Suchergebnisse

This book describes highly applicable mathematics without using calculus or limits in general. The study agrees with the opinion that the traditional calculus/analysis is not necessarily the only proper grounding for academics who wish to apply mathematics. The choice of topics is based on a desire to present those facets of mathematics which will be useful to economists and social/behavioral scientists. The volume is divided into seven chapters. Chapter I presents a brief review of the solution of systems of linear equations by the use of matrices. Chapter III introduces the theory of probability. The rest of the book deals with new developments in mathematics such as linear and dynamic programming, the theory of networks and the theory of games. These developments are generally recognized as the most important field in the `new mathematics' and they also have specific applications in the management sciences

AbstractAn inductive procedure is given for finding the nucleolus of an n‐person game in which all coalitions with less than n‐1 players are totally defeated. It is shown that, for such a game, one of three things may occur: (a) all players receive the same amount; (b) each player receives his quota, plus a certain constant (which may be positive, nerative, or zero); (c) the weakest player receives one half his quota, and the other players divide the remaining profit according to the nucleolus of a similar (n‐1)‐person game. It is also shown that the nucleolus of such a game yields directly the nucleolus of each derived game. An example is worked out in detail.

AbstractIt is shown that the Banzhaf value can be obtained by differentiating the multilinear extension of a game at the midpoint of the unit cube. This gives us a composition theorem for the value of compound games. As an example, the values of the electoral college and presidential election "games" are approximated by the method of extensions.

The Shapley value of the presidential election "game" is approximated by the method of multilinear extensions; the likely error in this approximation is computed by studying the error in the electoral college game. A comparison with the Banzhaf ratio is also given. The several biases inherent in the electoral system are analyzed from a mathematical point of view. It is shown that the system incorporates a strong bias in favor of the larger states, and a small bias in favor of the smallest states.