Showing papers on "Stochastic game published in 1973"

Additional aspects of the Stackelberg strategy in nonzero-sum games

Abstract: The Stackelberg strategy in nonzero-sum games is a reasonable solution concept for games where, either due to lack of information on the part of one player about the performance function of the other, or due to different speeds in computing the strategies, or due to differences in size or strength, one player dominates the entire game by imposing a solution which is favorable to himself. This paper discusses some properties of this solution concept when the players use controls that are functions of the state variables of the game in addition to time. The difficulties in determining such controls are also pointed out. A simple two-stage finite state discrete game is used to illustrate these properties.

Oddness of the number of equilibrium points: A new proof

Abstract: A new proof is offered for the theorem that, in “almost all” finite games, the number of equilibrium points is finite and odd. The proof is based on constructing a one-parameter family of games with logarithmic payoff functions, and studying the topological properties of the graph of a certain algebraic function, related to the graph of the set of equilibrium points for the games belonging to this family. In the last section of the paper, it is shown that, in the space of all games of a given size, those “exceptional” games which fail to satisfy the theorem (by having an even number or an infinity of equilibrium points) is a closed set of measure zero.

Discounted, positive, and noncooperative stochastic games

Abstract: In this paper, we consider the stochastic games ofShapley and prove under certain conditions the stochastic game has a value and both players have optimal strategies. We also prove a similar result for noncooperative stochastic games.

Algorithms for discounted stochastic games

Abstract: In this paper, a two-person zero-sum discounted stochastic game with a finite state space is considered. The movement of the game from state to state is jointly controlled by the two players with a finite number of alternatives available to each player in each of the states. We present two convergent algorithms for arriving at minimax strategies for the players and the value of the game. The two algorithms are compared with respect to computational efficiency. Finally, a possible extension to nonzero sum stochastic game is suggested.

On a Class of Secretary Problems

Abstract: Asymptotic forms for the optimal expected payoffs (minimal costs) for a generalized class of "Secretary" problems are investigated by an analysis of a related family of differential equations. A class of unbounded payoff functions with bounded expected payoffs is determined and methods for generating the expected payoffs are developed.

Coalitions and payoffs in three‐person sequential games: Initial tests of two formal models

Abstract: In this paper we develop two formal models predicting coalitions and payoffs among rank striving players in a sequential three‐person game. We test the models’ predictions with data from a laboratory study of eleven male triads. Each triad plays a sequence of games; in each game a two‐person coalition forms and divides the coalition's point value between the two coalition partners. Participants know that the sequence of games will end without warning at a randomly chosen time; at the sequence's end each player's monetary payoff is a linear function of the rank of his accumulated point score, relative to those of the other members of his triad. The complexity of this situation prevents players and analysts from representing it as a single game; thus they are unable to use n‐person game theory to identify optimal strategies. Consequently, we assume that players, unable to develop strategies that are demonstrably optimal in the long run, adopt certain bargaining heuristics and surrogate short run objectives....

Point Estimation and Risk Preferences

Abstract: The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic payoff function and a lower estimate with a piecewise linear payoff function, for example.

Collective bargaining: A differential game

Abstract: Earlier differential game models of collective bargaining deal only with the strike phase. The model proposed here allows for negotiations without and with a strike.

Continuous stochastic games

Abstract: Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness. NONCOOPERATIVE STOCHASTIC GAME; DISCOUNTED MARKOVIAN DECISION PROCESS; EQUILIBRIUM POINT; DYNAMIC PROGRAMMING

Two-sided search games

Abstract: NUMBER 4 A target hides himself in one of the boxes 1,2, ... , m with probability distribution X=

A simple example on informativeness and performance

Abstract: We present in this note a simple example in which increasing informativeness leads to decreasing performance.

Theory of Measures of Effectiveness for General-Purpose Military Forces: Part I. A Zero-Sum Payoff Appropriate for Evaluating Combat Strategies

Abstract: Difficulties encountered in a number of past force-procurement studies for general-purpose forces have been traced, in large measure, to problems in defining the objectives for such forces. In this paper, we develop a rationale for selecting a specific zero-sum payoff function to compare alternative combat strategies. The development is motivated by analogy with the Nash bargaining solution for non-zero-sum two-person games [Econometrica 21, 128–140 (1953)]. Our objectives are to relate known mathematical techniques and game-theory concepts to the problem of evaluating the effectiveness of general-purpose military forces, and thus to bridge the gap between theoreticians and practical planners. This paper is the first in a series devoted to the development of a more consistent analysis framework.

Nonzero-sum differential games with bargaining solutions

Abstract: Nash's bargaining solution for finite games is extended to differential games with nonzero-sum integral payoffs. Sufficient conditions for the optimality of a strategy pair are established. An example is given.

On repeated games with general information function

Abstract: For a class of repeated two-person zero-sum games with incomplete information it was proved byAumann andMaschler that limν n exists,ν n being the value of the game withn repetitions. If the players know at each stage the moves done by both players at all previous stages,Aumann andMaschler could prove that the error termδ n=¦ν n — limν n ¦ satisfiesδ n≤c/√n for somec>0. It was then shown byZamir that this bound is the lowest possible. In this paper it is shown that if previous moves are not always announced,δ n may be of higher order of magnitude e.g.δ n≥c/n 1/3 for somec>0. New upper bounds forδ n are given for two classes of games.

Using fishburn's techniques for analysis of decision trees: some examples*

Abstract: Techniques for the analysis of payoff matrices with “imprecise” assessment of state probabilities are applied to the analysis of decision trees in normal form. The analytical techniques, due to Peter Fishburn, are summarized and then illustrated by analyzing two decision trees. Problem I, a two state problem, provides geometrical analogs for the two most “imprecise” probability assessments and simple analytical and geometrical analogs for the least “imprecise” probability assessment. Problem II, a three state problem, illustrates the application of all four precision levels, from most “imprecise” through least “imprecise.”

Technical Note-Risk Aversion in Stochastic Programming with Recourse

Abstract: In stochastic programming with recourse the objective is to maximize the expected net payoff. This assumes implicitly no aversion to risk. With risk aversion, the objective becomes to maximize the expected concave utility of the net payoffs. Because of the special structure of the problem with risk aversion, a number of computational short cuts are possible in the mathematical program that results. All the second-stage problems can be solved as linear programs. Unfortunately, whether with or without risk aversion, it is necessary to solve the first-stage problem as a nonlinear program. This note shows that the latest representation of the gradient is but a simple modification of the latest representation of the linear objective function without risk aversion.

Open Loop Approximation of Differential Games

Abstract: A linear differential game of fixed duration where the players are restricted to compact control sets is approximated in value by a game in which the players may choose any square-integrable functions as admissible controls. This is done by appending penalty terms to the payoff. The general approximation result leads to a computational method of estimating the value of a certain class of games in which the approximating games have open-loop solutions.

Optimal strategies for a class of adaptive search processes

Abstract: Optimal strategies are investigated for a class of one-dimensional search processes in which the objective is to find a point which is near, but not beyond, a boundary of uncertain location. Problems of this type are encountered in the analysis of mining operations. Upper and lower bounds for the optimal expected payoff are derived, and the optimal search strategies are described explicitly for a large subclass of these processes. Results are obtained by formulating the search as a multistage decision process and using a dynamic programming approach.

The associated maximization problem for separable games

Abstract: Two-person zero-sum games with separable payoff functions are examined using the geometric concept of dual cones. It is shown that the value of such games may be found by solving an associated maximization problem. Some numerical implications, particularly the application of linear programming to finding approximate solutions, are discussed. With the value known, optimal mixed strategies may, in principle, be readily determined.

Note on “The Value of Two-Person Zero-Sum Repeated Games the extensive case”

Abstract: The purpose of this note 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.

Games against p -intelligent players

Abstract: The player who chooses his strategies after a logical analysis of the conflict such as to maximize his payoff is called normatively intelligent (n.i.). The player who chooses his strategies in accordance with a probability distribution disregarding the amount of his payoff is called a random mechanism (r.m.). The player who behaves with the probabilityp like an n.i. player and with the probability 1−p like a r. m. is calledp-intelligent (p.i.) player. For the matrix games, general two-player noncooperative games and forN-player games with at least one p.i. player it is defined the notion of the optimal strategy for the n.i. player. The strategy is generally a function of the parameterp, which characterizes the p.i. player (p being a vector if there are more p.i. players). It is shown that for an n.i. player it is profitable to adjust its strategies to the true nature of the p.i. opponents rather then use the equilibrium strategies. Several examples illustrating the theory are presented and the problem of estimation ofp from observed choices of the p.i. player is also briefly discussed. The motivation for the study consists in the easily verifiable fact that the real players usually behave neither like the n.i. players nor like the r.m.