Papers by Krzysztof J. Szajowski
Birkhäuser Boston eBooks, 1999
This paper treats stochastic games. A nonzero-sum average payoff stochastic games with arbitrary ... more This paper treats stochastic games. A nonzero-sum average payoff stochastic games with arbitrary state spaces and the stopping games are considered. Such models of games very well fit in some studies in economic theory and operations research. A correlation of strategies of the players, involving "public signals", is allowed in the nonzero-sum average payoff stochastic games. The main result is an extension of the correlated equilibrium theorem proved recently by Nowak and Raghavan for dynamic games with discounting to the average payoff stochastic games. The stopping games are special model of stochastic games. The version of Dynkin's game related to observation of Markov process with random priority assignment mechanism of states is presented in the paper. The zero-sum and nonzero-sum games are considered. The paper also provides a brief overview of the theory of nonzero-sum stochastic games and stopping games which are very far from being complete.
The paper extends the results on the problem of change point detection for Markov processes gener... more The paper extends the results on the problem of change point detection for Markov processes generalizing the results contained in the publications [11], [15] and [8]. The short description are as follows. A random sequence having segments being the homogeneous Markov processes is registered. Each segment has his own transition probability law and the length of the segment is unknown and random. The transition probabilities of each process are known and joint a priori distribution of the disorder moments are given. The detection of the disorder rarely is precise. The decision maker accepts some deviation in estimation of the disorder moments. In this note the aim is to indicate the segment of given length between disorders with maximal probabilities. The case with various precision for over and under estimation of the middle point is analyzed including situation when the disorders do not appear with positive probability is also included. The observed sequence, when the change point i...
We register a random sequence constructed based on Markov processes by switching between them. At... more We register a random sequence constructed based on Markov processes by switching between them. At two random moments θ1, θ2, where 0 ≤ θ1 ≤ θ2, the source of observations is changed. In effect the number of homogeneous segments is random. The transition probabilities of each process are known and a priori distribution of the disorder moments is given. Two cases are presented in details. In the first one the objective is to stop on between the disorder moments and in the second one our objective is to find the strategy which immediately detects the distribution changes. Both problems are reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.
Static & Dynamic Game Theory: Foundations & Applications, 2018
Many decision problems in economics, information technology and industry can be transformed to an... more Many decision problems in economics, information technology and industry can be transformed to an optimal stopping of adapted random vectors with some utility function over the set of Markov times with respect to filtration build by the decision maker's knowledge. The optimal stopping problem formulation is to find a stopping time which maximizes the expected value of the accepted (stopped) random vector's utility. There are natural extensions of optimal stopping problem to stopping gamesthe problem of stopping random vectors by two or more decision makers. Various approaches dependent on the information scheme and the aims of the agents in a considered model. This report unifies a group of non-cooperative stopping game models with forced cooperation by the role of the agents, their aims and aspirations (v. Assaf and Samuel-Cahn(1998), Szajowski and Yasuda(1997)) or extensions of the strategy sets (v. ).
Annals of the International Society of Dynamic Games
A mathematical model of competitive selection of the applicants for a post is considered. There a... more A mathematical model of competitive selection of the applicants for a post is considered. There are N applicants with similar qualifications on an interview list. The applicants come in random order and their salary demands are distinct. Two managers, I and II, interview them one at a time. The aim of the manager is to obtain the applicant who demands minimal salary. The candidate can be accepted only at the moment of his appearance. When both managers want to accept the same candidate, then some rule of assignment to one of the managers is applied. Any candidate hired by a manager will accept the offer with some given probability. A candidate can be hired only at the moment of his appear-Au: statement is repeated. ance. At each moment n one candidate is presented. The considered problem is a generalization of the best choice problem with uncertain employment and its game version with priority or random priority. The general stopping game model is constructed. The algorithms of construction of the game value and the equilibrium strategies are given. An example is solved.
Journal of Mathematical Sciences
Stochastic and Differential Games, 1999
This paper treats stochastic games. A nonzero-sum average payoff stochastic games with arbitrary ... more This paper treats stochastic games. A nonzero-sum average payoff stochastic games with arbitrary state spaces and the stopping games are considered. Such models of games very well fit in some studies in economic theory and operations research. A correlation of strategies of the players, involving "public signals", is allowed in the nonzero-sum average payoff stochastic games. The main result is an extension of the correlated equilibrium theorem proved recently by Nowak and Raghavan for dynamic games with discounting to the average payoff stochastic games. The stopping games are special model of stochastic games. The version of Dynkin's game related to observation of Markov process with random priority assignment mechanism of states is presented in the paper. The zero-sum and nonzero-sum games are considered. The paper also provides a brief overview of the theory of nonzero-sum stochastic games and stopping games which are very far from being complete.
Lecture Notes in Computer Science, 2013
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
arXiv (Cornell University), Apr 25, 2013
The sero-sum stopping game for the stochastic sequences has been formulated in late sixties of th... more The sero-sum stopping game for the stochastic sequences has been formulated in late sixties of the twenty century by Dynkin . The formulation had the assumption about separability of decision moment of the players which simplified the construction of the solution. Further research by Neveu [22] extended the model by admitting more general behaviour of the players and their pay-offs. In new formulation there is the problem with existence of the equilibrium. The proper approach to solution of the problem without restriction of former models was developed by Yasuda . The results was crucial in these research. The author made often reference to the Yasuda's result in his works (see ) as well as see results of others stimulated by this paper. Withal, in this note another stopping game model, developed by Yasuda with coauthors (see e.g. [14] and [40]) is recalled. The application of the model to an analysis of system of detectors shows the power of the game theory methods. In the last part of the paper I would like to express my personal relation to the Masami Yasuda game.
We register a random sequence which has the following properties: it has three segments being the... more We register a random sequence which has the following properties: it has three segments being the homogeneous Markov processes. Each segment has his own one step transition probability law and the length of the segment is unknown and random. It means that at two random moments θ1, θ2, where 0 θ1 θ2, the source of observations is changed and the first observation in new segment is chosen according to new transition probability starting from the last state of the previous segment. In effect the number of homogeneous segments is random. The transition probabilities of each process are known and a priori distribution of the disorder moments is given. The former research on such problem has been devoted to various questions concerning the distribution changes. The random number of distributional segments creates new problems in solutions with relation to analysis of the model with deterministic number of segments. Two cases are presented in details. In the first one the objectives is to stop on or between the disorder moments while in the second one our objective is to find the strategy which immediately detects the distribution changes. Both problems are reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.
Applicationes Mathematicae, 1997
Suppose that the process X = {X n , n ∈ N} is observed sequentially. There are two random moments... more Suppose that the process X = {X n , n ∈ N} is observed sequentially. There are two random moments of time θ 1 and θ 2 , independent of X, and X is a Markov process given θ 1 and θ 2 . The transition probabilities of X change for the first time at time θ 1 and for the second time at time θ 2 . Our objective is to find a strategy which immediately detects the distribution changes with maximal probability based on observation of X. The corresponding problem of double optimal stopping is constructed. The optimal strategy is found and the corresponding maximal probability is calculated.
A random sequence having segments being the homogeneous Markov processes is registered. Each segm... more A random sequence having segments being the homogeneous Markov processes is registered. Each segment has his own transition probability law and the length of the segment is unknown and random. The transition probabilities of each process are known and a priori distribution of the disorder moment is given. The former research on such problem has been devoted to various questions concerning the distribution changes when more than one homogeneous segment is expected. The detection of the disorder rarely is precise. The decision maker accepts some deviation in estimation of the disorder moment. In the models taken into account the aim is to indicate the change point with fixed, bounded error with maximal probability. The case with various precision for over and under estimation of this point is analysed. The case when the disorder does not appears with positive probability is also included. The observed sequence, when the change point is known, has the Markovian properties. The results ...
ArXiv, 2018
Let us consider two companies A and B. Both of them are interested in buying a set of some goods.... more Let us consider two companies A and B. Both of them are interested in buying a set of some goods. The company A is a big corporation and it knows the actual value of the good on the market and is able to observe the previous values of them. The company B has no information about the actual value of the good but it can compare the actual position of the good on the market with the previous position of the good offered. Both of the players want to choose the very best object overall. The recall is not allowed. The number of the objects is fixed and finite. One can think about these two types of buyers a business customer vs. an individual customer. The mathematical model of the competition between them is presented and the solution is defined and constructed.
ArXiv, 2020
The paper deals with disorders detection in the multivariate stochastic process. We consider the ... more The paper deals with disorders detection in the multivariate stochastic process. We consider the multidimensional Poisson process or the multivariate renewal process. This class of processes can be used as a description of the distributed detection system. The multivariate renewal process can be seen as the sequence of random vectors, where parts of its coordinates are holding times, others are the size of jumps and the index of stream, at which the new event appears. It is assumed that at each stream two kinds of changes are possible: in the holding time or in the size of jumps distribution. The various specific mutual relations between the change points are possible. The aim of the research is to derive the detectors which realize the optimal value of the specified criterion. The change point moment estimates have been obtained in some cases. The difficulties have appeared for the dependent streams with unspecified order of change points. The presented results suggest further rese...
Mathematica Applicanda, 2016
We consider a sequence of independent random variables with the known distribution observed seque... more We consider a sequence of independent random variables with the known distribution observed sequentially. The observation n is assumed to be a value of one order statistics such as s : n-th, where 1 s n. It the instances following the nth observation it may remain of the s : m or it will be the value of the order statistics r : m (of m > n observations). Changing the rank of the observation, along with expanding a set of observations there is a random phenomenon that is difficult to predict. From practical reasons it is of great interest. Among others, we pose the question of the moment in which the observation appears and whose rank will not change significantly until the end of sampling of a certain size. We also attempt to answer which observation should be kept to have the "good quality observation" as long as possible. This last question was analysed by Ferguson, Hardwick and Tamaki (1991) in the abstract form which they called the problem of duration. This article gives a systematical presentation of the known duration models and a new modifications. We collect results from different papers on the duration of the extremal observation in the no-information (denoted as rank based) case and the full-information case. In the case of non-extremal observation duration models the most appealing are various settings related to the two extremal order statistics. In the no-information case it will be the maximizing duration of owning the relatively best or the second best object. The idea was formulated and the problem was solved by Szajowski and Tamaki (2006). The full-information duration problem with special requirement was presented by Kurushima and Ano (2010).
Applicationes Mathematicae, 2017
A random sequence having two segments being the homogeneous Markov processes is registered. Each ... more A random sequence having two segments being the homogeneous Markov processes is registered. Each segment has his own transition probability law and the length of the segment is unknown and random. The transition probabilities of each process are known and a priori distribution of the disorder moment is given. The decision maker aim is to detect the moment of the transition probabilities change. The detection of the disorder rarely is precise. The decision maker accepts some deviation in estimation of the disorder moment. In the considered model the aim is to indicate the change point with fixed, bounded error with maximal probability. The case with various precision for over and under estimation of this point is analyzed. The case when the disorder does not appears with positive probability is also included. The results insignificantly extends range of application, explain the structure of optimal detector in various circumstances and shows new details of the solution construction. The motivation for this investigation is the modelling of the attacks in the node of networks. The objectives is to detect one of the attack immediately or in very short time before or after it appearance with highest probability. The problem is reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.
The aim of the paper is to extend the model of "fishing problem". The simple formulatio... more The aim of the paper is to extend the model of "fishing problem". The simple formulation is following. The angler goes to fishing. He buys fishing ticket for a fixed time. There are two places for fishing at the lake. The fishes are caught according to renewal processes which are different at both places. The fishes' weights and the inter-arrival times are given by the sequences of i.i.d. random variables with known distribution functions. These distributions are different for the first and second fishing place. The angler's satisfaction measure is given by difference between the utility function dependent on size of the caught fishes and the cost function connected with time. On each place the angler has another utility functions and another cost functions. In this way, the angler's relative opinion about these two places is modeled. For example, on the one place better sort of fish can be caught with bigger probability or one of the places is more comfortable...
Advances in Dynamic Games, 2012
The considered model will be formulated as related to "the fishing problem" even if the other app... more The considered model will be formulated as related to "the fishing problem" even if the other applications of it are much more obvious. The angler goes fishing. He uses various techniques and he has at most two fishing rods. He buys a fishing ticket for a fixed time. The fishes are caught with the use of different methods according to the renewal processes. The fishes' value and the inter arrival times are given by the sequences of independent, identically distributed (i.i.d.) random variables with the known distribution functions. It forms the marked renewal-reward process. The angler's measure of satisfaction is given by the difference between the utility function, depending on the value of the fishes caught, and the cost function connected with the time of fishing. In this way, the angler's relative opinion about the methods of fishing is modelled. The angler's aim is to have as much satisfaction as possible and additionally he has to leave the lake before a fixed moment. Therefore his goal is to find two optimal stopping times in order to maximize his satisfaction. At the first moment, he changes the technique of fishing e.g. by excluding one rod and intensifying on the rest. Next, he decides when he should stop the expedition. These stopping times have to be shorter than the fixed time of fishing. The dynamic programming methods have been used to find these two optimal stopping times and to specify the expected satisfaction of the angler at these times.
Stochastics, 2011
We register a random sequence constructed based on Markov processes by switching between them. At... more We register a random sequence constructed based on Markov processes by switching between them. At unobservable random moment a change in distribution of observed sequence takes place. Using probability maximizing approach the optimal stopping rule for detecting the disorder is identified. Some explicit solution for example is also obtained. The result is generalization of Bojdecki's model where before and after the change independent processes are observed.
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Papers by Krzysztof J. Szajowski