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.
Automation and Remote Control, Aug 1, 2022
Mathematical models of street traffic allowing assessment of the importance of their individual s... more Mathematical models of street traffic allowing assessment of the importance of their individual segments for the functionality of the street system is considering. Based on methods of cooperative games and the reliability theory the suitable measure is constructed. The main goal is to analyze methods for assessing the importance (rank) of road fragments, including their functions. A relevance of these elements for effective accessibility for the entire system will be considered.
The main objective of this article is to present Bayesian optimal control over a class of non-aut... more The main objective of this article is to present Bayesian optimal control over a class of non-autonomous linear stochastic discrete time systems with disturbances belonging to a family of the one parameter uniform distributions. It is proved that the Bayes control for the Pareto priors is the solution of a linear system of algebraic equations. For the case that this linear system is singular, we apply optimization techniques to gain the Bayesian optimal control. These results are extended to generalized linear stochastic systems of difference equations and provide the Bayesian optimal control for the case where the coefficients of these type of systems are non-square matrices. The paper extends the results of the authors developed for system with disturbances belonging to the exponential family.
Mathematical Game Theory and Applications, 2021
Mathematical models of street traffic allowing assessment of the importance of their individual s... more Mathematical models of street traffic allowing assessment of the importance of their individual segments for the functionality of the street system is considering. Based on methods of cooperative games and the reliability theory the suitable measure is constructed. The main goal is to analyze methods for assessing the importance (rank) of road fragments, including their functions. A relevance of these elements for effective accessibility for the entire system will be considered.
Mathematics, 2020
The aim of the work is to connect individual behavior of drivers with traffic intensity. By diver... more The aim of the work is to connect individual behavior of drivers with traffic intensity. By diversifying the populations of drivers into two categories, often considered in this type of an analysis, CO (cooperative) and DE (defective), the tendency of drivers from each of these groups to deviate from compliance with traffic rules is established. The effective driver behavior translates into disrupting traffic by slowing it down. Participant interactions are described using game theories that provide information for simulations algorithms based on cellular automata. Three different ways of using this combination of descriptions of traffic participants to examine the impact of their behavior on the traffic dynamics are shown. Directions of the further, detailed analysis are indicated, which requires basic research in the field of game theory models.
Applied Mathematics and Computation, 2016
The main objective of this article is to develop Bayesian optimal control for a class of non-auto... more The main objective of this article is to develop Bayesian optimal control for a class of non-autonomous linear stochastic discrete time systems with a random horizon of a control. By taking into consideration that the disturbances in the system are given by a random vector with components belonging to an exponential family with a natural parameter, we determine the Bayes control as the solution of a singular linear system. In addition we extend these results to generalized linear stochastic systems of difference equations.
Journal of Applied Probability, 2009
In this paper we consider the following problem. An angler buys a fishing ticket that allows him/... more In this paper we consider the following problem. An angler buys a fishing ticket that allows him/her to fish for a fixed time. There are two locations to fish at the lake. The fish are caught according to a renewal process, which is different for each fishing location. The angler's success is defined as the difference between the utility function, which is dependent on the size of the fish caught, and the time-dependent cost function. These functions are different for each fishing location. The goal of the angler is to find two optimal stopping times that maximize his/her success: when to change fishing location and when to stop fishing. Dynamic programming methods are used to find these two optimal stopping times and to specify the expected success of the angler at these times.
Journal of Mathematical Psychology
The decision-maker (DM) sequentially evaluates up to N of different, rankable options. DM must se... more The decision-maker (DM) sequentially evaluates up to N of different, rankable options. DM must select exactly the best one at the moment of its appearance. In the process of searching, DM finds out with each applicant whether she is the best applicant among those assessed so far (we call him a candidate). DM cannot return to rejected candidates. We discuss the psychological aspects of this selection problem, known in the literature as the secretary problem. The analysis is based on knowledge of the chances, and a subjective assessment of acceptance of the positive and negative effects DM's decision. The acceptance assessment of success and failure is presented on subjective scales. We set an optimal policy that recommends analyzing applicants up to a certain point in time (a threshold time) without selecting any of them and then selecting the next encountered candidate. The determined optimal threshold depends on the level of acceptance of the positive and negative effects of the choice. This issue is discussed in the article.
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 ...
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.
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.
Mathematica Applicanda, 2021
The paper focuses on portraying importance measures that are reasonably helpful in analyzing syst... more The paper focuses on portraying importance measures that are reasonably helpful in analyzing system reliability and its development. The presented measures concern coherent binary and multistate systems and help to distinguish the most influential elements of the system, which require more attention. Importance measures are presented for systems with known structure (e.g. parallel, series) and for repairable or nonrepairable components
Applied Stochastic Models in Business and Industry, 2018
We register a stochastic sequence affected by one disorder. Monitoring of the sequence is made in... more We register a stochastic sequence affected by one disorder. Monitoring of the sequence is made in the circumstances when not full information about distributions before and after the change is available. The initial problem of disorder detection is transformed to optimal stopping of observed sequence. Formula for optimal decision functions is derived.
We register a stochastic sequence affected by one or two disorders (two cases are considered). Mo... more We register a stochastic sequence affected by one or two disorders (two cases are considered). Monitoring is made in the circumstances when not full information about distributions between changes is available. Our aim is to detect the disorder or to localize the segment between changes (depending on the case). Both problems are transformed to optimal stopping of observed sequence and formulas of optimal decision functions are derived.
Frontiers of Dynamic Games, 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. Ramsey and Szajowski(2008)).
Applicationes Mathematicae, 2000
The following version of the two-player best choice problem is considered. Two players observe a ... more The following version of the two-player best choice problem is considered. Two players observe a sequence of i.i.d. random variables with a known continuous distribution. The random variables cannot be perfectly observed. Each time a random variable is sampled, the sampler is only informed whether it is greater than or less than some level specified by him. The aim of the players is to choose the best observation in the sequence (the maximal one). Each player can accept at most one realization of the process. If both want to accept the same observation then a random assignment mechanism is used. The zero-sum game approach is adopted. The normal form of the game is derived. It is shown that in the fixed horizon case the game has a solution in pure strategies whereas in the random horizon case with a geometric number of observations one player has a pure strategy and the other one has a mixed strategy from two pure strategies. The asymptotic behaviour of the solution is also studied.
Statistical Papers, 2020
We will analyze the importance of elements of a complex structure on the availability of the syst... more We will analyze the importance of elements of a complex structure on the availability of the system. The basis for the element assessment are the importance measures for multi-state systems introduced by Birnbaum (in: Krishaiah, Econometrics, principal components, reliability, and applications, Academic Press, New York, 1969) and Barlow and Proshan (Stoch Process Appl 3:153–173, 1975). The availability depends not only on the reliability, but also on the difficulty of maintenance, the ability to diagnose the need for service and its efficient implementation. If we assume that the need for maintenance is the result of deregulation, then determining the key elements to detect the moment of disorder of the system will be the basis for assessing the importance of the element for the system maintenance process.
In the sequential decision making task known as the best choice problem, n items are presented in... more In the sequential decision making task known as the best choice problem, n items are presented in a random order one at a time. After each item, the decision maker can determine only their relative ranks. The goal of decision maker is to select the best of all n items without the possibility of recalling previously observed items. Many important business decisions such as choosing a venture partner, adopting technologyical innovation, or hiring an employee can be modeled using a sequential decison approach. The best choice problem is the name for a class of sequential decision problems where the goal is to select the best one of n items. The decision maker either to stop the search with the current item or to continue depends only on the relative ranks of the items that have been observed so far. The paper deal with the continuous-time two person non-zero sum game extension of the best choice problem. The objects appear according to the Poisson process with unknown intensity assumed to be exponentially distributed. Each player can choose only one applicant. If both players would like to select the same one, then the priority is assigned randomly. The aim of the players is to choose the best candidate. A construction of Nash equilibria for such game is given. The optimal stopping problem for such stream of option has been presented and solved by Bruss (1987). The considered game is a generalization of the discrete time finite horizon two person non-zero sum game with stopping of Markov process solved by Szajowski (1993). The extension of the game to admit the sets of the randomized stopping times is taken into account. The Nash values for the randomized Nash equilibria is obtained. Analysis of the solutions for different lotteries are given.
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...
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Papers by Krzysztof J. Szajowski