Approximate solution of singularly perturbed nonlinear pursuit-evasion games

Several errors in Ref. 1 are corrected.     

A class of trilinear differential games

This paper characterizes a class ofN-person, general sum differential games for which the optimal strategies only depend upon remaining playing time. Such strategies can be easily characterized and determined, and the optimal play can be easily analyzed.We acknowledge the helpful comments of G. Leitmann and an anonymous referee.     

On the axiomatization of the τ-value

Summary The τ-value is a solution concept for a subclass of games with transferable utility introduced and axiomatized by Tijs (1981, 1987). In this note an alternative characterization of the τ-value by means of five axioms is offered. Two of them are well-known: efficiency and translation equivalence; the other three relate the solution of a game with the minimal and maximal aspiration vectors involved in the definition of the τ-value. This work was partially supported by Universidad del País Vasco (projects UPV 036.321-H061/91 and UPV 036.321-HA186/92).     

On a linear pursuit game with an unknown trap

A linear pursuit game with a trap, the location of which is unknown to the evader, is defined and investigated. The cases in which one of the players has complete energy dominance over his adversary are solved completely. In the general case, when no player dominates, the solution is indicated for the two-stage game.This research was supported in part by the Technion Fund for promotion of research.     

Analysis of the N-Card Version of the Game Le Her

We present a complete solution to a card game with historical origins. Our analysis exploits the convexity properties in the payoff matrix, allowing this discrete game to be resolved by continuous methods.     

Repeated proximity games

We consider repeated games with complete information and imperfect monitoring, where each player is assigned a fixed subset of players and only observes the moves chosen by the players in this subset. This structure is naturally represented by a directed graph. We prove that a generalized folk theorem holds for any payoff function if and only if the graph is 2-connected, and then extend this result to the context of finitely repeated games. Received June 1997/Revised version March 1998     

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.     

Simple-motion pursuit-evasion differential games,part 1: Stroboscopic strategies in collision-course guidance and proportional navigation

We consider pursuit-evasion differential games in the plane in which the players, i.e., the pursuer and the evader, have simple motion and are pedestrians à la Isaacs. Two information patterns are considered, namely the classical feedback strategy and the stroboscopic pursuit strategy; loosely speaking, the latter incorporates the instantaneous control employed by the evader, which we assume to be known to the pursuer. Within this framework, the question of modelling a pursuit-evasion encounter is addressed, and we examine three well-known guidance schemes of the line-of-sight, collision course, and proportional navigation types.     

Asymmetric probabilistic prospects of Stackelberg players

Alkanet al. (Ref. 1) consider the family of all bimatrix games with ordinal payoffs and conclude that the average leader and follower enjoy symmetric prospects under the Stackelberg solution concept. In contrast, economics lore stresses the asymmetry between leader and follower, the leader generally enjoying the more favored position. We replace the computational analysis of Ref. 1 by a simple probabilistic combinatorial argument. We then impose monotonicity conditions on the player preferences. With this regularity condition, the symmetry between leader and follower breaks down, and most of the resultant advantage accrues to the leader. Thus, the monotonicity largely restores the advantage ascribed by economics folklore to the leader. Our analysis extends to nonordinal payoff matrices.This work was supported by National Science Foundation Grant No. ECS-84-51032.The author thanks A. Alkan for showing him the interesting paradox. He thanks B. Foley, S. Hackman, and K. Calvin for helpful comments. The referee's remarks and suggestions, which improved the paper, are also appreciated.     

On equilibrium points of diagonaln-person games

In this paper, we compute explicitly the equilibrium points of diagonaln-person games when all of them have the same number of strategies. This number is arbitrary. A wide generalization of two-person games is immediately obtained.The author is grateful to Professor Joel Cohen who visited IMASL during the winter of 1987 and commented on the paper.