A MATHEMATICAL APPROACH TO PROPORTIONAL REPRESENTATION: Duncan Black on Lewis Carroll A MATHEMATICAL APPROACH TO PROPORTIONAL REPRESENTATION: Duncan Black on Lewis Carroll edited by lain McLean (Official Fellow in Politics, Nuffield College, Oxford) Alistair McMillan (Research Officer, Nuffield College, Oxford) and Burt L. Monroe (Assistant Professor of Political Science, Indiana University) l1li... Springer Science+Business" Media, LLC Library of Congress Cataloging-in-Publication Data A C.P.1. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-3735-8 ISBN 978-94-007-0824-2 (eBook) DOI 10.1007/978-94-007-0824-2 Copyright @ 1996 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1996 Softcover reprint of the hardcover 1st edition 1996 AlI 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, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, NorwelI, Massachusetts 02061. Printed an acid-free paper. v Contents Editors' Preface vii Introduction IX l. What is the Theory of Voting? ix 2. Duncan Black and the Study of Theoretical Politics xiii 3. The Duncan Black Archive at the University of Glasgow xiv 4. Black's Struggles for Recognition xvi 5. Black and Lewis Carroll XIX 6. Carroll's Principles ofParliamentary Representation xxi 7. The Making of This Book xxviii References for the Introduction xxxi Part 1. The Life and Logic of Lewis Carroll 1 1.1. Government by Logic 12 Part 2. The Principles of Parliamentary Representation 45 2.1. An Outline of Carroll's Argument. 45 2.2. The Central Argument in The Principles ofParliamentary Representation 51 2.3. Carroll and the Cambridge Mathematical School ofPR: Arthur Cohen and Edith Denman 63 Part 3. An Analysis of Carroll's Argument 91 3.1. Carroll's Scheme of Proportional Representation 91 3.2. The Desiderata: In at the Deep End 93 3.3 . The Droop Quota in a Two-Party System 98 3.4. The Representation of the Droop Quota 105 3.5. Walter Baily and the Number of Voters Unrepresented 112 3.6. J G Marshall and the Two-Person Zero-Sum Game 114 3.7. Demand Curves, Maximin, and the d'Hondt Scheme ofPR 120 3.8. The Fourfold Table and Carroll's Quota, Q(s) 129 3.9. Carroll's Practical Scheme and the Single Transferable Vote 139 3.10. Allocating Members to Districts 144 vi Part 4. Reprints of Original Material 151 4.1. Lewis Carroll: The Principles of Parliamentary Representation, with Supplement and Postscript to Supplement 151 4.2. James Garth Marshall: Minorities and Majorities; Their Relative Rights 175 4.3. Walter Baily: Proportional Representation in Large Constituencies 180 Index 185 vii Editors' preface This book has been thirty years in the making. As we explain in our Introduction, the main part of it is the posthumous work of the Scottish economist Duncan Black (1908-91). Black was an original and neglected thinker who had unusual difficulty in getting his work published. He was also a perfectionist. He told his friends that this book was ready to publish in the late 1960s, but he was never satisfied and continued to make additions and changes throughout the rest of his life. The small circle of people who knew the importance and originality of this work waited eagerly but in vain. Black wanted Kluwer Academic Publishers, who had published the facsimile reprint of his classic The Theory of Committees and Elections (1958; Kluwer reprint 1987) to publish this book; but it was far from being in a fit state to publish at his death. Black was a widower with no immediate family, and at his death his books would have been sold and his papers dispersed had it not been for the intervention of Richard Alexander, an economist at the Royal Naval College, Dartmouth, Devon, and a neighbour of the agent charged with disposing of Black's real estate after his death. On fmding that his house contained a substantial quantity of academic books and papers, Mr Alexander alerted Black's almae matres the University College of North Wales, Bangor (which declined them) and the University of Glasgow (which accepted them for its Archives). One of us (1M) visited Black's house with Zachary Rolnik of Kluwer. Amidst four roomfuls of miscellaneous papers, one of the fITst things we found was a chapter plan for this book. This helped us to decide that it was feasible to carry out Black's wishes. (Later, we discovered many other chapter plans, all different). The papers were hurriedly removed from Black's house and taken to Glasgow. Work on cataloguing them, with the help of a grant from the Economic and Social Research Council, began in 1993 and was completed in 1994. A grant from the Newlands fund of the University of Glasgow enabled BLM to join AM in Glasgow during the cataloguing. We were also assisted by Gordon Tullock (University of Arizona) and Gordon Brady (Sweet Briar College), who both visited Glasgow during the cataloguing process. The University of Warwick and Nuffield College, Oxford, successively administered viii the grants. Michael Moss and his staff at the University Archives in Glasgow were extremely generous and helpful hosts to the project. Duncan Black was elusive and self-effacing. We are very grateful to Nick Baigent, Alec Caimcross, Ronald Coase, and Donald Stokes for their help in enabling us to learn more about him as a person. For expert advice on Lewis Carroll we wish to thank Morton N. Cohen. The Bodleian Library and the library of Christ Church, Oxford, contain some of our primary sources and we are grateful for their help. Parts of the Introduction have appeared in the Journal of Theoretical Politics, 7(2), 1995. We are grateful to the publishers for permission to reproduce that material here. To all those who have helped rescue Duncan Black's papers, we give our heartfelt thanks. Oxford Bloomington, Indiana April 1995 Introduction ix Introduction 1. What is the Theory of Voting? This book is about a well-known writer, Lewis Carroll, and about a little-known subject, the theory of voting. It has been edited from the manuscripts of a writer who is still not as well known as he should be, Duncan Black. There is no need to explain who Lewis Carroll was, but every need to explain who Duncan Black was, and what the theory of voting is, as it is a generally misunderstood subject. It is best to begin by explaining what the theory of voting is not. It is not about the content of voting, nor yet about whether (and if so, how) a rational voter would ever vote. It is purely about voting procedures. A voting procedure has two main components. First, it prescribes how each voter should cast her vote: perhaps for just one candidate and perhaps for more than one; perhaps by ranking the candidates from best to worst; perhaps by dividing them into the two categories 'acceptable' and 'unacceptable'; perhaps for individual candidates, perhaps for a party list. Secondly, it prescribes how the individual votes are added up ('aggregated') to determine the group's choice, or the group's ordering. This may sound both routine and as dry as dust. It turns out that it is far from routine, and as a consequence that it is not at all dry. There are a number of introductions to the theory of voting. Barry and Hardin (1982, Part II), Mueller (1989) and Saari (1994) are introductions to the concepts. McLean and Urken (1995) is an introduction to the history, with texts. Duncan Black's own The Theory of Committees and Elections (Black 1958) is an excellent introduction to both topics, up to the date when he wrote. Here, we will present only enough of the theory as is needed to provide a context for this book. The theory of voting has two main subdivisions: the theory ofmajority rule, and the theory ofproportional representation. 'Majority rule' has a clear meaning when only two candidates, or issues, are in contention. If one option wins more than half of the votes cast, it is unambiguously the majority's choice. Simple majority rule satisfies some classically desirable properties of fairness; furthermore, it is the only voting procedure that does so (May, 1952). But as soon as there are more than two candidates, simple majority voting may perform very badly. For instance, a x A Mathematical Approach to Proportional Representation candidate may be the plurality winner (that is, get the largest number of votes of any of the candidate, but fewer than half of the votes cast), while being the absolute majority loser (that is, ranked last of all by more than half of the voters). An obvious way to avoid this problem is to conduct exhaustive pairwise voting. Each candidate is compared with each other. If the voters are asked to rank the candidates, this may be done at the second stage in the procedure, the aggregation stage. There is no need for the voters actually to vote on each pair. Exhaustive pairwise voting was first proposed by Ramon Lull in 1283. If it yields a clear winner, that is a candidate who has won a majority against every other, taken one at a time, that candidate has an obvious claim to be considered the best. Such a candidate is called the Condorcet winner after the Marquis de Condorcet (1785), the greatest figure in voting theory before this century.