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

TL;DR

This paper develops a framework for time-consistent equilibrium stopping decisions for groups with diverse discount rates, using aggregation preferences and iterative methods to identify optimal and mild equilibria in diffusion processes.

Contribution

It introduces an iterative approach for time-consistent stopping under diverse discount rates and characterizes all mild equilibria as fixed points, extending to uncertain discount scenarios.

Findings

Smallest mild equilibrium can be optimal under certain conditions.

Optimal equilibrium is also a weak equilibrium.

Characterization of equilibria depends on the attitude function and discount diversity.

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.