Correlated Equilibria in Competitive Staff Selection Problem

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 equilibria selection are used.Comment: The idea of this paper was presented at Game Theory and Mathematical Economics, International Conference in Memory of Jerzy Los(1920 - 1998), Warsaw, September 200

Multi-variate quickest detection of significant change process

The paper deals with a mathematical model of a surveillance system based on a net of sensors. The signals acquired by each node of the net are Markovian process, have two different transition probabilities, which depends on the presence or absence of a intruder nearby. The detection of the transition probability change at one node should be confirmed by a detection of similar change at some other sensors. Based on a simple game the model of a fusion center is then constructed. The aggregate function defined on the net is the background of the definition of a non-cooperative stopping game which is a model of the multivariate disorder detectionvoting stopping rule, majority voting rule, monotone voting strategy, change-point problems, quickest detection, sequential detection, simple game

On a random number of disorders

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 successive moments (they can be equal also and equal zero too) 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.disorder problem, sequential detection, optimal stopping, Markov process, change point, double optimal stopping

Duration problem: basic concept and some extensions

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 is less than s is less than n. It the instances following the $n$th 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)

Nonzero-sum Stochastic Games

This paper treats of stochastic games. We focus on nonzero-sum games and provide a detailed survey of selected recent results. In Section 1, we consider stochastic Markov games. A correlation of strategies of the players, involving ``public signals'', is described, and a correlated equilibrium theorem proved recently by Nowak and Raghavan for discounted stochastic games with general state space is presented. We also report an extension of this result to a class of undiscounted stochastic games, satisfying some uniform ergodicity condition. Stopping games are related to stochastic Markov games. In Section 2, we describe a version of Dynkin's game related to observation of a Markov process with random assignment mechanism of states to the players. Some recent contributions of the second author in this area are reported. The paper also contains a brief overview of the theory of nonzero-sum stochastic games and stopping games which is very far from being complete.average payoff stochastic games, correlated stationary equilibria, nonzero-sum games, stopping time, stopping games

Optimal detection of homogeneous segment of observations in stochastic sequence

A Markov process is registered. At random moment $\theta$ the distribution of observed sequence changes. Using probability maximizing approach the optimal stopping rule for detecting the change is identified. Some explicit solution is obtained.Comment: 13 page

Full vs. no information best choice game with finite horizon

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.Comment: Submitted to: Stochastic Operations Research in Business and Industry (eds. by Tadashi Dohi, Katsunori Ano and Shoji Kasahara), World Scientific Publishe