Linear-quadratic two-person zero-sum differential games,necessary and sufficient conditions: Comment

We give a simpler, easier-to-check, version of the theorem of the paper referred to, i.e., a necessary and sufficient condition for the existence of a saddle point to the linear-quadratic two-person zero-sum perfect information differential game.     

Necessary and sufficient conditions for multistage two-person zero-sum games

After previously obtained necessary conditions for optimal strategies in multistage two-person zero-sum games are summarized, necessary and sufficient conditions are derived. The utilization of these conditions is illustrated by means of a simple example.This research was supported by the Office of Naval Research under Grant No. Nonr 3656-31.     

A note on two-person zero-sum communicating stochastic games

For undiscounted two-person zero-sum communicating stochastic games with finite state and action spaces, a solution procedure is proposed that exploits the communication property, i.e., working with irreducible games over restricted strategy spaces. The proposed procedure gives the value of the communicating game with an arbitrarily small error when the value is independent of the initial state.     

Solution concepts in two-person multicriteria games

In this paper, we propose new solution concepts for multicriteria games and compare them with existing ones. The general setting is that of two-person finite games in normal form (matrix games) with pure and mixed strategy sets for the players. The notions of efficiency (Pareto optimality), security levels, and response strategies have all been used in defining solutions ranging from equilibrium points to Pareto saddle points. Methods for obtaining strategies that yield Pareto security levels to the players or Pareto saddle points to the game, when they exist, are presented. Finally, we study games with more than two qualitative outcomes such as combat games. Using the notion of guaranteed outcomes, we obtain saddle-point solutions in mixed strategies for a number of cases. Examples illustrating the concepts, methods, and solutions are included.     

Some thoughts on saddle-point conditions and information structures in zero-sum differential games

For a very simple two-stage, linear-quadratic, zero-sum difference game with dynamic information structure, we show that (i) there exist nonlinear saddle-point strategies which require the same existence conditions as the well-known linear, closed-loop, no-memory solution and (ii) there exist both linear and nonlinear saddle-point strategies which require more stringent conditions than the unique open-loop solution. We then discuss the implication of this result with respect to the existence of saddle points in zero-sum differential games for different information patterns.     

Formulation and analysis of combat problems as zero-sum bicriterion differential games

In this paper, we present a formulation and analysis of a combat game between two players as a zero-sum bicriterion differential game. Each player's twin objectives of terminating the game on his own target set, while simultaneously avoiding his opponent's target set, are quantified in this approach. The solution in open-loop pure strategies is sought from among the Pareto-optimal security strategies of the players. A specific preference ordering on the outcomes is used to classify initial events in the assured win, draw, and mutual kill regions for the players. The method is compared with the event-constrained differential game approach, recently proposed by others. Finally, a simple example of the turret game is solved to illustrate the use of this method.     

Sufficient conditions for optimality in two-person zero-sum differential games with state and strategy constraints

A sufficiency theorem for optimal feedback strategies in two-person zero-sum differential games is given. The theorem is applicable to a wide class of such games for which strategies are Borel measurable functions on a subset of the state space. The theorem generalizes those of [1, 2, and 5].     

Sufficient conditions for Nash bargaining in differential games

Sufficient conditions for Nash bargaining in differential games are given. These conditions are compared with the sufficient conditions given by Liu (Ref. 1).     

Relevant aspects in two-person games

The present paper is concerned with characterizing in a nonusual form the equilibrium points for the mixed extension of a two-person game. We study interesting properties about such equilibrium points which are concerned with different pairs of them. Finally, we introduce an elimination procedure for pure strategies and relate in a general way the complete set of equilibrium points.This work has been partially supported by the Consejo Nacional de Investigaciones Cientificas y Tecnicas, Buenos Aires, Argentina.     

On saddlepoints of two-person zero-sum games with applications to data verification tests

A set of sufficient conditions for the existence of saddlepoint strategies of two-person zero-sum games is given which may be described as follows: The set of pure strategies for one of the two players is a compact metric space. His opponent has a best answer to each randomized strategy. The payoff function satisfies a continuity condition concerning the weak convergence of probability measures. These conditions are neither covered by those of well known existence theorems for saddlepoints nor do they generalize them. They are applied to treat important practical problems that have not been solved before. Received November 1995/Revised version August 1998     

Statistical linear differential games: A treatment of incomplete information

A zero-sum, two-player linear differential game of fixed duration is considered in the case when the information is incomplete but a statistical structure gives both players the possibility tospy the value of an unknown parameter in the payoff. Considerations of topological vector spaces and functional analysis allow one to demonstrate, via a classical Sion's theorem, sufficient conditions for the existence of a value.The author is indebted to Professor J. Fichefet for his helpful remarks and indications.     

Application of stochastic differential games to medium-range air-to-air missiles

Stochastic differential game techniques are applied to compare the performance of a medium-range air-to-air missile for three different thrust-mass profiles. The measure of the performance of the missile is the probability that it will reach a lock-on point with a favorable range of guidance and flight parameters during a fixed time interval [0,t _f ].     

Saddle-point equilibrium sequence in one class of singular infinite horizon zero-sum linear-quadratic differential games with state delays

ABSTRACT

We consider an infinite horizon zero-sum linear-quadratic differential game with state delays in the dynamics. The cost functional of this game does not contain a control cost of the minimizing player (the minimizer), meaning that the considered game is singular. For this game, definitions of the saddle-point equilibrium and the game value are proposed. These saddle-point equilibrium and game value are obtained by a regularization of the singular game. Namely, we associate this game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. An asymptotic analysis of this cheap control game is carried out. Using this asymptotic analysis, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.     

Risk-sensitive stochastic differential games with reflecting diffusions

We study risk-sensitive differential games for controlled reflecting diffusion processes in a bounded domain. We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria; namely, discounted cost and ergodic cost. Under certain assumptions we establish the existence of Nash/saddle-point equilibria for relevant cases.     

Stochastic two-target pursuit-evasion differential games in the plane

A stochastic version of a two-target homicidal chauffeur, pursuit-evasion differential game (using polar coordinates) is considered. This is used to model a dogfight between a very agile playerQ and a less maneuverable playerP. First, the case where both players have complete observation of the state of the game is considered. A numerical study is conducted, by solving numerically a nonlinear partial differential equation on a torus in ², to investigate the role of the parameters of speed, maneuverability, and performance of the weapon systems, in the encounter. Second, the model is extended to include the case where playerP is jamming playerQ's measurements of , where denotes the bearing ofQ fromP. A numerical study is conducted, by solving numerically a nonlinear partial differential equation on a generalized torus in ³, to investigate the role of the jamming parameter on the outcome of the combat.     

Stackelberg strategies in linear-quadratic stochastic differential games

This paper obtains the Stackelberg solution to a class of two-player stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that the players make independent noisy measurements of the initial state and are permitted to utilize only this information in constructing their controls. Furthermore, by the very nature of the Stackelberg solution concept, one of the players is assumed to know, in advance, the strategy of the other player (the leader). For this class of problems, we first establish existence and uniqueness of the Stackelberg solution and then relate the derivation of the leader's Stackelberg solution to the optimal solution of a nonstandard stochastic control problem. This stochastic control problem is solved in a more general context, and its solution is utilized in constructing the Stackelberg strategy of the leader. For the special case Gaussian statistics, it is shown that this optimal strategy is affine in observation of the leader. The paper also discusses numerical aspects of the Stackelberg solution under general statistics and develops algorithms which converge to the unique Stackelberg solution.This work was performed while the second author was on sabbatical leave at the Department of Applied Mathematics, Twente University of Technology, Enschede, Holland.     

Necessary and sufficient conditions for oscillation of first order nonlinear neutral differential equations

In this paper, we prove that every solution of the first order nonlinear neutral differential equation

     

A sufficient condition for the evadability of differential evasion games

A sufficient condition for the strict evadability of nonlinear differential evasion games is obtained. The result complements, in some sense, the relevant results obtained by the author in a previous paper. An illustrative example is discussed as well. The author thanks Professor L. D. Berkovitz for some discussions.