On time-inconsistent stopping problems and mixed strategy stopping times

Abstract

A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean-variance and a variance problem are studied. The state process is a general one-dimensional Itô diffusion.

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

“…Stopping problems with (i) and (ii) can be formulated and studied in the framework of the present paper whereas stopping problems of type (iii) can be dealt with in the framework studied in [10]. Stopping problems with (ii)-(iii) are described below.…”

“…Mean-variance problems are, however, time-inconsistent for the fundamentally different reason that the expression to be maximized is a non-linear function of the expected value of a reward. Hence, mean-variance problems cannot be studied in the present framework (a mean-variance problem is however studied in [10]). A version of the mean-variance stopping problem is to find a stopping time τ that maximizes…”

“…Furthermore, we demand that the decision whether to stop or not should depend directly only on the preferences of each agent x and not, for example, on the outcome of some randomization procedure, or on events from the past -that is, we consider pure stopping strategies cf. Definition 2.1 and Remark 2.5 (while mixed stopping strategies are considered in [10]). These conditions are, in reverse order, formalized in the following definitions.…”

“…In the present paper we consider only pure stopping strategies, which we define as entry times into sets in the state space, see Remark 2.5 for a motivation. In [10] we consider mixed strategies for time-inconsistent stopping problems of a different class compared to the present paper. Remark 2.5 contains an explanation of some of the game-theory terms used in the present paper.…”

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

“…The second notion of equilibrium, which we call weak equilibrium in this paper, is proposed in [3] and further investigated in [2]. Following [3], we have the definition of weak equilibrium (we adapt the definition slightly for our setting).…”