Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time

Abstract

A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined and allows for unambiguous economic interpretation. The terms “strong equilibria” and “weak equilibria” are coined for controls under the new and standard definitions, respectively. When the state process is a time-homogeneous continuous-time Markov chain, a careful asymptotic analysis gives complete characterizations of weak and strong equilibria. Thanks to the Kakutani–Fan fixed-point theorem, the general existence of weak and strong equilibria is also established under an additional compactness assumption. Our theoretic results are applied to a two-state model under nonexponential discounting. In particular, we demonstrate explicitly that there can be incentive to deviate from a weak equilibrium, which justifies the need for strong equilibria. Our analysis also provides new results for the existence and characterization of discrete-time equilibria under infinite horizon.

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

“…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.…”

“…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.…”

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

“…As ρ ε S ≥ ε > 0, the condition (1.7) does capture the deviation from stopping to continuing, and is much stronger than (1.3). However, there is still a drawback for (1.7): when the limit is equal to zero, it is possible that for all ε > 0 we have f (x) < E x [δ(ρ ε S )f (X ρ ε S )], and thus there is an incentive to deviate (see [2,Remark 3.5] and [13,1,6] for more details). Roughly speaking, this is similar to a critical point not necessarily being a local maximum in calculus.…”

“…This remedies the issue of weak equilibria mention in the above, and captures the economic meaning of "equilibrium" more accurately. Such kind of equilibria is also studied in [13,6] for time inconsistent control. Obviously, a strong equilibrium must be weak.…”

Consistent investment of sophisticated rank‐dependent utility agents in continuous time

“…22Definition 2.1 entails a “first‐order” condition, which is weaker than a more natural “zeroth‐order” condition that demands there should be no strategy locally performing better. Recently, Huang and Zhou (2019) and He and Jiang (2019) propose and study the notion of a strong equilibrium , which corresponds to the zeroth‐order condition.…”