Paper: arXiv 2302.07470

Authors: Shuoqing Deng, Xiang Yu, Jiacheng Zhang

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework for time-consistent optimal stopping under diverse discount rates using diffusion processes and fixed-point operators. While highly novel and mathematically sophisticated on the theoretical front, it is purely theoretical and lacks empirical validation or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve time-inconsistent optimal stopping for aggregated diverse discount rates] --> B[Input Data: Diffusion process, diversity distribution of rates, and planner's attitude function]
    B --> C[Framework: Formulate aggregation preference & smooth ambiguity model]
    C --> D[Methodology: Apply consistent planning via an iterative operator approach]
    D --> E[Computation: Characterize time-consistent mild equilibria as operator fixed points]
    E --> F[Findings: Smallest mild equilibrium is optimal & weak equilibrium under sufficient conditions]
    E --> G[Findings: Violating conditions reveals high sensitivity to attitude function and rate diversity]