Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems

Abstract

We study an optimal stopping problem under non-exponential discounting, where the state process is a multi-dimensional continuous strong Markov process. The discount function is taken to be log sub-additive, capturing decreasing impatience in behavioral economics. On strength of probabilistic potential theory, we establish the existence of an optimal equilibrium among a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfying a boundary condition. This generalizes the existence of optimal equilibria for one-dimensional stopping problems in prior literature.

Who Are I: Time Inconsistency and Intrapersonal Conflict and Reconciliation

“…As discussed in the above, immediate stop is always a mild equilibrium; so it is expected that there exist multiple mild intra-personal equilibrium stopping rules; see Huang and Nguyen-Huu (2018) and . To address the issue of multiplicity, and Huang and Wang (2020) consider, in the setting of an infinite-horizon, continuoustime optimal stopping under nonexponential discounting, the "optimal" mild intra-personal equilibrium stopping rule τ * which achieves the maximum of J(t, x; τ ) over τ ∈ E for all t ∈ [0, T ), x ∈ X, where E is the set of all mild intra-personal equilibrium stopping rules. Bayraktar et al (2021) compare mild intra-personal equilibrium stopping rules with weak (respectively strong) intra-personal equilibrium stopping rules obtained by embedding optimal stopping into stochastic control and then applying Definition 2 (respectively Definition 3).…”