Showing papers on "Game tree published in 1972"
TL;DR: In this article, a third-order pursuit-evasion game with both pursuers and evaders having the same speed and minimum turn radius is described. And the game of kind is first solved for the barrier or envelope of capturable states, and then solved for optimal controls of the two pursuers as functions of the relative position.
Abstract: This paper describes a third-order pursuit—evasion game in which both players have the same speed and minimum turn radius. The game of kind is first solved for thebarrier or envelope of capturable states. When capture is possible, the game of degree is then solved for the optimal controls of the two players as functions of the relative position. The solution is found to include a universal surface for the pursuer and a dispersal surface for the evader.
122 citations
TL;DR: In this paper, the minimax solution for a game in which player I chooses a real number and player II seeks it by choosing a trajectory represented by a positive function was found.
Abstract: The minimax solution is found for a game in which player I chooses a real number and player II seeks it by choosing a trajectory represented by a positive function.
42 citations
01 Jan 1972
TL;DR: The counting vector of a simple game is the vector f = (f(1),f(2), *, f (n)) wheref(i) is the number of winning coalitions containing the player i as discussed by the authors.
Abstract: The counting vector of a simple game is the vector f = (f(1),f(2), *, f (n)) wheref(i) is the number of winning coalitions containing the player i. In this paper, we show that the counting vector of a weighted majority game determines the game uniquely. With the aid of the counting vector we find an upper bound on the number of weighted majority games. 1. Preliminaries on simple games. A simple game is a pair G= (N; W), where N = {1, 2, * * *, n} is a set of n members and W is a set of subsets of N. The members of N are called players; subsets of N are called coalitions. The elements of W are called winning coalitions. A simple game is called monotone if every superset of a winning coalition is itself a winning coalition. A weighted majority game is a simple game for which there exist n nonnegative numbers w., w2, , w. and a positive number q, such that, S is a winning coalition if and only if w(S) = i,eS w, > q. w = [wl, w2, * * *, w,; q] is called the representation of the game. A weighted majority game is denoted by G = (N; w) where w is its representation. G is called constant-sum if for each coalition S exactly one of the two coalitions S and N S is winning. 2. The counting vector theorem. Given a simple game G = (N; W), we denote by f (i) the number of winning coalitions containing player i, and by f the vector f = (f(l), f (2),*** f (n)) called the counting vector of the game. THEOREM 2.1. Let G= (N; [W1), W(1) .. WM; q(l)]) and G = (N; [w(2), W(2) ,.. w(2); q(2)]) Received by the editors November 23. 1970. AMS 1970 subject classifications. Primary 90D12.
16 citations