Showing papers on "Stochastic game published in 1975"

Games with multiple payoffs

Abstract: The traditional theories of decision making and games are based on an assumption that prevents their broader practical utilization:a single dimensional payoff In reality, any alternative is likely to imply more than one payoff, eg not only costs but also time, price, quality, safety, maintainability, productivity, etc Similarly, the Theory of Games faces difficulties as we attempt to apply it to the conflict situations of the social and business environment The assumption that a single dimensional payoff, like money, points or chips, is a realistic consequence of individual moves or strategies, is difficult to sustain In this short paper, concepts of nondominated solutions and a decomposition of parametric spaces are used to formulate and resolvedecision problems with vector payoffs Both Games against Nature and Two-Person, Zero-Sum frameworks are considered A concept of compromise solutions is introduced to help the decision maker with further reduction of the nondominated set Some numerical examples are given

Optimal Control of Stochastic Linear Distributed Parameter Systems

Abstract: In this article we give necessary and sufficient conditions of optimality for linear stochastic distributed parameter systems, with convex differentiable payoffs and partial observation. They are obtained through variational methods, which can be applied only in the case of fixed information (i.e., not dependent on the control or the state). However, by a density argument it is proven that an optimal control adapted to the observation is also optimal for a space of controls adapted to some fixed information. Therefore we can get the necessary and sufficient conditions also in the case of feedback controls. We then prove, as a consequence, the separation principle for distributed parameter systems in the case of a quadratic payoff.

The Sequential Truel

Abstract: The Sequential Truel is a three-person game which generalizes the simple duel. The players' positions are fixed at the vertices of an equilateral triangle, and they fire, in sequence, until there is only one survivor or until each survivor has fired a pre-specified number of times. The rules of the particular game may or may not permit the tactic of abstention, i.e. firing into the air. Several versions of Sequential Truel (with and without abstention) are examined here. It is found that, often, there is a single equilibrium point which can be called the solution of the truel for rational players. Quite frequently, the poorest marksman of the three has the greatest payoff at this equilibrium point.

A generalized choice problem

Abstract: This paper deals with a generalization of a class of optimal stopping problems often referred to as the secretary problem. In the secretary problem, a decision maker views a group ofN candidates sequentially, each of which can be ranked according to some quality, although he does not know any distribution for the quality being measured. Generalizations pursued in this paper include randomN and a more elaborate payoff structure. Conditions on the payoff structure are given which result in a special subclass of stopping rules.

A Tale of Four Information Structures

Abstract: From a decision-theoretic viewpoint, the value of information in a stochastic optimization problem is roughly characterized as Value of Information = the Best the decision maker can do with the information - the best the decision maker can do without the information.

Alternatives to the Tree Model for Extensive Games

Abstract: The interaction between the moves of the players, the information available for each move and the outcome of chance moves, is described for finite games by the well known tree model due to Kuhn [2]. Among the reasons for considering alternative models of extensive games are (i) A desire to weaken the finiteness requirement and the need to handle the measurability problems which then arise. (ii) The natural way in which extensive games often present themselves is an input-output description. Even in the finite case, to derive a tree model from such a description requires the general solution of the closed loop relations, which can be demanding. (iii) For several simple but important conclusions concerning the effect of information changes in games, the tree model is already more detailed than necessary and may even hide the simplicity of the situation. On the other hand, the matrix form of the game does not contain enough detail to permit formulation of even such a concept as “open-loop strategy”.

Lower Semicontinuous Stochastic Games with Imperfect Information

Abstract: Shapely's stochastic game is considered in a more general setting, with the accumulated payoff being regarded as a function on the space of infinite trajectories, and the set of states of the system taken as a compact metric space. It has been shown that any game with a lower semicontinuous payoff has value and one of the players has an optimal strategy. As a consequence, in Shapley's game both players have optimal strategies.

The pursuit-evasion problem of two aircraft in a horizontal plane

Abstract: The pursuit-evasion problem of two aircraft in a horizontal plane is modelled as a zerosum differential game with capture time as payoff. The aircraft are modelled as point masses with thrust and bank angle controls. The games of kind and degree for this differential game are solved.

Pareto optimum by independent trials

Abstract: We consider a method by which players in a continuous N -person game can arrive at a Pareto optimal solution by a trial process. The process has a number of novel features. Firstly, it is assumed that the players do not know the payoff functions . Secondly, the players are assumed to act quite independently . In spite of this lack of information and lack of cooperation, the players eventually arrive at what is usually regarded as a cooperative solution. The process is a model of the accounting procedures used by firms, and the results predict approach to an equilibrium state of a market model. Proofs are given only in outline here.

Dynamic Time-Staged Model for R&D Portfolio Planning—A Real World Case

Abstract: — A dynamic, time-staged mixed integer model is currently being used for the selection of industrial long-range R&D portfolio. Input data from research and marketing are processed through logical programs to provide the discounted payoffs, probabilities (Bayesian, parallel strategies), subjective nonmonetary multiple criteria weights, and the matrix for the mathematical programming model. The multiperiod portfolio is revised sequentially for resources exhaustion throughout the planning horizon by the staged introduction of additional projects. Marginal payoff function (additional investments vs. expected payoff) is a by-product for the support of additional resources justification. Mathematical and heuristic techniques used to overcome common difficulties confronted by previously discussed models are presented. Experience with model introduction to management is also discussed.

Incentive game with incomplete information: an application to a public input model

Abstract: The purpose of this communication is to extend some results of [1] concerning an incentive game with incomplete information to the public input model of Groves and Loeb (see [2]) This model involves a group of firms using a public input which is made available to the group by a central agency, knowing only the “revealed” revenue functions of the firms Groves and Loeb formulate the associated incentive problem in the framework of a non-cooperative n-person game with complete information in which the strategy space of each player (each firm) is the set of plausible revenue functions which he may reveal as his “true” revenue function The payoff of each player is the profit he realizes after sharing the cost of the public input according to some given rule The main result of [2] is to exhibit a rule for which the true revenue functions of the players form a Nash Equilibrium This is the notion of incentive compatibility introduced by [4] It should be noted however, that this notion is rather demanding since it requires that, a priori, each firm considers each possible revenue function of each other firm as if it was the true function of this other firm But each firm may have “beliefs” concerning the likelihood of the Colloque sur la Theorie des Jeux, Institut des Hautes Etudes de Belgique, 1975

Stochastic pursuit-evasion games with information lag. i. perfect observation

Abstract: A discrete time, scalar, pursuit-evasion game is presented. in which an evader, moving according to a stationary stochastic process, is continually being followed by a pursuer. Both players have perfect observations of the evader's positions, but the observations of the pursuer are subject to a time lag. It is assumed that the strategy of the evader can be represented as an infinite moving average, and that he is restricted by a velocity constraint. The pursuer is limited to strategies linear in his information, and the payoff is taken to be the mean-square distance between pursuer and evader. Under these conditions it is shown that the game does not have a value, and subsequently the lower and upper values and corresponding strategies are found. PURSUIT-EVASION GAMES; PREDICTION GAMES; LAGGED INFORMATION

A Note on Generalized Pursuit-Evasion Games

Abstract: Two player zero sum differential games are an extension of optimal control problems When the cost or payoff is the integral of some function h up to the first time the trajectory enters a “terminal set” the differential game is one of survival If $h \equiv 1$, the payoff is just the time elapsed up to the “capture time” and the game is one of pursuit and evasion If $h \geqq 1$, the game is called a generalized pursuit-evasion game, and in previous papers it has been shown- that when the Isaacs condition is satisfied the upper and lower “extended” values of such a differential game are equal—that is, the game “has extended value” In the present note this result is proved under the weaker condition that $\max _y \min _z h(t,x,y,z) \geqq 0$

Stochastic Differential Games and Alternate Play

Abstract: Consider a static two-person zero sum competitive situation where a continuous real valued function ϕ (y,z) is given. A player, or controller, J 1 is to choose y from a compact space Y with the object of maximising ϕ, and a second player J 2 is at the same time to choose z from a similar space Z with the object of minimising ϕ. If Y and Z are finite the situation is that of the classic zero sum matrix game.

A median solution for a two-person, non-zero sum game

Abstract: Various solutions have been suggested for the two-person non-zero sum game, prominent among them being the maximin strategy and the equilibrium point. The median solution suggested in this paper benefits both players, provided they agree to cooperate.

Some General Properties of Non-Cooperative Games

Abstract: Let there be given two strategy sets Σ1 and Σ2, and two payoff functions f1 and f2 on Σ1 × Σ2. This is the usual setting of two-persons-non-cooperative game theory. We are looking for situations (σ1, σ2) ∈ Σ1 × Σ2 which exhibit some kind of stability when each player strives to maximize his own payoff. Such situations will be called “equilibria” or “solutions” of the game, and many different kinds have been discovered since the beginning of game theory.

Introduction to Differential Games II: Stochastic Games and Parabolic Equations

Abstract: In section 6 of the first lecture we stated that the non-linear parabolic equation related to the Isaacs-Bellman equation has a unique solution, and we mentioned how Fleming [5], [6], by considering a related stochastic difference equation, established the convergence of the upper and lower values of his approximating games However, we did not explicitly introduce any stochastic differential games; this was because in a stochastic situation certain measurability problems enter and care has to be taken when defining strategies and value

Note on the optimal strategies for the finite-stage Markov game

Abstract: In this no te we consider the finite-stage Markov game with finitely many states and actions as described by Zachrisson [5]. Zachrisson proves that this game has a value and shows that value and optimal strategies may be determined with a dynamic programming approach. However, he silently assumed that both players would use only Markov strategies. Here we will give a simple proof which shows this restriction to be irrelevant.