Dynkin games with heterogeneous beliefs

Copyright © 2017 Applied Probability Trust. We study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times

Dynkin games with incomplete and asymmetric information

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian set-up, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result which allows us to find the value of the game and players' optimal strategies by solving suitable quasi-variational inequalities with some non-standard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.Comment: 31 pages, 5 figures, small changes in the terminology from game theor

Zero-sum stopping games with asymmetric information

We study a model of two-player, zero-sum, stopping games with asymmetric information. We assume that the payoff depends on two continuous-time Markov chains (X, Y), where X is only observed by player 1 and Y only by player 2, implying that the players have access to stopping times with respect to different filtrations. We show the existence of a value in mixed stopping times and provide a variational characterization for the value as a function of the initial distribution of the Markov chains. We also prove a verification theorem for optimal stopping rules which allows to construct optimal stopping times. Finally we use our results to solve explicitly two generic examples

Zero-sum stopping games with asymmetric information

We study a model of two-player, zero-sum, stopping games with asymmetric information. We assume that the payoff depends on two continuous-time Markov chains (X, Y), where X is only observed by player 1 and Y only by player 2, implying that the players have access to stopping times with respect to different filtrations. We show the existence of a value in mixed stopping times and provide a variational characterization for the value as a function of the initial distribution of the Markov chains. We also prove a verification theorem for optimal stopping rules which allows to construct optimal stopping times. Finally we use our results to solve explicitly two generic examples

Learning to Disagree in a Game of Experimentation

We analyse strategic experimentation in which information arrives through fully revealing, publicly observable “breakdowns.” With hidden actions, there exists a unique equilibrium that involves randomization over stopping times. This randomization induces belief disagreement on the equilibrium path. When actions are observable, the equilibrium is pure, and welfare improves. We analyse the role of policy interventions such as subsidies for experimentation and risk-sharing agreements. We show that the optimal risk-sharing agreement restores the first-best outcome, independent of the monitoring structure

Value and Nash Equilibrium in Games of Optimal Stopping

We study games of optimal stopping (Dynkin games). A Dynkin game is a mathematical model involving several competing players, each of them interested in capturing the moments when certain stochastic processes are at an extremum. Actions of players are referred to as “stopping” (of an underlying process), and the outcome for every player depends on the stopping decisions of the other players. Our focus is on Dynkin games with asymmetric information. Asymmetry of information refers to the situation in which different players have different (possibly incomplete) knowledge of the underlying world. Observations of the underlying processes (or of a more general information flow) and of the actions of competitors allow the players to make optimal stopping choices. An important aspect of our framework is a possibility of randomising these choices: for example, in order to avoid revealing private information to competitors. We develop a general stochastic framework for studying Dynkin games with asymmetric information. In particular, we provide conditions for the existence of the value in such games. Separately, we study issues arising in games with mixed first-mover advantage, in which sometimes it is beneficial for the players to act as soon as possible, and sometimes to wait for another player to act

Operational Decision Making under Uncertainty: Inferential, Sequential, and Adversarial Approaches

Modern security threats are characterized by a stochastic, dynamic, partially observable, and ambiguous operational environment. This dissertation addresses such complex security threats using operations research techniques for decision making under uncertainty in operations planning, analysis, and assessment. First, this research develops a new method for robust queue inference with partially observable, stochastic arrival and departure times, motivated by cybersecurity and terrorism applications. In the dynamic setting, this work develops a new variant of Markov decision processes and an algorithm for robust information collection in dynamic, partially observable and ambiguous environments, with an application to a cybersecurity detection problem. In the adversarial setting, this work presents a new application of counterfactual regret minimization and robust optimization to a multi-domain cyber and air defense problem in a partially observable environment

Monotone equimeasurable rearrangements with non-additive probabilities

In the classical theory of monotone equimeasurable rearrangements of functions, “equimeasurability” (i.e. the fact the two functions have the same distribution) is defined relative to a given additive probability measure. These rearrangement tools have been successfully used in many problems in economic theory dealing with uncertainty where the monotonicity of a solution is desired. However, in all of these problems, uncertainty refers to the classical Bayesian understanding of the term, where the idea of ambiguity is absent. Arguably, Knighitan uncertainty, or ambiguity is one of the cornerstones of modern decision theory. It is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to situations of ambiguity. This paper introduces the idea of a monotone equimeasurable rearrangement in the context of non-additive probabilities, or capacities that satisfy a property that I call strong nonatomicity. The latter is a strengthening of the notion of nonatomicity, and these two properties coincide for additive measures and for submodular (i.e. concave) capacities. To illustrate the usefulness of these tools in economic theory, I consider an application to a problem arising in the theory of production under uncertainty