William F. Lucas

Measuring Power in Weighted Voting Systems

There are a large number of voting situations in which some individuals or blocs of voters effectively cast more ballots than others. Such weighted voting systems are found in governmental bodies such as the U.S. Congress, some state legislatures and county boards, in the Electoral College, in voting by stockholders in a corporation, in several university senates, in many other multimember electoral districts in which several representatives are elected at-large from a single district, as well as when strictly disciplined political parties vote as a single bloc.

Applied Game Theory

This paper is concerned with the formalization of the notion of power in the context of two person negotiations. The notion of power is defined as follows: the amount of power of A over B is related to the level of achievement of Bs objectives in the interaction A-B. As for the origin of power, it is taken as related to the strategies available to both parties in their interaction huch as rewards, punishments, etc.). The development of these ideas relies mainly on the concepts of game theory but specific factors drawn from the psychological and psychosociologicalliterature, such as the notion of representation, arc explicitely introduced in the model. Besides the formalization, the paper includes a detailed discussion of two examples enhancing the role of representation in power analysis. An axiomatic treatment of the model is briefly reported in the appendix.

Political and related models

The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modern mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, they should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and suggestions for the teacher are provided. Several of these chapters have been tested in a variety of classroom settings, and all have undergone extensive peer review and revision. Illustrations and exercises are included in most chapters. Some units can be covered in one class, whereas others provide sufficient material for a few weeks of class time. Volume 1 contains 23 chapters and deals with differential equations and, in the last four chapters, problems leading to partial differential equations. Applications are taken from medicine, biology, traffic systems and several other fields. The 14 chapters in Volume 2 are devoted mostly to problems arising in political science, but they also address questions appearing in sociology and ecology. Topics covered include voting systems, weighted voting, proportional representation, coalitional values, and committees. The 14 chapters in Volume 3 emphasize discrete mathematical methods such as those which arise in graph theory, combinatorics, and networks.

The Apportionment Problem

One of the first equity problems that arises in the distribution of resources is the apportionment problem. It is concerned with distributing available personnel or other resources in “integral parts” to different subdivisions or tasks. One may be distributing seats in a legislature among different political constituencies, allocating the number of available teachers for a high school or college to the different departments, or determining the number of ships to be assigned to the different fleets in the Navy given certain priorities and goals. In practice this problem frequently arises even before one considers the classical assignment problem that is concerned with the existence of feasible and efficient assignments of resources to various units, such as assigning particular individuals to certain jobs. Several different methods for solving the apportionment problem as well as many of the relevant properties of the various methods will be presented in this chapter, along with examples that indicate a few of the more obvious applications.

Growth and New Intuitions: Can We Meet the Challenge?

The mathematical sciences have changed significantly during the past few decades. The most obvious change is the enormous growth of mathematics. However, the most exciting and potentially beneficial movement may well be the extensive mathematization of many traditional as well as newly emerging disciplines. The consequent influx of rich new ideas and alternative intuitional sources can greatly rejuvenate and invigorate mathematics itself. It would thus appear that mathematics should, for some time into the future, exhibit a truly great scientific advance that would bring reasonable prosperity to both its individual practitioners and its supporting institutions. Nevertheless, there is ample evidence to indicate that these possibilities are not being realized, and the prospects for the future are much less encouraging. The most obvious illustration of this is the gross mismatch between the mathematics that students are currently being taught and the skills that are marketable to most current users of the subject.

The Existence Problem for Solutions

In 1944 John von Neumann and Oskar Morgenstern [18] presented an extensive development of a theory of n-person cooperative games in characteristic function form. The most basic and most challenging theoretical question regarding this theory concerns whether their solution sets exist. It has been known [5] since 1967 that such solutions need not exist in the most general case of all games. However, all known counterexamples are of a rather specialized nature, and there still remains large classes of games for which this question is unanswered. So this fundamental problem continues to be among the most important and intriguing problems in cooperative game theory, as well as one that appears most difficult to solve. The answer should prove of interest in both theory and applications.

Modeling Coalitional Values

The idea of a set of elements along with elementary notions about subsets are fundamental concepts in modern mathematics and are well-known to contemporary mathematics students. These elementary concepts, together with some method for assigning numbers to various subsets of a given set, are often sufficient to begin applying the techniques of mathematical modeling to a good number of interesting and nontrivial applications. Many important situations are characterized to a large extent by describing the set of participants involved and the values achievable by certain subsets of these participants. Such applications occur in economics and politics, business and operations research, the social and environmental sciences, and elsewhere.

Fair Division: From Cake-Cutting to Dispute Resolution.

Introduction 1. Proportionality for n=2 2. Proportionality for n>2: the divisible case 3. Proportionality for n>2: the indivisible case 4. Envy-freeness and equitability for n=2 5. Applications for the point-allocation procedures 6. Envy-free procedures for n=3 and n=4 7. Envy-free procedures for arbitrary n 8. Divide-the-dollar 9. Fair division by auctions 10. Fair divisions by elections 11. Conclusions Glossary Bibliography Index.

Problem Solving and Modeling in the First Two Years

Mathematical education is currently confronted with a variety of substantial problems. This paper discusses three of these problems which are related to the role of finite mathematics and its appropriate place in the undergraduate curriculum. First, there is the pressing problem of how to incorporate important new mathematical subjects and discoveries into an already crowded curriculum, including discrete topics at the elementary level. Second, there is the recently emerging problem that the mathematical community, except for computer science, is very rapidly losing its base of talented young people. Third, there is the question of how to design courses which give due attention to the revolution and needs in discrete mathematics and are worthy enough to compete with traditional calculus courses. It is important in addressing this first problem of the “curriculum crunch” to not overlook the second problem of “diminishing base” which could be devastating to the health of the general mathematical sciences if allowed to continue for long. It is also not clear to many that we have yet come up with the appropriate finite mathematics courses to take a major slot in the freshman-sophomore years of college.