Showing papers on "Stochastic game published in 1971"

On stochastic games, II

Abstract: In this paper, we consider positive stochastic games, when the state and action spaces are all infinite. We prove that, under certain conditions, the positive stochastic game has a value and that the maximizing player has an e-optimal stationary strategy and the minimizing player has an optimal stationary strategy.

The Value of Two-Person Zero-Sum Repeated Games the extensive case

Abstract: The purpose of this article is to extend the results of J. F.Mertens and S.Zamir, The Value of Two-Person Zero-Sum Repeated Games with Lack of Information on Both Sides (Intern. Journal of Game Theory,1, 39–64, 1971) to the case where both players are not necessarily informed of each other's pure strategy choices at each stage.

Optimal threat strategies of bimatrix games

Abstract: A bi-matrix threat game is defined as a triple (A,B,S) whereA andB arem×n payoff matrices, andS is a closed convex subset of the plane, with (aij,Bij) eS for eachi,j. Given (threat) mixed strategiesx andy,Nash's model suggests that the eventual outcome will be that point (u, v) eS which maximizes the product (u −xAyt) (v −xByt) subject tou ≥xAyt,v ≥xByt. Optimality of the threat strategies is then defined in the obvious way.

A Generalized Lagrange-Multiplier Method for Constrained Matrix Games

Abstract: This paper presents a general method for solving constrained matrix games of a type occurring frequently in military and industrial operations research. The usual context is the optimal allocation of constrained resources by two opposing sides among a series of independent cells such that the payoff overall is the sum of the payoffs at each cell. The cells themselves might represent separate invasion or supply routes, battlefields, individual targets, or marketing ventures. Generalized Lagrange multipliers Everett multipliers are used to effect the solution. It is shown that the basic properties of the games preclude the possibilities of nonconcavities or extraneous solutions.

The absolute maximum payoff in differential games and optimal control

Abstract: This paper deals with a differential game or optimal control problem in which the payoff is the maximum (or minimum), during play, of some scalar functionK of the statex. This unconventional payoff has many practical applications. By defining certain auxiliary games for a significant class of problems, one can show how to solve the general case where more than one maximum ofk(t)=K[x(t)] occurs under optimal play. For a subclass of such problems, it is found thatclosed optimal solutions can exist on certain surfaces in the playing space. As the playing interval becomes indefinitely long, the open optimal trajectories converge to (or diverge from) such surfaces. In particular, for two-dimensional problems of this subclass, the closed optimal trajectories are periodic and called periodic barriers. They are analogous to limit cycles in uncontrolled nonlinear systems.

Random payoff games with partial information : one person games against nature

Abstract: — In this paper we analyze the game situation in which the player has incomplete or partial information concerning the random payoff from the game. Several criteria for such a décision maker are formulated with solutions based on the techniques of linear programming. Our results extend considerably the previous results which had been obtained for the solution of random pay off games with partial information, and give the optimizing player a higher expected payoff from the game.

Discrete Stochastic Differential Games

Abstract: Publisher Summary This chapter focuses on discrete stochastic differential games. The term differential games is a derivative of the mathematical theory of games. Games can have many forms depending on the number of players and the way in which winnings and losses are computed. Differential games involve only two players, where the loser pays the winner a specified amount after the play of a given game. As the algebraic sum of the winner's game (positive) and loser's gain (negative) is zero, this type of game is known as two-player zero sum game. Games can be presented either in extensive form as a set of rules and a succession of choices for each player or in normal form as a matrix or function, which relates the amount to the winner to the choices made by the two players. The amount, as a function of the choices, is known as the payoff of the game. An important concept in game theory is information. In games of perfect information, each player knows the exact value of the payoff and all that has occurred in the past. Differential games involve a payoff, which is in some way related to a dynamical system. The two players attempt to optimize this payoff by choosing game optimal control strategies. Differential games can be divided into classes, namely, those where observations of the state are perfect and those where they are not.