Catálogo de publicaciones - libros

Astopping game problem is formulated by cooperating with fuzzy stopping time in a decision environment. The dynamic fuzzy system is a fuzzification version of a deterministic dynamic system and the move of the game is a fuzzy relation connecting between two fuzzy states.We define a fuzzy stopping time using several degrees of levels and instances under a monotonicity property, then an “expectation” of the terminal fuzzy state via the stopping time. By inducing a scalarization function (a linear ranking function) as a payoff for the game problem we will evaluate the expectation of the terminal fuzzy state. In particular, a two-person zero-sum game is considered in case its state space is a fuzzy set and a payoff is ordered in a sense of the fuzzy max order. For both players, our aim is to find the equilibrium point of a payoff function. The approach depends on the interval analysis, that is, manipulating a class of sets arising from α-cut of fuzzy sets. We construct an equilibrium fuzzy stopping time under some conditions.

The paper is concerned with two-person nonzero-sum stopping games in which pairs of randomized stopping times are game strategies. For a general form of reward functions, existence of Nash equilibrium strategies is proved under some restrictions for three types of games: quasi-finite-horizon, random-horizon and infinite-horizon games.

We survey recent results on the existence of the value in zero-sum stopping games with discrete and continuous time, and on the existence of -equilibria in nonzero-sum games with discrete time.

We consider stopping games for Markov chains in the formulation introduced by Dynkin [6]. Two players observe a Markov sequence and may stop it at any stage. When the chain is stopped the game terminates and Player 1 receives from Player 2 a sum depending on the player who stopped the chain and on its current state. If the game continues infinitely, then Player 1 gets “the payoff at infinity” depending on the “limiting” behavior of the chain trajectory.

We describe the structure of solutions for a class of stopping games with a countable state space and nonnegative payoffs. The payoff is equal to zero if Player 1 stops the chain but not Player 2. These solutions require using of randomized stopping rules. We study an extent of dependence of the value for these games on the “the payoff at infinity”. It turns out that this extent is determined with the “limiting” behavior of payoffs and with the transition structure of the chain.

A zero-sum game version of the full-information best choice problem is considered. Two players observe sequentially a stream of random variables (objects) from a known continuous distribution appearing according to some renewal process with the object of choosing the largest one. The horizon of observation is a positive random variable independent of objects. The observation of the random variables is imperfect and the players are informed only whether the object is greater than or less than some levels specified by both of them. Each player can choose at most one object. If both want to accept the same object, a random assignment mechanism is used. If some Player accepts an object, the other Player can change his level and continues the game alone. A similar game with discrete time and random number of objects is considered as a dual problem. The normal form of the game is derived. For the Poisson stream and the exponential horizon the value of the game and the form of the equilibrium strategy are obtained. In discrete-time case a game with geometric number of objects is completely solved.

A mathematical model of competitive selection of the applicants for a post is considered. There are 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. 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 appearance and can be accepted at that moment. At each moment 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.

A non-zero-sum -stage game version of a full-information best-choice problem under expected net value (ENV) maximization is analyzed and the solutions are obtained in some special cases of 2-person and 3-person games. The essential feature contained in this multistage game is the fact that the players have their own weights by which at each stage one player’s desired decision is preferred to the opponent’s one by drawing a lottery.

A bargaining problem with two players, Labor (player ) and Management (player ) is considered. The players must decide the monthly wage paid to by . At the beginning players and submit their offers and . If there is an agreement at + /2. If not, the arbiter is called in and he chooses the offer which is nearest to his solution α. The arbiter imposes a solution α which is concentrated in two points with probabilities = 1/2. The equilibrium in the arbitration game among pure and mixed strategies is derived.

Queueing phenomena, along with many related decision problems, are well known to all of us from daily situations. We often need to answer questions concerning whether to queue or not, where to queue, how long to queue etc. In networking applications, both in road traffic as well as in telecommunication networks, individuals or some central controllers are frequently faced with similar questions. These decisions have often to be taken in a randomly changing environment, in a decentralized way, and with partial information. This gives rise to many challenging problems in dynamic games. We shall describe in this overview some generic queueing problems (both static and dynamic) that require game theoretic models and solutions.

We study optimal static routing problems in open multiclass networks with state-independent arrival and service rates. Our goal is to study the uniqueness of optimal routing under different scenarios. We consider first the overall optimal policy, that is the routing policy whereby the overall mean cost of a job is minimized. We then consider an individually optimal policy whereby jobs are routed so that each job may feel that its own expected cost is minimized if it knows the mean cost for each path. This is related to the Wardrop equilibrium concept in a multiclass framework. We finally study the case of class optimization, in which each of several classes of jobs tries to minimize the averaged cost per job within that class; this is related to the Nash equilibrium concept. For all three settings, we show that the routing decisions at optimum need not be unique, but that the utilizations in some large class of links are uniquely determined.