the assumptions that the virtual value $J$ is concave in the allocation and that allocations lie in a compact interval, and it proves a second result by induction that the algorithm converges across the set of critical points. Importantly, these results hold without requiring continuity, differentiability, or concavity of $J$, and extend to discrete allocations, broadening ironing-like methods beyond classical concavification.

Abstract

I propose a new approach to solving standard screening problems when the monotonicity constraint binds. A simple geometric argument shows that when virtual values are quasi-concave, the optimal allocation can be found by appropriately truncating the solution to the relaxed problem. I provide an algorithm for finding this optimal truncation when virtual values are concave.

Ironing Without Concavification

TL;DR

This note addresses standard single-agent screening with a monotonicity constraint by solving a relaxed problem without monotonicity and then optimally truncating the resulting allocation to enforce monotone behavior, with the objective . It shows that, when a solution exists, the optimum can be obtained from a truncated, piecewise-monotone family determined by the monotonicity-change points of the relaxed solution , and it reduces the search to a finite-dimensional problem over . The paper provides a constructive algorithm under the assumptions that the virtual value is concave in the allocation and that allocations lie in a compact interval, and it proves a second result by induction that the algorithm converges across the set of critical points. Importantly, these results hold without requiring continuity, differentiability, or concavity of , and extend to discrete allocations, broadening ironing-like methods beyond classical concavification.

Abstract

I propose a new approach to solving standard screening problems when the monotonicity constraint binds. A simple geometric argument shows that when virtual values are quasi-concave, the optimal allocation can be found by appropriately truncating the solution to the relaxed problem. I provide an algorithm for finding this optimal truncation when virtual values are concave.
Paper Structure (6 sections, 6 theorems, 20 equations, 2 figures, 1 algorithm)

This paper contains 6 sections, 6 theorems, 20 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem
  3. Structure of the solution
  4. Proof of Theorem \ref{['th:1']}.
  5. Solution algorithm
  6. Proof of Theorem \ref{['th:2']}.

Key Result

Theorem 1

If the planner's problem has a solution, it has a solution of the form $x^*_v$ for some $v\in \mathcal{X}^{|\mathcal{I}|-1}$. This solution can be recovered by solving:

Figures (2)

  • Figure 1: Algorithm \ref{['alg:1']} recursively transforming $x_R$ (blue) into subsequent $\mathop{\mathrm{T}}\nolimits[x,v^*,n]$ (red).
  • Figure :

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • Proposition 1
  • ...and 1 more