Philipp Strack

Skorokhod Embeddings In Bounded Time

This article deals with the Skorokhod embedding problem in bounded time for the Brownian motion with drift Xt = κt + Wt: Given a probability measure μ we aim at finding a stopping time τ such that the law of Xτ is μ, and τ is almost surely smaller than some given fixed time horizon T > 0. We provide necessary and sufficient conditions on the distribution μ for the existence of such bounded stopping times.

Strategic Experimentation with Private Payoffs

We consider a game of strategic experimentation in which players face identical discrete-time bandit problems with a safe and a risky arm. In any period, the risky arm yields either a success or a failure, and the first success reveals the risky arm to dominate the safe one. When payoffs are public information, the ensuing free-rider problem is so severe that equilibrium experimentation ceases at the same threshold belief at which a single agent would stop, even if players can coordinate their actions through mediated communication. When payoffs are private information and the success probability on the risky arm is not too high, however, the socially optimal symmetric experimentation profile can be supported as a perfect Bayesian equilibrium for sufficiently optimistic prior beliefs, even if players can only communicate via binary cheap-talk messages.

Optimal stopping with private information

Many economic situations are modeled as stopping problems. Examples include job search, timing of market entry decisions, irreversible investment or the pricing of American options. This paper analyzes optimal stopping as a mechanism design problem with transfers. We show that under a dynamic single crossing condition a stopping rule can be implemented by a transfer that only depends on the realized stopping decision if and only if it is a cut-off rule. We characterize the transfer implementing a given stopping rule using a novel technique based on constrained stochastic processes.

Gambling in contests

This paper presents a strategic model of risk-taking behavior in contests. Formally, we analyze an n-player winner-take-all contest in which each player decides when to stop a privately observed Brownian motion with drift. A player whose process reaches zero has to stop. The player with the highest stopping point wins. Unlike the explicit cost for a higher stopping time in a war of attrition, here, higher stopping times are riskier, because players can go bankrupt. We derive a closed-form solution of a Nash equilibrium outcome. In equilibrium, highest expected losses occur at an intermediate negative value of the drift.

Stochastic Choice and Optimal Sequential Sampling

We model the joint distribution of choice probabilities and decision times in binary choice tasks as the solution to a problem of optimal sequential sampling, where the agent is uncertain of the utility of each action and pays a constant cost per unit time for gathering information. In the resulting optimal policy, the agents choices are more likely to be correct when the agent chooses to decide quickly, provided that the agents prior beliefs are correct. For this reason it better matches the observed correlation between decision time and choice probability than does the classical drift-diffusion model, where the agent is uncertain which of two actions is best but knows the utility difference between them.

Dynamic Revenue Maximization: A Continuous Time Approach ∗

We characterize the profit-maximizing mechanism for repeatedly selling a non-durable good in continuous time. The valuation of each agent is private information and changes over time. At the time of contracting every agent privately observes his initial type which influences the evolution of his valuation process. In the profit-maximizing mechanism the allocation is distorted in favor of agents with high initial types. We derive the optimal mechanism in closed form, which enables us to compare the distortion in various examples. The case where the valuation of the agents follows an arithmetic/geometric Brownian motion, Ornstein-Uhlenbeck process, or is derived from a Bayesian learning model are discussed. We show that depending on the nature of the private information and the valuation process the distortion might increase or decrease over time.

Finite, integrable and bounded time embeddings for diffusions

We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous diffusion X: given a distribution \rho, we construct a stopping time T such that the stopped process X_T has the distribution \rho? Our solution method makes use of martingale representations (in a similar way to Bass [3] who solves the SEP for Brownian motion) and draws on law uniqueness of weak solutions of SDEs. Then we ask if there exist solutions of the SEP which are respectively finite almost surely, integrable or bounded, and when does our proposed construction have these properties. We provide conditions that guarantee existence of finite time solutions. Then, we fully characterize the distributions that can be embedded with integrable stopping times. Finally, we derive necessary, respectively sufficient, conditions under which there exists a bounded embedding.

When More Information Reduces the Speed of Learning

We study how effectively long-lived rational agents learn from repeatedly observing each others’ actions. We find that in the long run, information aggregation fails, and the fraction of private information transmitted goes to zero as the number of agents gets large. With Normal signals, in the long-run, agents learn less from observing the actions of any number of other agents than they learn from seeing three other agents’ signals. We identify rational groupthink — in which agents ignore their private signals and choose the same action for long periods of time — as the cause of this failure of information aggregation.

Continuous time contests with private information

This paper introduces a class of contest models in which each player decides when to stop a privately observed Brownian motion with drift and incurs costs depending on his stopping time. The player who stops his process at the highest value wins a prize. We prove existence and uniqueness of a Nash equilibrium outcome and derive the equilibrium distribution in closed form. As the variance tends to zero, the equilibrium outcome converges to the symmetric equilibrium of an all-pay auction. For two players and constant costs, each player’s equilibrium profit decreases if the drift increases, the variance decreases, or the costs decrease.

Gambling in Dynamic Contests

This paper presents a strategic model of risk-taking behavior in the framework of a continuous time contest. Formally, we analyze a dynamic game in which each player decides when to stop a privately observed Brownian Motion with drift. Only the player who stops his process at the highest value wins a prize. We derive a closed-form solution for the unique Nash equilibrium outcome in mixed strategies and we establish that the expected value of the stopped stochastic processes is non-monotone in the drift. In particular, the highest losses in terms of expected value occur if the drift is only moderately negative. Thus, relative performance payments, while suitable to provide the right incentives in good times, induce socially undesirable gambling activities if times are moderately bad. Possible applications of the model include contests for status, job promotion contests, competition between mutual funds, and relative payment schemes of CEOs.