Optimal equilibria for time‐inconsistent stopping problems in continuous time

Abstract

For an infinite-horizon continuous-time optimal stopping problem under non-exponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log sub-additive and the state process is one-dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. While there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula. MSC (2010): 93E20, 91G80.

On time-inconsistent stopping problems and mixed strategy stopping times

“…The development of the literature on the game-theoretic approach to time-inconsistent stopping problems is in an earlier stage. Recent papers include [1,9,11,20,21,22,24]. Section 1.2 describes references studying particular time-inconsistent stopping problems, while Section 2.1 contains a further review focusing on the choice of definition for pure and mixed strategies and equilibria.…”

On the Notions of Equilibria for Time-Inconsistent Stopping Problems in Continuous Time

“…Remark 2.1. Assumption 2.2 (i) is closely related to decreasing impatience in Behavioral Economics and Finance and commonly used when studying non-exponential discounting problems; see e.g., [8,7,4].…”

“…Note that x ∨ J(x, S) represents the value associated with the stopping region/strategy S. In [7] the existence of optimal equilibrium is established. A discrete-time version is in [8].…”

“…Moreover, we provide a new iteration method which directly constructs an optimal mild equilibrium and thus also establish its existence (see Thoerem 2.2). In [8,7], an optimal equilibrium is constructed by the intersection of all (mild) equilibriums. In principle, this requires us to first find all (mild) equilibria in order to get the optimal one, which may not be implementable in many cases.…”

Conditional Optimal Stopping: A Time-Inconsistent Optimization

“…Nevertheless, this concept of equilibrium is not the only one possible; in particular, first-order conditions are not sufficient for optimality in general. The recent study [21] introduces a stronger concept of optimality and highlights the differences. In [3,4] the authors study time-inconsistent control in discrete and continuous time, respectively, and the relation between them, for a general class of objectives that are a sum of an expected utility and a nonlinear function of an expected utility with possible dependence on the initial condition.…”