Monotone stopping games

We consider the extension of optimal stopping problems to non-zero-sum strategic settings called stopping games. By imposing a monotone structure on the pay-offs of the game we establish the existence of a Nash equilibrium in non-randomized stopping times. As a corollary, we identify a class of games for which there are Nash equilibria in myopic stopping times. These games satisfy the strategic equivalent of the classical ‘monotone case' assumptions of the optimal stopping problem.

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Dynkin games with heterogeneous beliefs

Journal of Applied Probability ◽

10.1017/jpr.2016.97 ◽

2017 ◽

Vol 54 (1) ◽

pp. 236-251 ◽

Cited By ~ 2

Author(s):

Erik Ekström ◽

Kristoffer Glover ◽

Marta Leniec

Keyword(s):

Optimal Stopping ◽

Nash Equilibria ◽

Heterogeneous Beliefs ◽

Stopping Times ◽

Value Functions ◽

Natural Conditions ◽

Dynkin Games ◽

Zero Sum ◽

Stopping Games ◽

The Given

AbstractWe 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.

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On wald-type optimal stopping for Brownian motion

Journal of Applied Probability ◽

10.2307/3215175 ◽

1997 ◽

Vol 34 (1) ◽

pp. 66-73 ◽

Cited By ~ 4

Author(s):

S. E. Graversen ◽

G. Peškir

Keyword(s):

Brownian Motion ◽

Optimal Stopping ◽

Stopping Time ◽

Maximal Inequality ◽

Stopping Times ◽

Standard Brownian Motion ◽

Square Root ◽

Real Analysis ◽

Optimal Stopping Problems ◽

Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

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A Forward Algorithm for Solving Optimal Stopping Problems

Journal of Applied Probability ◽

10.1017/s002190020000139x ◽

2006 ◽

Vol 43 (01) ◽

pp. 102-113

Author(s):

Albrecht Irle

Keyword(s):

State Space ◽

Optimal Stopping ◽

Stopping Time ◽

The State ◽

Optimal Stopping Problem ◽

Forward Algorithm ◽

Optimal Stopping Time ◽

Entrance Time ◽

Markov Sequence ◽

Optimal Stopping Problems

We consider the optimal stopping problem for g(Z n ), where Z n , n = 1, 2, …, is a homogeneous Markov sequence. An algorithm, called forward improvement iteration, is presented by which an optimal stopping time can be computed. Using an iterative step, this algorithm computes a sequence B 0 ⊇ B 1 ⊇ B 2 ⊇ · · · of subsets of the state space such that the first entrance time into the intersection F of these sets is an optimal stopping time. Various applications are given.

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An Algorithmic Procedure for Finding Nash Equilibrium

GANIT Journal of Bangladesh Mathematical Society ◽

10.3329/ganit.v40i1.48196 ◽

2020 ◽

Vol 40 (1) ◽

pp. 71-85

Author(s):

HK Das ◽

T Saha

Keyword(s):

Nash Equilibrium ◽

Nash Equilibria ◽

Payoff Matrix ◽

Matrix Game ◽

Perfect Equilibrium ◽

Repeated Use ◽

Definition Of ◽

Zero Sum ◽

Mathematical Construction ◽

Algorithmic Procedure

This paper proposes a heuristic algorithm for the computation of Nash equilibrium of a bi-matrix game, which extends the idea of a single payoff matrix of two-person zero-sum game problems. As for auxiliary but making the comparison, we also introduce here the well-known definition of Nash equilibrium and a mathematical construction via a set-valued map for finding the Nash equilibrium and illustrates them. An important feature of our algorithm is that it finds a perfect equilibrium when at the start of all actions are played. Furthermore, we can find all Nash equilibria of repeated use of this algorithm. It is found from our illustrative examples and extensive experiment on the current phenomenon that some games have a single Nash equilibrium, some possess no Nash equilibrium, and others had many Nash equilibria. These suggest that our proposed algorithm is capable of solving all types of problems. Finally, we explore the economic behaviour of game theory and its social implications to draw a conclusion stating the privilege of our algorithm. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 71-85

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Optimal stopping with discount and observation costs

Journal of Applied Probability ◽

10.1017/s0021900200015254 ◽

2000 ◽

Vol 37 (01) ◽

pp. 64-72 ◽

Cited By ~ 2

Author(s):

Robert Kühne ◽

Ludger Rüschendorf

Keyword(s):

Differential Equations ◽

Optimal Stopping ◽

Point Processes ◽

Numerical Solutions ◽

Domain Of Attraction ◽

Approximate Solutions ◽

Poisson Approximation ◽

Stopping Times ◽

Optimal Stopping Problem ◽

Optimal Stopping Times

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

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Variance Optimal Stopping for Geometric Lévy Processes

Advances in Applied Probability ◽

10.1017/s0001867800007734 ◽

2015 ◽

Vol 47 (01) ◽

pp. 128-145 ◽

Cited By ~ 3

Author(s):

Kamille Sofie Tågholt Gad ◽

Jesper Lund Pedersen

Keyword(s):

Optimal Stopping ◽

Lévy Process ◽

Lévy Processes ◽

Levy Processes ◽

Levy Process ◽

Stopping Times ◽

Optimal Stopping Problem ◽

Maximum Variance ◽

Geometric Levy Process

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

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Optimal stopping with discount and observation costs

Journal of Applied Probability ◽

10.1239/jap/1014842268 ◽

2000 ◽

Vol 37 (1) ◽

pp. 64-72 ◽

Cited By ~ 4

Author(s):

Robert Kühne ◽

Ludger Rüschendorf

Keyword(s):

Differential Equations ◽

Optimal Stopping ◽

Point Processes ◽

Numerical Solutions ◽

Domain Of Attraction ◽

Approximate Solutions ◽

Poisson Approximation ◽

Stopping Times ◽

Optimal Stopping Problem ◽

Optimal Stopping Times

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

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