Equilibrium concepts for time‐inconsistent stopping problems in continuous time

Abstract

A new notion of equilibrium, which we call strong equilibrium, is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐consistent stopping under decreasing impatience. Finance and Stochastics, 22(1), 69–95 and Christensen, S., & Lindensjö, K. (2018). On finding equilibrium stopping times for time‐inconsistent markovian problems. SIAM Journal on Control and Optimization, 56(6), 4228–4255, which in this paper are called mild equilibrium and weak equilibrium, respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous‐time Markov chain and the discount function is log subadditive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.

Optimal stopping under model ambiguity: A time‐consistent equilibrium approach

“…For the current real options valuation problem under model ambiguity, we will show that an optimal equilibrium also exists under appropriate conditions.Remark For time‐inconsistent stopping problems, an equilibrium can be defined as in the present paper (i.e., Definition 2.10, based on the fixed‐point approach in Huang and Nguyen‐Huu (2018)), as in Christensen and Lindensjö (2018) (based on the standard definition of an equilibrium for control problems in Ekeland and Lazrak (2006)), or as in Bayraktar et al. (2020) (based on “strong equilibria” for control problems in Huang and Zhou (2020a)). As argued in Bayraktar et al.…”

“…As argued in Bayraktar et al. (2020) and Huang and Zhou (2020a), the third kind of definition captures the idea of subgame perfect Nash equilibrium most accurately: it prevents deviation from the present strategy in a however small time interval starting from today—an ideal property that may not be achieved by an equilibrium of the first or the second kind. In a continuous‐time Markov chain model, Bayraktar et al.…”

“…In a continuous‐time Markov chain model, Bayraktar et al. (2020) analyze these three types of equilibria in detail, showing that an optimal equilibrium of the first kind (i.e., defined as in Definition 4.1) is in fact an equilibrium of the third kind. If such a result could be generalized to a diffusion model (which remains an open problem), an optimal equilibrium in Definition 4.1 would automatically possess the ideal property mentioned above.…”

Equilibria of Time-inconsistent Stopping for One-dimensional Diffusion Processes

“…However, this is not really captured in (1.3) after a second thought: In one dimensional setting, under very mild condition we have ρ S = 0 a.s., and thus (1.3) holds trivially. 1 That is, there is no actual deviation from stopping to continuing captured in (1.3).…”

“…This kind of equilibria is first proposed and studied in stopping problems in the context of non-exponential discounting in [7]. It is called mild equilibrium in [1] to distinguish from other equilibrium concepts. Mild equilibria are further considered in [8] and [10] where the time inconsistency is caused by probability distortion and model uncertainty respectively.…”