Double stopping by two decision-makers

1993 ◽  
Vol 25 (2) ◽  
pp. 438-452 ◽  
Author(s):  
K. Szajowski
Keyword(s):  
Markov Process ◽  
Optimal Stopping ◽  
Value Function ◽  
Optimal Strategies ◽  
Decision Makers ◽  
Secretary Problem ◽  
Gain Function ◽  
Competitive Situation ◽  
Zero Sum ◽  
The Value Function

A problem of optimal stopping of the discrete-time Markov process by two decision-makers (Player 1 and Player 2) in a competitive situation is considered. The zero-sum game structure is adopted. The gain function depends on states chosen by both decision-makers. When both players want to accept the realization of the Markov process at the same moment, the priority is given to Player 1. The construction of the value function and the optimal strategies for the players are given. The Markov chain case is considered in detail. An example related to the generalized secretary problem is solved.

Double stopping by two decision-makers

1993 ◽  
Vol 25 (02) ◽  
pp. 438-452 ◽  
Author(s):  
K. Szajowski
Keyword(s):  
Markov Process ◽  
Optimal Stopping ◽  
Value Function ◽  
Optimal Strategies ◽  
Decision Makers ◽  
Secretary Problem ◽  
Gain Function ◽  
Competitive Situation ◽  
Zero Sum ◽  
The Value Function

A problem of optimal stopping of the discrete-time Markov process by two decision-makers (Player 1 and Player 2) in a competitive situation is considered. The zero-sum game structure is adopted. The gain function depends on states chosen by both decision-makers. When both players want to accept the realization of the Markov process at the same moment, the priority is given to Player 1. The construction of the value function and the optimal strategies for the players are given. The Markov chain case is considered in detail. An example related to the generalized secretary problem is solved.

On Optimal Stopping of Inhomogeneous Standard Markov Processes

1995 ◽  
Vol 2 (4) ◽  
pp. 335-346
Author(s):  
B. Dochviri
Keyword(s):  
Markov Process ◽  
Optimal Stopping ◽  
Value Function ◽  
Stopping Times ◽  
Homogeneous Markov Process ◽  
Limit Procedure ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
The Value Function

Abstract The connection between the optimal stopping problems for inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. An excessive characterization of the value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous standard Markov process is given. The form of ε-optimal (optimal) stopping times is also found.

scholarly journals Optimal stopping for the exponential of a Brownian bridge

2020 ◽  
Vol 57 (1) ◽  
pp. 361-384
Author(s):  
Tiziano de Angelis ◽  
Alessandro Milazzo
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Brownian Bridge ◽  
Linear Structure ◽  
Gain Function ◽  
Closed Form Solutions ◽  
Optimal Stopping Rule ◽  
Optimal Stopping Theory ◽  
Inhomogeneous Diffusion ◽  
The Value Function

AbstractWe study the problem of stopping a Brownian bridge X in order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

Metric for Disassembly and Reuse Decisions: Formulation and Validation

2008 ◽  
Author(s):  
Vijitashwa Pandey ◽  
Deborah Thurston
Keyword(s):  
Maximum Likelihood ◽  
Value Function ◽  
Choice Theory ◽  
Decision Makers ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Maximum Value ◽  
Long Range Planning ◽  
The Relationship ◽  
The Value Function

Design for disassembly and reuse focuses on developing methods to minimize difficulty in disassembly for maintenance or reuse. These methods can gain substantially if the relationship between component attributes (material mix, ease of disassembly etc.) and their likelihood of reuse or disposal is understood. For products already in the marketplace, a feedback approach that evaluates willingness of manufacturers or customers (decision makers) to reuse a component can reveal how attributes of a component affect reuse decisions. This paper introduces some metrics and combines them with ones proposed in literature into a measure that captures the overall value of a decision made by the decision makers. The premise is that the decision makers would choose a decision that has the maximum value. Four decisions are considered regarding a component’s fate after recovery ranging from direct reuse to disposal. A method on the lines of discrete choice theory is utilized that uses maximum likelihood estimates to determine the parameters that define the value function. The maximum likelihood method can take inputs from actual decisions made by the decision makers to assess the value function. This function can be used to determine the likelihood that the component takes a certain path (one of the four decisions), taking as input its attributes, which can facilitate long range planning and also help determine ways reuse decisions can be influenced.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (1) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

scholarly journals A Two Person Zero Sum Game Oriented to Integration of Objectives

2014 ◽  
Vol 5 (2) ◽  
pp. 65-85 ◽  
Author(s):  
Hakan Aplak ◽  
Mehmet Kabak ◽  
Erkan Köse
Keyword(s):  
Environmental Factors ◽  
Optimal Strategies ◽  
Payoff Matrix ◽  
Decision Makers ◽  
Decision Making Process ◽  
Final Decision ◽  
Competitive Environments ◽  
Conflicting Objectives ◽  
Zero Sum

Abstract Decision making process is a process which includes decision makers, actors, environmental factors, objectives, strategies and criteria. In competitive environments, effectiveness of decision process depends on determining all environmental factors and evaluating them according to objectives. Decision makers aim to find optimal strategies for conflicting objectives. Game theory is an approach based on mathematics in which strategies of players are evaluated reciprocally by considering environmental effects.In this study, a two-person zero-sum game approach is presented for choosing optimal strategies of actors in competitive environment by balancing objectives reciprocally. This approach refers to evaluation of each objective, creation of decision payoff matrixes by using fuzzy logic mathematical applications and their transformation to final decision payoff matrix subsequently. Finally, optimal strategies and their probabilities are found. A military case study is presented for illustrating the application of proposed approach.

scholarly journals American options in an imperfect complete market with default

2018 ◽  
Vol 64 ◽  
pp. 93-110 ◽  
Author(s):  
Roxana Dumitrescu ◽  
Marie-Claire Quenez ◽  
Agnès Sulem
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
American Options ◽  
American Option ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Nonlinear Expectation ◽  
Portfolio Strategy ◽  
Reflected Bsde ◽  
The Value Function

We study pricing and hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process (ξt). We define the seller's price of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with a nonlinear expectation 𝓔g (induced by a BSDE), which corresponds to the solution of a nonlinear reflected BSDE with obstacle (ξt). Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer's price of the American option, which is defined as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy φ so that he/she is superhedged. We show that the buyer's price is equal to the value function of an optimal stopping problem with a nonlinear expectation, and that it can be characterized via the solution of a reflected BSDE with obstacle (ξt). Under the additional assumption of left upper semicontinuity along stopping times of (ξt), we show the existence of a super-hedge (τ, φ) for the buyer.

A secretary problem with two decision makers

1997 ◽  
Vol 34 (04) ◽  
pp. 1068-1074 ◽  
Author(s):  
Robert W. Chen ◽  
Burton Rosenberg ◽  
Larry A. Shepp
Keyword(s):  
Continuous Distribution ◽  
Optimal Strategies ◽  
Decision Makers ◽  
Secretary Problem ◽  
The Other ◽  
Winning Probability ◽  
Interview Process ◽  
The One

n applicants of similar qualification are on an interview list and their salary demands are from a known and continuous distribution. Two managers, I and II, will interview them one at a time. Right after each interview, manager I always has the first opportunity to decide to hire the applicant or not unless he has hired one already. If manager I decides not to hire the current applicant, then manager II can decide to hire the applicant or not unless he has hired one already. If both managers fail to hire the current applicant, they interview the next applicant, but both lose the chance of hiring the current applicant. If one of the managers does hire the current one, then they proceed with interviews until the other manager also hires an applicant. The interview process continues until both managers hire an applicant each. However, at the end of the process, each manager must have hired an applicant. In this paper, we first derive the optimal strategies for them so that the probability that the one he hired demands less salary than the one hired by the other does is maximized. Then we derive an algorithm for computing manager II's winning probability when both managers play optimally. Finally, we show that manager II's winning probability is strictly increasing in n, is always less than c, and converges to c as n →∞, where c = 0.3275624139 · ·· is a solution of the equation ln(2) + x ln(x) = x.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (01) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

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