The Theory of Committees and Elections

252 Pages·1986·32.116 MB·English

by  Duncan Black M.A., Ph.D. (auth.)

The Theory of Committees and Elections
The Theory of
Committees and
Elections
by
Duncan Black, M.A., Ph.D.
Emeritus Professor of Economics
University College of North Wales
...
K1uwer Academic Publishers
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Library of Congress Cataloging in Publication Data
Black, Duncan.
The theory of committees and elections.
Bibliography: p.
Includes index.
1. Voting. 2. Social choice. I. Title.
JF100I.B49 1986 328'.2 86-22452
ISBN-I 3:978-94-0 10 -8375-1 elSBN-13:978-94-009-4225-7
DOl: 10 .1 007/978-94-009-4225-7
Copyright © 1987 by Kluwer Academic Publishers
First published in 1958 by Cambridge University Press. Reprinted in 1963, 1968 and 1971. This
reprinting made by arrangement with the author.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip
Drive, Assinippi Park, Norwell, Massachusetts 02061.
IN MEMORIAM
M.B.
CONTENTS
Preface page xi
Acknowledgements xiii
PART I
THE THEORY OF COMMITTEES
AND ELECTIONS
I. A Committee and Motions 1
II. Independent Valuation 4
III. Can a Motion be Represented by the same Symbol
on Different Schedules 1 11
IV. A Committee using a Simple Majority: Single-
peaked Preference Curves 14
V. A Committee using a Simple Majority: other
Shapes of Preference Curves 25
1. Curves either single-peaked or single-peaked with a
plateau on top
2. Other classes of curves
VI. A Committee using a Simple Majority: any Shapes
of Preference Curves, Number of Motions
Finite 35
VII. Cyclical Majorities 46
VIII. When the Ordinary Committee Procedure is in use
the Members' Scales of Valuation may be
Incomplete 51
IX. Which Candidate O'U{/ht to be Elected 1 55
X. Examination of some Methods of Election in
Single-member Constituencies 66
XI. Proportional Representation 75
[ vii ]
XII. The Decisions of a Committee using a Special
Majority page 84
1. When the members' preference curves are single
peaked
2. When the members' preference curves are subject to
no restriction
XIII. The Elasticity of Committee Decisions with an
Altering Size of Majority 99
1. When the members' preference curves are single
peaked
2. When the members' preference curves are subject to
no restriction
XIV. The Elasticity of Committee Decisions with altera-
tions in the Members' Preference Schedules 109
1. When the members' preference curves are single-
peaked
2. When the members' preference curves are subject to
no restriction
XV. The Converse Problem: the Group of Schedules to
Correspond to a Given Voting Matrix 119
XVI. A Committee using a Simple Majority: Com-
plementary Motions 125
XVII. International Agreements, Sovereignty and the
Cabinet 140
PART II
HISTORY OF THE MATHEMATICAL THEORY
OF COMMITTEES AND ELECTIONS (EXCLUDING
PROPORTIONAL REPRESENTATION)
XVIII. Borda, Condorcet and Laplace 156
1. Jean-Charles de Borda (1733-1799)
2. Marie Jean Antoine Nicolas Caritat, Marquis de
Condorcet (1743-1794)
3. Pierre-Simon, Marquis de Laplace (1749-1827)
4. Conclusions
XIX. E. J. Nanson and Francis Galton 185
[ viii ]
xx. The Circumstances in which Rev. C. L. Dodgson
(Lewis Carroll) wrote his Three Pamphlets page 189
Appendix. Text of Dodgson's Three Pamphlets and of
'The Cyclostyled Sheet' 214
A Discussion of the Various Methods of Procedure in Conducting
Elections (1873)
Suggestions as to the Best Method of Taking Votes, Where More
than Two Issues are to be Voted on (1874)
A Method of Taking Votes on More than Two Issues (1876)
'The CycloBtyled Sheet' (7 Dec. 1877)
Notes on Dodgson's Third Pamphlet' A Method ... ' (1876) 234
Index 239
[ix ]
PREFACE
THIS book or some related work has occupied me spasmodically
over rather a long period, in fact ever since I listened to the class
lectures of Professor A. K. White on the possibility of forming a
pure science of Politics. Mter an earlier version of Part I had
failed to obtain publication in 1947, some chapters appeared as
articles, and I am obliged to the editors of the journals mentioned
below for permission to reprint this material, sometimes in a
modified form.
When I first attempted publication I was unacquainted with the
earlier history of the theory, and, indeed, did not even know that
it had a history; and the later additions to the book have largely
been by way of writing the present Part II. This historical section
does not include the important recent work, Social Ohoice and
Individual Values (1951), of Professor Kenneth J. Arrow; but it
does include all the mathematical work on committees and
elections appearing before the middle of this century which has
come to my notice, although the last item in it is dated 1907. No
doubt there is much important material which I have failed to see.
The theorizing of the book grew out of a reading of the English
political philosophers and of the Italian writers on Public Finance.
At a very early stage I was helped to find the general lines of
development by discussion with my colleague Professor Ronald H.
Coase on his view of the nature of the firm; and later talks with
Dr R. A. Newing did much to clear my ideas on some ofthe mathe
matical issues. Dr Newing suggested to me the use of a matrix
notation which, some years afterwards, I discovered had already
been employed by Rev. C. L. Dodgson (Lewis Carroll).
Dodgson's pamphlet A Di8cussion of the Various Methods of
Procedure in Oonducting Elections is reproduced by courtesy of the
Library of the University of Princeton in which is the only known
copy, and his pamphlet A Method of Taking Votes on More Than
Two I88ue8 and 'The Cyclostyled Sheet' by courtesy of the
Bodleian Library, Oxford. At Dodgson's old college of Christ
Church, Dr T. B. Heaton, Professor J. O. Urmson and Mr Geoffrey
Bill were helpful in a number of ways; and I have been permitted
[ xi]
to quote from the minutes of the Governing Body of the college
a.nd also from the 'Ve re Bayne Memora.ndum Book' which is the
property of the Senior Common Room. It was a pleasure to
correspond with Miss F. Menella Dodgson who has enabled me to
quote two entries from the diaries of Lewis Carroll, which, owing
to their specialized interest, had been excluded from the published
version.
The Academie des Sciences permitted access to its minutes in
connexion with the work of Borda a.nd Condorcet.
The University of Wales has added to my already great
obligations to it by a generous grant towards publication, made
by its Press Board.
D.B.
BA.NGOR,
August 1956
[ xii ]
ACKNOWLEDGEMENTS
I am indebted to the editors for permission to incorporate, wholly
or in part, material which has already appeared as articles in the
following journals.
'On the Rationale of Group Decision-Making', Journal of Political
Economy, February 1948.
'Un approccio aUa teoria delle decisioni di comitato', Giornale degli
Economi8ti, May-June 1948.
'The Decisions of a Committee Using a Special Majority', Econo
m.etrica, July 1948.
'The Elasticity of Committee Decisions with an Altering Size of
Majority', Econometrica, July 1948.
'The Elasticity of Committee Decisions with Alterations in the
Members' Preference Schedules', South African Journal of EconomiCB,
March 1949.
'The Theory of Elections in Single-Member Constituencies', Oanadian
Journal of EconomiCB and Political Science, May 1949.
'Some Theoretical Schemes of Proportional Representation' ,
Oanadian Journal of EconomiCB and Political Science, August 1949.
[ xiii ]