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1983 | Buch

The Process of Applied Mathematics Kapitel lesen Erstes Kapitel lesen

Political and Related Models

herausgegeben von: Steven J. Brams, William F. Lucas, Philip D. Straffin Jr.

Verlag: Springer New York

Buchreihe : Modules in Applied Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. The Process of Applied Mathematics
Abstract
In discussing the applications of mathematics in the undergraduate curriculum, one is inclined to begin with a precise definition of what is meant by applied mathematics. There are many informative discussions of the nature of what is commonly referred to as applied mathematics, for example, [4], [10], [12], and [13], and it is not our purpose here to expand on these. Instead, we shall initially adopt a relatively informal approach, avoiding definitions and relying on comments and examples to convey ideas. It is our hope that the reader will acquire from this chapter an appreciation of the process that is exemplified in the modules that comprise this book and its three companion volumes.
Maynard Thompson
Chapter 2. Proportional Representation
Abstract
Suppose the voters of a certain school district are to elect five of their members to represent them on the local school board. Suppose also that approximately one-fifth of the voters are women and one-fifth are black males. What type of voting system should be used if we wish to give the women and the blacks the opportunity to elect at least one woman and one black (assuming there are women and blacks among the nominees)? A widely used voting method in the United States is the method of plurality voting in which each voter may cast up to a total of five votes (if five are to be elected). The voter casts one vote or no votes for a given candidate. The five candidates with the largest vote totals are declared elected.
Edward Bolger
Chapter 3. Comparison Voting
Abstract
As a preview of the subsequent analysis, imagine the following hypothetical situation. Assume that the five most recent Democratic presidents all lived at the same time (the present). All decide to seek their party’s presidential nomination in the first presidential primary in New Hampshire. You are a registered Democratic voter in New Hampshire and can vote for any candidate. If the candidate with the most votes wins in a plurality election, which candidate would you vote for? (Your judgment should be based not on the specific policies advocated by these presidents at the time they served, but instead on their competence to solve problems and their effectiveness as political leaders.)
Steven J. Brams
Chapter 4. Modeling Coalitional Values
Abstract
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.
William F. Lucas, Louis J. Billera
Chapter 5. Urban Wastewater Management Planning
Abstract
By way of a hypothetical example, we shall describe how decisions are made to expend public monies to control water pollution in a river system. Federal guidelines require that the technical, economic, financial, environmental, and socio-political aspects of the problem be evaluated. Thus, multiple and often conflicting objectives exist. Also, planning is undertaken on an area-wide or river basin level, wherein we must deal with numerous political jurisdictions. The analyst, especially if he is trained only in one discipline, has a strong tendency to define the “problem” as being one that requires primary input from his area of specialization. The hard question is to try to allocate successfully the available resources in such a way that the highest assurance of making the correct decision is obtained. The quality of the analysts’ efforts are gauged by the quality of the information for making the decisions and not on their independent value as worthwhile scientific investigations.
James P. Heaney
Chapter 6. An Everyday Approach to Matrix Operations
Abstract
From a historical point of view, it is clear that interaction between mathematics and other disciplines has been twofold. First, mathematics has been applied to other disciplines, and second, problems from other fields have helped stimulate the development of centain types of mathematics. This two-way interaction has long been evident in the case of physics, where application of mathematics to certain physical problems, such as the relationship between velocity, distance, and acceleration, in turn spurred new developments in mathematics, particularly in calculus and differential equations.
Robert M. Thrall, E. L. Perry
Chapter 7. Sources of Applications of Mathematics in Ecological and Environmental Subject Areas, Suitable for Classroom Use
Abstract
Mathematics teachers are frequently dissatisfied with their sources of appropriate, realistic examples for classroom use. This paper, which is essentially a review and annotated bibliography (with the annotations appearing in the text itself) of selected articles and books that use mathematics in an ecological or “environmental” context, is a partial remedy for that dissatisfaction, I hope. The variety of mathematics that appears in biological, ecological, environmental, and resource economics literature is amazing—mathematical areas as diverse as Boolean algebra, optimal control theory, and Fourier analysis are used.
Homer T. Hayslett Jr.
Chapter 8. How to Ask Sensitive Questions without Getting Punched in the Nose
Abstract
Do you regularly smoke pot? Have you ever had a homosexual experience? Are you a Communist? Do you cheat on your income tax? A “yes” answer to any of the preceding questions may be embarrassing to an individual. In fact, many people would become irate at being asked one of these questions and might well refuse to answer. Some would feel that such matters were personal or confidential and no one’s business but their own. However, such questions may be other people’s business. For example, consider the case in the armed forces where military officials are interested in estimating the percentage of military men using hard drugs. Based on this estimate they will decide whether or not there is a need to expand the drug rehabilitation program. The officials will need to interview the men about their use of drugs and must be equipped with methods that will elicit truthful answers to sensitive or personal questions about drug use.
John C. Maceli
Chapter 9. Measuring Power in Weighted Voting Systems
Abstract
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.
William F. Lucas
Chapter 10. To the (Minimal Winning) Victors Go the (Equally Divided) Spoils: A New Power Index for Simple n-Person Games
Abstract
Our intent in this module is to motivate the development of a new index of power in n-person simple games that models the collective decision-making process of players whose explicit behavioral objective is to effect change. Our concern is with games that closely correspond to the character of interpersonal interaction found in organizations, committees, and legislatures; situations in which power and authority, rather than monetary forms of payoff, are the motivating force (i.e., the interaction may be characterized as being “political” in nature).
John Deegan Jr., Edward W. Packel
Chapter 11. Power Indices in Politics
Abstract
“Political science, as an empirical discipline, is the study of the shaping and sharing of power” (Lasswell and Kaplan in Power and Society, 1950).
Philip D. Straffin Jr.
Chapter 12. Committee Decision Making
Abstract
This teaching module develops a normative spatial model of the committee decision-making process. Meant for use in the undergraduate classroom, it develops the requisite notions from Political Science from the beginning and uses only basic mathematics (cf. Appendix II). Three different suggestions for models are explained instead of presenting only one model. It is recommended that the students write a paper at the end of the module in which they describe, support, and criticize one of these models, or one of their own construction.
Peter Rice
Chapter 13. Stochastic Difference Equations with Sociological Applications
Abstract
For the sake of simplicity, all the variables in all of the examples given in these notes will refer to “the number of people of a certain type in a population with stationary age structure.” Variables (e.g., X5,Y0,Zt) will be written in capitals with a subscript indicating time (measured in years), and constants (e.g., a, b, υ) will be written in lower case letters.
Loren Cobb
Chapter 14. The Apportionment Problem
Abstract
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.
William F. Lucas
Metadaten
Titel
Political and Related Models
herausgegeben von
Steven J. Brams
William F. Lucas
Philip D. Straffin Jr.
Copyright-Jahr
1983
Verlag
Springer New York
Electronic ISBN
978-1-4612-5430-0
Print ISBN
978-1-4612-5432-4
DOI
https://doi.org/10.1007/978-1-4612-5430-0