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On time-consistent equilibrium stopping under aggregation of diverse discount rates

Shuoqing Deng, Xiang Yu, Jiacheng Zhang

TL;DR

This paper develops a framework for time-consistent equilibrium stopping when a group aggregates diverse discount rates through a random rate $\rho$ distributed by $F_{\rho}$ and an aggregation attitude $\phi$. By embedding a consistent-planning principle via a policy-improvement operator $\Theta$, the authors characterize time-consistent mild equilibria as fixed points in the one-dimensional diffusion setting, showing that the smallest barrier equilibrium $[0,a^*]$ is optimal under natural sufficient conditions, and that the optimal equilibrium is a weak equilibrium when $\phi$ is concave. They provide explicit constructions of $a^*$ and discuss invariance properties with respect to the aggregation function, illustrating how $\phi$ and the distribution of discount rates shape outcomes. Through concrete examples with geometric Brownian motion and a 3D Bessel process, they highlight cases where the optimal equilibrium coincides with or diverges from the smallest equilibrium, including the possibility of nonexistence of a global optimum depending on model parameters. These results illuminate the trade-offs in collective decision making under model uncertainty and diverse time preferences, with implications for intergenerational investment and long-horizon stopping problems.

Abstract

This paper studies a central planner's decision making on behalf of a group of members with diverse discount rates. In the context of optimal stopping, we work with an aggregation preference to incorporate all discount rates via an attitude function that reflects the aggregation rule chosen by the central planner. The problem formulation is also applicable to single agent's stopping problem with uncertain discount rate, where our aggregation preference coincides with the conventional smooth ambiguity preference. The resulting optimal stopping problem is time inconsistent, for which we develop an iterative approach using consistent planning and characterize all time-consistent mild equilibria as fixed points of an operator in the setting of one-dimensional diffusion processes. We provide some sufficient conditions on the underlying models and the attitude function such that the smallest mild equilibrium attains the optimal equilibrium. In addition, we show that the optimal equilibrium is a weak equilibrium. When the sufficient condition of the attitude function is violated, we illustrate by various examples that the characterization of the optimal equilibrium may differ significantly from some existing results for a single agent, which now sensitively depends on the attitude function and the diversity distribution of discount rates within the group.

On time-consistent equilibrium stopping under aggregation of diverse discount rates

TL;DR

This paper develops a framework for time-consistent equilibrium stopping when a group aggregates diverse discount rates through a random rate distributed by and an aggregation attitude . By embedding a consistent-planning principle via a policy-improvement operator , the authors characterize time-consistent mild equilibria as fixed points in the one-dimensional diffusion setting, showing that the smallest barrier equilibrium is optimal under natural sufficient conditions, and that the optimal equilibrium is a weak equilibrium when is concave. They provide explicit constructions of and discuss invariance properties with respect to the aggregation function, illustrating how and the distribution of discount rates shape outcomes. Through concrete examples with geometric Brownian motion and a 3D Bessel process, they highlight cases where the optimal equilibrium coincides with or diverges from the smallest equilibrium, including the possibility of nonexistence of a global optimum depending on model parameters. These results illuminate the trade-offs in collective decision making under model uncertainty and diverse time preferences, with implications for intergenerational investment and long-horizon stopping problems.

Abstract

This paper studies a central planner's decision making on behalf of a group of members with diverse discount rates. In the context of optimal stopping, we work with an aggregation preference to incorporate all discount rates via an attitude function that reflects the aggregation rule chosen by the central planner. The problem formulation is also applicable to single agent's stopping problem with uncertain discount rate, where our aggregation preference coincides with the conventional smooth ambiguity preference. The resulting optimal stopping problem is time inconsistent, for which we develop an iterative approach using consistent planning and characterize all time-consistent mild equilibria as fixed points of an operator in the setting of one-dimensional diffusion processes. We provide some sufficient conditions on the underlying models and the attitude function such that the smallest mild equilibrium attains the optimal equilibrium. In addition, we show that the optimal equilibrium is a weak equilibrium. When the sufficient condition of the attitude function is violated, we illustrate by various examples that the characterization of the optimal equilibrium may differ significantly from some existing results for a single agent, which now sensitively depends on the attitude function and the diversity distribution of discount rates within the group.
Paper Structure (10 sections, 12 theorems, 161 equations, 3 figures)

This paper contains 10 sections, 12 theorems, 161 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Characterization of Time-consistent Equilibria
  4. Sufficient Conditions on the Existence of Optimal Equilibrium
  5. Discussions on the conditions imposed
  6. Main results
  7. Connection between the optimal equilibrium and weak equilibrium
  8. Impacts on the Optimal Equilibrium by Attitude Function and Diversity Distribution
  9. Geometric Brownian motion
  10. The case of Bessel process with $n=3$

Key Result

Lemma 2.2

If $X$ is a one dimension diffusion process, we have $\tau(x,R)=\tau(x,\bar{R})=0$ for any $x\in \bar{R}$ and $R\in\mathcal{B}(\mathbb{R})$, where $\bar{R}$ is the closure of $R$. Consequently, $\phi(g(x)) = J(x,R)$, for any $x\in \bar{R}$ and $R\in\mathcal{B}(\mathbb{R})$.

Figures (3)

  • Figure 1: Different regions to determine $\Lambda(x,a)$.
  • Figure 2: Optimal barrier $a^{**}(x)$ that attains the maximum of $\Lambda(x,a)$ for geometric Brownian Motion. (Red line and red region)
  • Figure 3: Optimal barrier $a^{**}(x)$ that attains the maximum of $\Lambda(x,a)$ for Bessel case. (Red line and red region)

Theorems & Definitions (33)

  • Remark 2.1
  • Definition 2.1: Stopping Policy
  • Lemma 2.2
  • Definition 3.1: Equilibrium
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • ...and 23 more