Time-Consistent Stopping Under Decreasing Impatience

arXiv (Cornell University), 2024

How do decisions change with the economic environment and with time? This paper studies general nonstationary stopping problems and provides the methodological tools to answer these questions. First, we identify conditions that ensure a monotone relation between decisions' timing and outcomes. These conditions apply to a prevalent class of economic environments. Second, we develop a theory of monotone comparative statics for stopping problems, offering general and unifying qualitative insights into the decision-maker's value and stopping behavior. We apply our results to models of information acquisition, bankruptcy, irreversible investment, and option pricing to explain documented patterns at odds with current theories.

We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem, in which the decision maker uses a dynamic convex risk measure to evaluate future rewards. We also find a saddle point for an equivalent zero-sum game of control and stopping, between an agent (the "stopper") who chooses the termination time of the game, and an agent (the "controller", or "nature") who selects the probability measure.

IMA Journal of Mathematical Control and Information, 2015

We present a methodology for obtaining explicit solutions to infinite time horizon optimal stopping problems involving general, onedimensional, Itô diffusions, payoff functions that need not be smooth and state-dependent discounting. This is done within a framework based on dynamic programming techniques employing variational inequalities and links to the probabilistic approaches employing r-excessive functions and martingale theory. The aim of this paper is to facilitate the the solution of a wide variety of problems, particularly in finance or economics.

Advances in Applied Probability, 2011

In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

Stochastic Processes and their Applications, 2000

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a di erential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.

Siam Journal on Control and Optimization, 2000

Utility maximization problems of mixed optimal stopping/control type are considered, which can be solved by reduction to a family of related pure optimal stopping problems. Sufficient conditions for the existence of optimal strategies are provided in the context of continuoustime, Itô process models for complete markets. The mathematical tools used are those of optimal stopping theory, continuous-time martingales, convex analysis, and duality theory. Several examples are solved explicitly, including one which demonstrates that optimal strategies need not always exist.

SSRN Electronic Journal, 2020

In this paper, we provide evidence of history-dependent stopping behavior. Using data from an online chess platform, we estimate a dynamic discrete choice model in which an agent may have time non-separable preferences over the stochastic outcomes of their actions. We show that the agent's decisions cannot be reconciled in a model with time separable preferences and that there is substantial heterogeneity in preferences across players. In particular, there are two types of people: those who get discouraged by a loss and stop, and others, who get encouraged by failure and keep playing until a win. We show how to leverage the information about an agent's type in market design to achieve various welfare goals. A counterfactual analysis demonstrates that a matching algorithm that incorporates stopping behavior can significantly increase the length of play.

arXiv: Optimization and Control, 2019

In this paper we consider discrete and continuous time risk sensitive optimal stopping problem. Using suitable properties of the underlying Feller-Markov process we prove continuity of the optimal stopping value function and provide formula for the optimal stopping policy. Next, we show how to link continuous time framework with its discrete time analogue. By considering suitable approximations, we obtain uniform convergence of the corresponding value functions.

2004

This note derives the dynamic programming equation (DPE) to a differentiable Markov Perfect equilibrium in a problem with non-constant discounting and general functional forms. We begin with a discrete stage model and take the limit as the length of the stage goes to 0 to obtain the DPE corresponding to the continuous time problem. We characterize the multiplicity of equilibria under non-constant discounting and discuss the relation between a given equilibrium of that model and the unique equilibrium of a related problem with constant discounting. We calculate the bounds of the set of candidate steady states and we Pareto rank the equilibria.

Social Science Research Network, 2009

In many situations, such as trade in stock exchanges, agents have many opportunities to act within a short interval of time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium.