A differential game of approach with two pursuers and one evader

A differential game of approach with one evader and two pursuers with a nonconvex payoff function is considered. The duration of the game is fixed. The payoff functional is the distance between the object being pursued and the pursuer closest to it when the game terminates. An explicit form of the game value is found for all possible game positions. The paper is closely related to Refs. 1–12.The authors would like to thank Professor A. I. Subbotin for his valuable advice and encouragement.     

A markov chain game with dynamic information

Two players, not knowing each other's position, move in a domain and can flash a searchlight. The game terminates when one player is caught within the area illuminated by the flash of the other. However, if this first player is not in this area, then the other player has disclosed his position to the former one, who may be able to exploit this information. The game is considered on a finite state space and in discrete time.The work of the second author was supported by ZWO, The Netherlands Organization for the Advancement of Pure Research, Contract No. B62-239, by the US Air Force Office of Scientific Research, Grant No. AFOSR-85-0245, and by the National Science Foundation, Grant No. NSF-INT-8504097.Visiting Professor at Delft University of Technology during 1986.     

Equivalences between differential games and optimal controls

It is shown that linear differential pursuit games with linear targets, if player controls are required to be piecewise constant or if player controls areL ₁-functions (but pursuer control is bang-bang whenever quarry control is), are equivalent to a linear, autonomous control problem. As a byproduct, a sufficient condition for terminating the game, in Pontryagin's sense, is obtained.The present paper has been influenced by Prof. O. Hajek's work in differential games; the converse parts of the proofs presented here are very similar to those in Ref. 5. The author wishes to thank Dr. Hajek for his suggestions, comments, and critique. The Brasilian Government BNDE provided partial financial support.     

Unique winning policies for linear differential pursuit games

We study optimal control problems which are the duals, in a specified sense, to a certain class of linear differential games. Directly verifiable conditions, in terms of the data of the game, for uniqueness of solutions of the dual problem and thus for uniqueness of winning policies for the differential game, are derived. As a byproduct, in the particular context of two-dimensional problems, a strong result concerning normality is obtained. As a second byproduct, several geometrical and topological properties of thestar difference are derived. This set operation is of paramount importance for the study of rich classes of differential and difference games extending far beyond that treated here.Notation co(P) convex hull of a setPR ⁿ - ext(P) set of extreme points ofP - [x ₁,x ₂] line segment joining the pointsx ₁,x ₂ R ⁿ - S(P, c) Supporting closed halfspace ofP with exterior normalc - H(P, c) supporting hyperplane ofP with exterior normalc - F(P, c) face of the polytopeP with exterior normalc - span(P) linear span ofP - lin(P) linear closure ofP = smallest linear manifold containingP - relbd(P) relative boundary ofP - int(P) interior ofP with respect to the topology ofR ⁿ - ri(P) relative interior ofP with respect to lin(P) - a, b inner product (inR ⁿ) ofa andb - U/W {x:x U R ⁿ andx W R ⁿ} - cl(P) closure ofP This work was done during the year 1972, when the author was a student of Prof. O. Hajek at Case Western Reserve University, Cleveland, Ohio. The author wishes to thank Dr. Hajek for his many comments, suggestions, and critique.     

Differential game of optimal approach of two inertial pursuers to a noninertial evader

In this paper, the game of the optimal approach of two identical inertial pursuers to a noninertial evader is investigated. The duration of the game is fixed. The payoff functional is the distance between the evader and the closest pursuer when the game terminates. The value function is constructed for all possible positions of the game. The regions where the pursuit is one-to-one and the regions where it is essentially collective are described algorithmically. Some analogies between this game and the linear differential game with elliptical vectograms are indicated. It is noted that the focal surface and the dispersal surface are in proximity of one another.     

Sufficient conditions for optimality of threat strategies in a differential game

The problem of defining threat strategies in nonzero-sum games is considered, and a definition of optimal threat strategies is proposed in the static case. This definition is then extended to differential games, and sufficient conditions for optimality of threat strategies are derived. These are then applied to a simple example. The definition proposed here is then compared with the definition of threat strategies given by Nash.     

Evasion from many pursuers in simple motion differential game with integral constraints

We study a two dimensional evasion differential game with several pursuers and one evader with integral constraints on control functions of players. Assuming that the total resource of the pursuers does not exceed that of the evader, we solve the game by presenting explicit strategy for the evader which guarantees evasion.     

A stochastic homicidal chauffeur pursuit-evasion differential game

A certain stochastic pursuit-evasion problem of the homicidal chauffeur type is considered. The pursuer strategy synthesized in this paper is fairly simple in contrast to the less straightforward swerve maneuver employed in the deterministic model. The analysis may partially explain why relatively simple pursuit strategies are apparently always adopted in practice.This work was partially supported by a grant from Control Data.The authors wish to thank Dr. D. H. Martin for a very enlightening discussion.     

The gold-mine game

The paper considers the following two-person zero-sum game. The minimizing player chooses to hide his gold and a mine in two distinct boxes from an infinite number of boxes labelled 1, 2, 3,.... The maximizing player now chooses to open the boxes in some order, and if he finds the gold before the mine the payoff to him is 1; otherwise, the payoff is zero. The game is solved in the sense of Kindler.     

Two-dimensional pursuit-evasion game with penalty on turning rates

An approach, both analytical and numerical, is used to solve a two-dimensional pursuit-evasion game characterized by a difficulty level intermediate between that of thesimple motion game (with freely and instantaneously oriented velocities) and that of thegame of two cars (with lower bounds on curvature radii). Each player's velocity has a constant modulus. The maneuvers are penalized by introducing, in the performance index, an integral term for the squared velocity turning rate.The local problem solution is relatively easy to find: the equations of motion and the adjoint equations can be integrated by means of elliptic functions and integrals. The global problem is more delicate to solve, because of the existence of a dispersal singular surface requiring an important numerical search to be determined. Thesynthesis problem (how to express the optimal strategies as functions of state) is not explicitly solvable, but a numerical approach using successive approximations can be developed. Illustrative interception trajectories are given.The authors are grateful to Mr. J. P. Peltier, Head, Guidance Group, Aerospace Mechanics Division, Systems Department, ONERA, Châtillon, France, for his suggestions and his efficient assistance in the numerical aspect of this study.     

A differential oligopoly game with differentiated goods and sticky prices

We investigate a dynamic oligopoly game where goods are differentiated and prices are sticky. We study the open-loop and the closed-loop memoryless Nash equilibrium, and show that the latter equilibrium entails a larger level of steady state production as compared to the former; both equilibria entail a larger level of production in steady state than the static game. We also study the effects of price stickiness and product differentiation upon the steady state equilibrium allocation and profits. The per-firm equilibrium output is increasing in both product differentiation and price stickiness, while profits are increasing in both product differentiation and the speed of price adjustment. The steady state social welfare monotonically increases in the speed of price adjustment, and the overproduction entailed by dynamic competition has beneficial effect from a social standpoint.     

Construction of the value function in a pursuit-evasion game with three pursuers and one evader

A differential pursuit-evasion game is considered with three pursuers and one evader. It is assumed that all objects (players) have simple motions and that the game takes place in a plane. The control vectors satisfy geometrical constraints and the evader has a superiority in control resources. The game time is fixed. The value functional is the distance between the evader and the nearest pursuer at the end of the game. The problem of determining the value function of the game for any possible position is solved.

Three possible cases for the relative arrangement of the players at an arbitrary time are studied: “one-after-one”, “two-after-one”, “three-after-one-in-the-middle” and “three-after-one”. For each of the relative arrangements of the players a guaranteed result function is constructed. In the first three cases the function is expressed analytically. In the fourth case a piecewise-programmed construction is presented with one switchover, on the basis of which the value of the function is determined numerically. The guaranteed result function is shown to be identical with the game value function. When the initial pursuer positions are fixed in an arbitrary manner there are four game domains depending on their relative positions. The boundary between the “three-after-one-in-the-middle” domain and the “three-after-one” domain is found numerically, and the remaining boundaries are interior Nicomedean conchoids, lines and circles. Programs are written that construct singular manifolds and the value function level lines.     

On the game of two cars

Necessary and sufficient conditions for the existence of the evasion strategy in the so-called game of two cars are given.     

Nonsmooth semipermeable barriers, Isaacs’ equation, and application to a differential game with one target and two players

We study the target problem which is a differential game where one of the players aims at reaching a target while the other player aims at avoiding this target forever. We characterize the victory domains of the players by means of geometric conditions and prove that the boundary of the victory domains is a nonsmooth semipermeable surface, i.e., is a solution (in a weak sense) of the Isaacs equation: sup _u inf _v 〈f (x, u, v),p〉 = 0, wheref is the dynamic of the system,u andv are the respective controls of the players, andp is a normal to the boundary of the victory domains at the pointx.     

Analysis of a combat problem: The turret game

The turret game is defined and solved to illustrate the nature of games of combat. This game represents a highly simplified version of air combat, yet it is sufficiently complex so as to exhibit a rich variety of combat phenomena. A review of the formulation of -combat games is included in this paper.on sabbatical leave from Technion, Israel Institute of Technology, Haifa, Israel.     

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.     

A basic searchlight game

A searchlight game is a two-person zero-sum dynamic game of the pursuit-evasion type in which at least one of the two players has a searchlight. A searchlight can be flashed a given number of times within a fixed time period and the objective is to catch the opponent in the region illuminated by the flash. Olsder and Papavassilopoulos instituted the study of these games and, in this paper, we supplement their results, obtaining a closed formula for the value and optimal strategies for the players in their basic game.     

On stochastic games with additive reward and transition structure

In this paper, we introduce a new class of two-person stochastic games with nice properties. For games in this class, the payoffs as well as the transitions in each state consist of a part which depends only on the action of the first player and a part dependent only on the action of the second player.For the zero-sum games in this class, we prove that the orderfield property holds in the infinite-horizon case and that there exist optimal pure stationary strategies for the discounted as well as the undiscounted payoff criterion. For both criteria also, finite algorithms are given to solve the game. An example shows that, for nonzero sum games in this class, there are not necessarily pure stationary equilibria. But, if such a game possesses a stationary equilibrium point, then there also exists a stationary equilibrium point which uses in each state at most two pure actions for each player.     

Parental care as a differential game: A dynamic extension of the Houston–Davies game

An interpretation of the conflict between male and female parents during the process of caring for their common offspring by means of Game Theory was given in Houston and Davies. [A.I. Houston, N.B. Davies, The evolution of cooperation and life history in the dunnock Prunella modularis, in: R.M. Sibly, R.H. Smith (Eds.), Behavioral Ecology, Blackwell Scientific Publications, 1985, pp. 471–487]. Mathematically, this model represents a static game with continuous strategy sets. Recently, this model was reconsidered in a dynamic discrete time framework which also included state dependencies [J.M. McNamara et al., A dynamic game-theoretic model of parental care, J. Theor. Biol. 205 (2000) 605–623]. In this article, we give an interpretation of the parental care conflict in continuous time by means of a differential game with state dependent strategies.     

Singular surfaces in a linear pursuit-evasion game with elliptical vectograms

In a linear pursuit-evasion game with elliptical vectograms, singular surfaces appear in the plane of symmetry containing the ellipses minor axes and, as a consequence, locally nonsmooth isocost tubes are generated. Both dispersal and attractive focal surfaces are encountered. The dispersal surface is a zone of initial conditions for trajectories leaving the plane of symmetry. Optimal trajectories attracted by the focal surface merge tangentially to the plane of symmetry and remain there until the boundary of the dispersal zone is reached. Determination of the saddle-point strategies in the focal surface leads to constructing the isocost surfaces in the entire game space.