We study two-player zero-sum stopping games in continuous time and infinite horizon. We prove that the value in randomized stopping times exists as soon as the payoff processes are right-continuous. In particular, as opposed to existing literature, we do not assume any conditions on the relations between the payoff processes. We also show that both players have simple epsilon- optimal randomized stopping times; namely, randomized stopping times which are small perturbations of non-randomized stopping times.
In the nonzero-sum setting, we establish a connection between Nash equilibria in games of optimal stopping (Dynkin games) and generalised Nash equilibrium problems (GNEP). In the Dynkin game this reveals novel equilibria of threshold type and of more complex types, and leads to novel uniqueness and stability results.
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 sto…
We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further…
The paper is concerned with a variant of the continuous-time finite state Markov game of control and stopping where both players can affect transition rates, while only one player can choose a stopping time. We use the dynamic programming principle and reduce this problem to a system of ODEs with unilateral constraints. This system plays the role of the Bellman equation. We show that its solution provides the optimal strategies of the players. Additionally, we prove the existence and uniqueness theorem for the deduced system of ODEs with unilateral constraints.
We consider a zero-sum stochastic game for continuous-time Markov chain with countable state space and unbounded transition and pay-off rates. The additional feature of the game is that the controllers together with taking actions are also allowed to stop the process. Under suitable hypothesis we show that the game has a value and it is the unique solution of certain dynamic programming inequalities with bilateral constraints. In the process we also prescribe a saddle point equilibrium.
We consider a zero-sum continuous time stopping game in which the pay-off is revealed in the maximum of the two stopping times instead of the minimum, which is the case in Dynkin games.
This paper analyses two-player nonzero-sum games of optimal stopping on a class of linear regular diffusions with not non-singular boundary behaviour (in the sense of Itô and McKean (1974), p.\ 108). We provide sufficient conditions under which Nash equilibria are realised by each player stopping the diffusion at one of the two boundary points of an interval. The boundaries of this interval solve a system of algebraic equations. We also provide conditions sufficient for the uniqueness of the equilibrium in this class.
We consider multi-player stopping games in continuous time. Unlike Dynkin games, in our games the payoff of each player is revealed after all the players stop. Moreover, each player can adjust her own stopping strategy by observing other players' behaviors. Assuming the continuity of the payoff functions in time, we show that there always exists an $\eps$-Nash equilibrium in pure stopping strategies for any $\eps>0$.
A Dynkin game is a zero-sum, stochastic stopping game between two players where either player can stop the game at any time for an observable payoff. Typically the payoff process of the max-player is assumed to be smaller than the payoff process of the min-player, while the payoff process for simultaneous stopping is in between the two. In this paper, we study general Dynkin games whose payoff processes are in arbitrary positions. In both discrete and continuous time settings, we provide necessary and sufficient conditions for the existence of pure strategy Nash equilibria and epsilon-optimal …
In this paper we establish a new connection between a class of 2-player nonzero-sum games of optimal stopping and certain $2$-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover a differential link between the players' value functions holds across the two games.
The properties of value functions of time inhomogeneous optimal stopping problem and zero-sum game (Dynkin game) are studied through time dependent Dirichlet form. Under the absolute continuity condition on the transition function of the underlying diffusion process and some other assumptions, the refined solutions without exceptional starting points are proved to exist, and the value functions of the optimal stopping and zero-sum game, which are finely and cofinely continuous, are characterized as the solutions of some variational inequalities, respectively.
On a filtered probability space $(Ω,\mathcal{F}, (\mathcal{F}_t)_{t\in[0,\infty]}, \mathbb{P})$, we consider the two-player non-zero-sum stopping game $u^i := \mathbb{E}[U^i(ρ,τ)],\ i=1,2$, where the first player choose a stopping strategy $ρ$ to maximize $u^1$ and the second player chose a stopping strategy $τ$ to maximize $u^2$. Unlike the Dynkin game, here we assume that $U(s,t)$ is $\mathcal{F}_{s\vee t}$-measurable. Assuming the continuity of $U^i$ in $(s,t)$, we show that there exists an $ε$-Nash equilibrium for any $ε>0$.
This paper analyses a stochastic differential game of control and stopping in which one of the players modifies a diffusion process using impulse controls, an adversary then chooses a stopping time to end the game. The paper firstly establishes the regularity and boundedness of the upper and lower value functions from which an appropriate variant of the dynamic programming principle (DPP) is derived. It is then proven that the upper and lower value functions coincide so that the game admits a value and that the value of the game is a unique viscosity solution to a HJBI equation described by a …
We consider two-player non-zero-sum stopping games in discrete time. Unlike Dynkin games, in our games the payoff of each player is revealed after both players stop. Moreover, each player can adjust her own stopping strategy according to the other player's action. In the first part of the paper, we consider the game where players act simultaneously at each stage. We show that there exists a Nash equilibrium in mixed stopping strategies. In the second part, we assume that one player has to act first at each stage. In this case, we show the existence of a Nash equilibrium in pure stopping strate…
We develop the fictitious play algorithm in the context of the linear programming approach for mean field games of optimal stopping and mean field games with regular control and absorption. This algorithm allows to approximate the mean field game population dynamics without computing the value function by solving linear programming problems associated with the distributions of the players still in the game and their stopping times/controls. We show the convergence of the algorithm using the topology of convergence in measure in the space of subprobability measures, which is needed to deal with…
We show that the value function of an optimal stopping game driven by a one-dimensional diffusion can be characterised using a modification of the Legendre transformation if and only if the optimal stopping game exhibits a Nash equilibrium (i.e. a saddle point of the optimal stopping game exists). This result is an analytical complement to the results in Peskir, G. (2012) A Duality Principle for the Legendre Transform. Journal of Convex Analysis, 19(3), 609-630 where the `duality' between a concave-biconjugate which is modified to remain below an upper barrier and a convex-biconjugate which is…
We formulate a stochastic game of mean field type where the agents solve optimal stopping problems and interact through the proportion of players that have already stopped. Working with a continuum of agents, typical equilibria become functions of the common noise that all agents are exposed to, whereas idiosyncratic randomness can be eliminated by an Exact Law of Large Numbers. Under a structural monotonicity assumption, we can identify equilibria with solutions of a simple equation involving the distribution function of the idiosyncratic noise. Solvable examples allow us to gain insight into…
This paper studies a class of zero-sum stopping game in a regime switching model. A verification theorem as a sufficient criterion for Nash equilibriums is established based on a set of variational inequalities (VIs). Under an appropriate regularity condition for solutions to the VIs, a suitable system of algebraic equations is derived via the so-called smooth-fit principle. Explicit Nash equilibrium stopping rules of threshold-type for the two players and the corresponding value function of the game in closed form are obtained. Numerical experiments are reported to demonstrate the dependence …
We study the existence of optimal actions in a zero-sum game $\inf_τ\sup_PE^P[X_τ]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_τ\mathcal{E}(X_τ)$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ such as the $G$-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope $Y$ and prove that the first hitting time $\inf\{t:Y_t=X_t\}$ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition.…
This paper revisits the well-studied \emph{optimal stopping} problem but within the \emph{large-population} framework. In particular, two classes of optimal stopping problems are formulated by taking into account the \emph{relative performance criteria}. It is remarkable the relative performance criteria, also understood by the \emph{Joneses preference}, \emph{habit formation utility}, or \emph{relative wealth concern} in economics and finance, plays an important role in explaining various decision behaviors such as price bubbles. By introducing such criteria in large-population setting, a giv…
We study a zero-sum game where the evolution of a spectrally one-sided Levy process is modified by a singular controller and is terminated by the stopper. The singular controller minimizes the expected values of running, controlling and terminal costs while the stopper maximizes them. Using fluctuation theory and scale functions, we derive a saddle point and the value function of the game. Numerical examples under phase-type Levy processes are also given.
On a filtered probability space $(Ω,\mathcal{F},P,\mathbb{F}=(\mathcal{F}_t)_{t=0,\dotso,T})$, we consider stopper-stopper games $\overline V:=\inf_{\Rho\in\bT^{ii}}\sup_{τ\in\T}\E[U(\Rho(τ),τ)]$ and $\underline V:=\sup_{\Tau\in\bT^i}\inf_{ρ\in\T}\E[U(\Rho(τ),τ)]$ in discrete time, where $U(s,t)$ is $\mathcal{F}_{s\vee t}$-measurable instead of $\mathcal{F}_{s\wedge t}$-measurable as is often assumed in the literature, $\T$ is the set of stopping times, and $\bT^i$ and $\bT^{ii}$ are sets of mappings from $\T$ to $\T$ satisfying certain non-anticipativity conditions. We convert the problems in…
We consider a stochastic game of control and stopping specified in terms of a process $X_t=-θΛ_t+W_t$, representing the holdings of Player 1, where $W$ is a Brownian motion, $θ$ is a Bernoulli random variable indicating whether Player 2 is active or not, and $Λ$ is a non-decreasing process representing the accumulated "theft" or "fraud" performed by Player 2 (if active) against Player 1. Player 1 cannot observe $θ$ or $Λ$ directly, but can merely observe the path of the process $X$ and may choose a stopping rule $τ$ to deactivate Player 2 at a cost $M$. Player 1 thus does not know if she is th…
We consider an investment problem in which an investor performs capital injections to increase the liquidity of a firm for it to maximise profit from market operations. Each time the investor performs an injection, the investor incurs a fixed transaction cost. In addition to maximising their terminal reward, the investor seeks to minimise the risk of loss of their investment (from a possible firm ruin) by exiting the market at some point in time. We show that the problem can be reformulated in terms of a new stochastic differential game of control and stopping in which one of the players modif…
A multi-player competitive Dynkin stopping game is constructed. Each player can either exit the game for a fixed payoff, determined a priori, or stay and receive an adjusted payoff depending on the decision of other players. The single period case is shown to be "weakly unilaterally competitive". We present an explicit construction of the unique value at which Nash and optimal equilibria are attained. Multiple period generalisations are explored. The game has interpretations in economic and financial contexts, for example, as a consumption model with bounded resources. It also serves as a star…
We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear p…
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.
We formulate a notion of doubly reflected BSDE in the case where the barriers $ξ$ and $ζ$ do not satisfy any regularity assumption and with a general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $ξ$ is right upper-semicontinuous and $ζ$ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\mathcal{E}^f$-Dynkin game, i.e. a game problem over stopping times with (non-linear) $f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the g…
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 varianc…
We consider a class of infinite-time horizon optimal stopping problems for spectrally negative Levy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egam…
We study a discrete-time finite-horizon two-players nonzero-sum stopping game where the filtration of Player 1 is richer than the filtration of Player 2. A major difficulty which is caused by the information asymmetry is that Player 2 may not know whether Player 1 has already stopped the game or not. Furthermore, the classical backward-induction approach is not applicable in the current setup. This is because when the informed player decides not to stop, he reveals information to the uninformed player and hence the decision of the uninformed player at time $t$ may not be determined by the play…
We consider the game-theoretic approach to time-inconsistent stopping of a one-dimensional diffusion where the time-inconsistency is due to the presence of a non-exponential (weighted) discount function. In particular, we study (weak) equilibria for this problem in a novel class of mixed (i.e., randomized) stopping times based on a local time construction of the stopping intensity. For a general formulation of the problem we provide a verification theorem giving sufficient conditions for mixed (and pure) equilibria in terms of a set of variational inequalities, including a smooth fit condition…
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.
We study a class of optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that there exists a Nash equilibrium in randomized stopping times which is described explicitly in terms of the corresponding one-player game.
We obtain a probabilistic proof of the local Lipschitz continuity for the optimal stopping boundary of a class of problems with state space $[0,T]\times\mathbb{R}^d$, $d\ge 1$. To the best of our knowledge this is the only existing proof that relies exclusively upon stochastic calculus, all the other proofs making use of PDE techniques and integral equations. Thanks to our approach we obtain our result for a class of diffusions whose associated second order differential operator is not necessarily uniformly elliptic. The latter condition is normally assumed in the related PDE literature.
In this paper we study the optimization problem of an economic agent who chooses a job and the time of retirement as well as consumption and portfolio of assets. The agent is constrained in the ability to borrow against future income. We transform the problem into a dual two-person zero-sum game, which involves a controller, who is a minimizer and chooses a non-increasing process, and a stopper, who is a maximizer and chooses a stopping time. We derive the Hamilton-Jacobi- Bellman quasi-variational inequality(HJBQV) of a max-min type arising from the game. We provide a solution to the HJBQV an…
We provide an algorithm to find the value and an optimal strategy of the solitaire variant of the Ten Thousand dice game in the framework of Markov Control Processes. Once an optimal critical threshold is found, the set of non-stopping states of the game becomes finite, and the solution is found by a backwards algorithm that gives the values for each one of these states of the game. The algorithm is finite and exact.The idea to find the critical threshold comes from the continuous pasting condition used in optimal stopping problems for continuous-time processes with jumps.
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 wh…
We reveal an interesting convex duality relationship between two problems: (a) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (b) a controller-and-stopper problem, in which the controller controls the drift and volatility of a process in order to maximize a running reward based on that process, and the stopper chooses the time to stop the running reward and rewards the controller a final amount at that time. Our primary goal is to show that the minimal probability of ruin, whose stoc…
This paper introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. The paper further applies the model to study the optimal conversion and calling strategies of convertible bonds, and their asymptotics when the Poisson intensity goes to infinity.
This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players' decisions. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the staff selection competition in the case of two departments are given. Utilitarian, egalitarian, republican and libertarian concepts of correlated equi…
