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Choosing a consultant in a dynamic investment problem

Yuval Cornfeld, Ehud Lehrer, Eilon Solan

TL;DR

The paper investigates dynamic information acquisition in a two-state investment setting where a DM can pay a fixed cost to consult multiple experts before choosing between two investments. It provides a rigorous Bayesian framework with a Bellman-recursion structure, proving the existence and Markovian nature of optimal strategies, and showing that the value function $V_J(p_0,c)$ is continuous and convex in $(p_0,c)$. A key result is that if a consultant reveals the state with positive probability and $c$ is sufficiently small, that consultant is employed in all optimal policies; under a rational-ratio condition, the value function becomes piecewise bilinear in the prior and cost, enabling finite characterization of optimal strategies. The three-signal extension yields convex revealing regions and delineates when revealers or estimators dominate, along with an equivalence transformation that helps compare consultants with different information structures. These findings provide structural insights into sequential information acquisition with practical implications for investment screening and diagnostic decision-making.

Abstract

Consider a dynamic decision-making scenario where at every stage the investor has to choose between investing in one of two projects or gathering more information. At each stage, the investor may seek counsel from one of several consultants, who, for a fixed cost, provide partial information about the realized state. We explore the optimal strategy and its dependence on the belief and the consultation cost. Our analysis reveals that if one of the consultants discloses the state with a nonzero probability, this consultant will be used in any optimal strategy, provided the consultation cost is sufficiently small.

Choosing a consultant in a dynamic investment problem

TL;DR

The paper investigates dynamic information acquisition in a two-state investment setting where a DM can pay a fixed cost to consult multiple experts before choosing between two investments. It provides a rigorous Bayesian framework with a Bellman-recursion structure, proving the existence and Markovian nature of optimal strategies, and showing that the value function is continuous and convex in . A key result is that if a consultant reveals the state with positive probability and is sufficiently small, that consultant is employed in all optimal policies; under a rational-ratio condition, the value function becomes piecewise bilinear in the prior and cost, enabling finite characterization of optimal strategies. The three-signal extension yields convex revealing regions and delineates when revealers or estimators dominate, along with an equivalence transformation that helps compare consultants with different information structures. These findings provide structural insights into sequential information acquisition with practical implications for investment screening and diagnostic decision-making.

Abstract

Consider a dynamic decision-making scenario where at every stage the investor has to choose between investing in one of two projects or gathering more information. At each stage, the investor may seek counsel from one of several consultants, who, for a fixed cost, provide partial information about the realized state. We explore the optimal strategy and its dependence on the belief and the consultation cost. Our analysis reveals that if one of the consultants discloses the state with a nonzero probability, this consultant will be used in any optimal strategy, provided the consultation cost is sufficiently small.
Paper Structure (11 sections, 8 theorems, 18 equations)

This paper contains 11 sections, 8 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. The structure of the paper
  4. The Model and the Main Results
  5. The Model
  6. Fundamental properties of the investment problem
  7. Revealing consultants
  8. Consultants with a rational ratio
  9. Three-signal investment problem
  10. Proofs
  11. Proof of Lemma \ref{['theorem:simple:properties']}

Key Result

Lemma 1

For every investment problem, a Markovian optimal strategy exists. Moreover, $V_J$ is continuous, convex in $p_0$ for every fixed $c$, convex and monotonically decreasing in $c$ for every fixed $p_0$, and satisfies the following recursive equation: where $P_j(p_0,s)=p_0 S_j(s|\omega) + (1-p_0)S_j(s|\omega^c)$ is the probability that when the prior is $p_0$, the signal provided by consultant $j$ i

Theorems & Definitions (23)

  • Definition 1
  • Remark 1
  • Lemma 1
  • Example 1
  • Example 2
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Remark 2
  • ...and 13 more