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Matrices and Society

Matrices and Society: Matrix Algebra and Its Applications in the Social Sciences

Ian Bradley
Ronald L. Meek
Copyright Date: 1986
Pages: 240
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  • Book Info
    Matrices and Society
    Book Description:

    Matrices offer some of the most powerful techniques in modem mathematics. In the social sciences they provide fresh insights into an astonishing variety of topics. Dominance matrices can show how power struggles in offices or committees develop; Markov chains predict how fast news or gossip will spread in a village; permutation matrices illuminate kinship structures in tribal societies. All these invaluable techniques and many more are explained clearly and simply in this wide-ranging book.

    Originally published in 1986.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5853-8
    Subjects: Sociology

Table of Contents

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  1. Front Matter
    (pp. 1-4)
  2. Table of Contents
    (pp. 5-6)
  3. Preface
    (pp. 7-12)
    Ian Bradley
  4. 1 Matrices and How to Manipulate Them
    (pp. 13-29)

    You are at home in the evening; there is nothing good on television; and you are at a loose end. There are three possibilities open to you: to go out to the pub, to go out to the theatre, or to stay at home and invite some friends round for a game of cards. In order to weigh up the comparative advantages and disadvantages of these three alternatives, you decide to put certain basic facts about each of them down on paper. And, being an orderly, methodical type you put them down in the form of a table, like this:...

  5. 2 Matrix Inversion
    (pp. 30-41)

    Inverting matrices turns out to be of considerable importance to social scientists, and in the next chapter we shall look at the important role it plays in economic planning. It is necessary, therefore, that you should be happy with the concept, be able to compute simple matrix inverses and understand how one could approach the calculation in more complicated cases.

    The inverse of a square matrixA,as we have seen,* is defined as a matrixBsuch thatAB=BA= the identity matrixI.The example we used was of

    $A = \left( {\begin{array}{*{20}{c}} 1 & 2 \\ { 2} & { 3} \\ \end{array}} \right)$,

    where the inverse turned out to...

  6. 3 The Ins and Outs of Economic Planning
    (pp. 42-60)

    Let us begin our first illustration of the use of matrices in the social sciences by recalling an example used in the first chapter. We imagined, you may remember, a very simple three-industry economy, in which what we may now call the physicalinput–outputrelationships between the industries could be represented in the following matrix:

    Here therowstell us where each industry's output goes to: of the total output of 300 units produced every year by industryA, for example, we see that 50 units are used as input in its own production process; 100 units are sold to...

  7. 4 Matrices and Matrimony in Tribal Societies
    (pp. 61-78)

    In the first chapter of this book we were introduced to a type of matrix in which the components were all either Is or Os, representing not differentquantitiesof something or other, but rather the presence (= 1) or the absence (= 0) of some particularqualityorrelationship.What we are going to do in the present chapter is to see how matrices of this kind can be used to throw light on certain kinds ofkinship systems– such as those existing for example among Australian aboriginal tribes – which lie at the centre of the studies of many...

  8. 5 Dominance in Coops and Courts
    (pp. 79-91)

    Farmer Brown keeps five hens. He calls them by pet names, but we shall call them simply A, B, C, D and E. He notices, as caring hen-keepers apparently do, that there is a strict pecking order between his five charges. In the case of each of the ten possible pairs of hens (AB, AC, AD, AE, BC, BD, BE, CD, CE and DE), one of the hens is the pecker and the other is the pecked. Farmer Brown also notes that the pecking relationship is not necessarilytransitive,in the sense that (for example) hen A may peck hen...

  9. 6 The Simple Mathematics of Markov Chains
    (pp. 92-105)

    There is a professor in one of our universities who has three pet questions – let us call them A, B and C – one of which always turns up in every examination he sets his students. He never uses the same question twice in succession. If he used question A last time, he decides on the question he will use this time by tossing a coin; if it comes up heads he uses question B; if tails, he uses question C. If he used question B last time, he decides on the question he will use this time by tossingtwo...

  10. 7 Models of Mobility
    (pp. 106-121)

    Our examples in the last chapter were rather frivolous – and designedly so, because frivolity is often a good teacher. But suppose that our subject of study were not the question which some absurd professor was going to ask in an exam paper, but the division of the population of a nation between town and country. Suppose one could work out a matrix of transition probabilities which showed the probability of a town-dweller moving to the country, and that of a country-dweller moving to the town in a particular period of time – a year, say. One ought to be able to...

  11. 8 The Mathematics of Absorbing Markov Chains
    (pp. 122-129)

    In the Markov chains we have been considering processes which never come to a stop. We reach an equilibrium in the sense that the probabilities of the process being in each state remains constant as the process goes through further stages. No process gets stuck in a particular state because there is always a definite probability that it will leave that state. To study processes that do come to an end we need to look at the mathematics of a different kind of chain – anabsorbing Markov chain.

    Let us suppose we post a letter from Leicester to an address...

  12. 9 ‘Everywhere Man Is in Chains’
    (pp. 130-148)

    Absorbing Markov chains have been and are being used to study a variety of social processes. In this chapter we consider two examples which indicate not only the possible fruitfulness of the approach but also some of the difficulties that such studies may encounter.

    In order to estimate the prison facilities a society will require in the future one needs to predict many different things. How many and what types of crime will be committed? How effective will the police be at arresting suspects and bringing them to trial? What will be the likelihood of courts finding the accused guilty,...

  13. 10 The Seven Ages of Man and Population Problems
    (pp. 149-172)

    Life itself can readily be seen as an absorbing Markov chain. Sadly not everybody gets through all of Shakespeare’s seven ages,* but all people once born embark upon the process of moving from one age to the next or to the absorbing state of death. This is a rather special and simple chain, in that unless at any time we die we move through the states (or ages) in strict chronological order.†

    Let us construct a transition matrix that gives our chances of survival through the seven ages.

    ‘Age now’ is given by the columns and ‘age next’ is the...

  14. 11 Playing Games in Theory
    (pp. 173-193)

    There are many examples of conflict between individuals, groups, classes and nations. Individuals compete for jobs; political parties attempt to win elections; trades unions want higher wages for their members, while shareholders and managers would like wage cuts; wars are only too common. The understanding of how such important conflicts are resolved is obviously a major but difficult occupation of economists, political scientists and sociologists, and it is perhaps not surprising that new ideas on how to examine these conflicts are often greeted with much the same enthusiasm as would be caused by the discovery of the philosopher’s stone. It...

  15. 12 Magic, Fishing and Farming – Some Applications of Constant-sum Games Theory
    (pp. 194-206)

    At the beginning of the last chapter it was emphasized that the theory of constant-sum games gave a prescription for how rational people should play such games, and not necessarily a description of how people actually do play them. Psychologists have undertaken laboratory experiments, usually with supposedly intelligent students, to discuss whether or not people repeatedly playing a game find the optimal strategy.* It is fair to summarize the results of the studies by saying that in small games most students found the optimal strategy if the solution of the game required the playing of a pure strategy, but that...

  16. 13 Conflict or Co-operation
    (pp. 207-230)

    In the games considered so far the players’ interests have been diametri cally opposed. We turn now to games where the conflict is not of such an extreme form but in which it is possible for players sometimes to adopt strategies that are mutually advantageous. Many of these games do not have such definite solutions as constant-sum games but the analysis of them does help to see the structure of many conflicts and hence the difficulties that may arise in attempts at resolution.

    We have seen that the solution of some games is that players should adopt the deliberately random...

  17. Epilogue and Further Reading
    (pp. 231-239)

    ‘Give a small boy a hammer and he will discover that everything in sight needs pounding.’*

    There are considerable incentives for misusing a skill that one has spent time in mastering. Thus there is a temptation, having conquered the theory of Markov chains or the theory of games, to see all social situations or processes as games or chains. We noticed, for instance, that some attempts to apply game theory have been of a contrived nature. These examples were deliberately selected to illustrate the danger of letting one’s armoury of research techniques determine the way one approaches a problem. If...