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Decision Making under Costly Sequential Information Acquisition: the Paradigm of Reversible and Irreversible Decisions

Renyuan Xu, Thaleia Zariphopoulou, Luhao Zhang

TL;DR

This paper develops a novel framework for decision making under costly sequential information acquisition with reversible choices between a known product $A$ and an unknown product $B$. The DM learns via two information regimes and can revert a first choice at a cost, yielding a hierarchical, nested optimal stopping problem analyzed through viscosity solutions. The authors establish existence, uniqueness, and regularity results, characterize exploration vs. product-selection regions, and provide comparative insights relative to the standard irreversible model under both Poisson and Gaussian refined signals. The approach unifies sequential decisions, information costs, and evolving information sources, with broad implications for AI adoption, data service choices, healthcare, and finance, and offers a flexible methodological framework for future extensions.

Abstract

Decision making in modern stochastic systems, including e-commerce platforms, financial markets and healthcare systems, has evolved into a multifaceted process that combines information acquisition and adaptive information sources. This paper initiates a study on such integrated settings, where these elements are not only fundamental but, also, interact in a complex and stochastically intertwined manner. We introduce a relatively simple model, which, however, captures the involved novel elements. A decision maker (DM) may choose between an established product $A$ of known value and a new product $B$ whose value is unknown. In parallel, the DM observes signals about the unknown value of product $B$ and can, also, opt to exchange it for product $A$ if $B$ is initially chosen. Mathematically, the model gives rise to sequential optimal stopping problems with distinct informational regimes (before and after buying product $B$), differentiated by the initial, coarser signal and the subsequent, more accurate one. We analyze in detail the underlying problems using predominantly viscosity solution techniques, departing from the existing literature on information acquisition which is based on traditional optimal stopping arguments. More broadly, the modeling approach introduced herein offers a novel framework for developing more complex interactions among decisions, information sources and information costs in stochastic environments, through a sequence of nested obstacle problems.

Decision Making under Costly Sequential Information Acquisition: the Paradigm of Reversible and Irreversible Decisions

TL;DR

This paper develops a novel framework for decision making under costly sequential information acquisition with reversible choices between a known product and an unknown product . The DM learns via two information regimes and can revert a first choice at a cost, yielding a hierarchical, nested optimal stopping problem analyzed through viscosity solutions. The authors establish existence, uniqueness, and regularity results, characterize exploration vs. product-selection regions, and provide comparative insights relative to the standard irreversible model under both Poisson and Gaussian refined signals. The approach unifies sequential decisions, information costs, and evolving information sources, with broad implications for AI adoption, data service choices, healthcare, and finance, and offers a flexible methodological framework for future extensions.

Abstract

Decision making in modern stochastic systems, including e-commerce platforms, financial markets and healthcare systems, has evolved into a multifaceted process that combines information acquisition and adaptive information sources. This paper initiates a study on such integrated settings, where these elements are not only fundamental but, also, interact in a complex and stochastically intertwined manner. We introduce a relatively simple model, which, however, captures the involved novel elements. A decision maker (DM) may choose between an established product of known value and a new product whose value is unknown. In parallel, the DM observes signals about the unknown value of product and can, also, opt to exchange it for product if is initially chosen. Mathematically, the model gives rise to sequential optimal stopping problems with distinct informational regimes (before and after buying product ), differentiated by the initial, coarser signal and the subsequent, more accurate one. We analyze in detail the underlying problems using predominantly viscosity solution techniques, departing from the existing literature on information acquisition which is based on traditional optimal stopping arguments. More broadly, the modeling approach introduced herein offers a novel framework for developing more complex interactions among decisions, information sources and information costs in stochastic environments, through a sequence of nested obstacle problems.
Paper Structure (22 sections, 19 theorems, 181 equations, 8 figures)

This paper contains 22 sections, 19 theorems, 181 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Modeling framework.
  4. Technical contribution.
  5. Related Literature.
  6. Our Contributions.
  7. The general decision making model under costly sequential information
  8. Special case: The existing single (irreversible) decision model.
  9. The new general decision model with reversible decisions; nested optimal stopping problems
  10. General regularity results and sensitivity analysis
  11. The "exploration" and the "product - selection" regions
  12. The single (irreversible) decision problem with general information acquisition costs
  13. Sensitivity analysis for general information costs
  14. Constant information acquisition cost
  15. The general model: costly sequential information acquisition and (ir)reversibility of decisions
  16. ...and 7 more sections

Key Result

Lemma 2

There exists a constant $\rho_0>0$ such that for each $\rho \ge \rho_0$, the value function $V$ is Lipschitz continuous on $\left[ 0,1\right]$.

Figures (8)

  • Figure 1: Timeline of the decisions.
  • Figure 2: Timeline of the decisions when the initial choice is product $B$.
  • Figure 3: Viscosity supersolution $v$.
  • Figure 4: Viscosity subsolution $u$.
  • Figure 5: $V_1^B$ defined in \ref{['V-B']} (in orange) v.s. $V_1$ defined in \ref{['G-general']} (in dashed blue).
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 1: Nested Decision Making under Sequential Differential Information.
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 28 more