NUMBERS RULE TThhee VVeexxiinngg MMaatthheemmaattiiccss ooff DDeemmooccrraaccyy,, ffrroomm PPllaattoo ttoo tthhee PPrreesseenntt George G. Szpiro PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 0000--SSzzppiirroo__ffmm ii--002211.iinndddd iiiiii 1111//1122//22000099 88::0033::1144 AAMM.
Copyright 2010 © by George G. Szpiro Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Szpiro, George, 1950– Numbers rule : the vexing mathematics of democracy, from Plato to the present George G. Szpiro / George G. Szpiro.
CONTENTS PREFACE .ix ONE THE ANTI-DEMOCRAT .1 TWO THE LETTER WRITER .22 THREE THE MYSTIC .33 FOUR THE CARDINAL .50 FIVE THE OFFICER .60 SIX THE MARQUIS .73 SEVEN THE MATHEMATICIAN .89 EIGHT THE OXFORD DON .100 NINE THE FOUNDING FATHERS .119 TEN THE IVY LEAGUERS .134 ELEVEN THE PESSIMISTS .165 TWELVE THE QUOTARIANS .187 THIRTEEN THE POSTMODERNS .202 BIBLIOGRAPHY .215 INDEX .219 0000--SSzzppiirroo__ffmm ii--002211.iinndddd vviiii 1111//1122//22000099 88::0033::1144 AAMM.
PREFACE I t may come as a surprise to many readers that our democratic institu- tions and the instruments to implement the will of the people are by no means foolproof. In fact, they may have strange consequences. One ex- ample is the so-called Condorcet Paradox. Named after the eighteenth- century French nobleman Jean-Marie Marquis de Condorcet, it refers to the surprising fact that majority voting, dear to us since times immemo- rial, can lead to seemingly paradoxical behavior. I do not want to let the cat out of the bag just yet by giving away what this paradox is. Sufﬁ ce it to say for now that this conundrum has kept mathematicians, statisticians, political scientists, and economists busy for two centuries—to no avail. Worse, toward the middle of the twentieth century, the Nobel Prize win- ner Kenneth Arrow proved mathematically that paradoxes are unavoid- able and that every voting mechanism, except one, has inconsistencies. As if that were not enough, a few years later, Allan Gibbard and Mark Sat- terthwaite showed that every voting mechanism, except one, can be ma- nipulated. Unfortunately, the only method of government that avoids par- adoxes, inconsistencies, and manipulations is a dictatorship.
CHAPTER ONE THE ANTI-DEMOCRAT P lato, son of Ariston and Perictione, has been called the greatest of Greek philosophers by his admirers and chastised as the worst anti-democrat by his detractors. Socrates’ most brilliant student, Plato devoted his life to studying and teaching, to exploring the meaning of life, inquiring into the nature of justice, and pondering how to be a better person.
CHAPTER TWO THE LETTER WRITER L ike the administrators in the ancient Greek cities, the civil servants of the Roman Empire were concerned with governing well and dispensing justice. The magistrate and state ofﬁ cial Gaius Plinius Caecilius Secun- dus, generally known as Pliny the Younger, raised a profound question about the proper way to vote on a particular issue. Born in AD 61 or 62 in what is now the Italian city of Como, Pliny had lost his father, a land- owner, while still a child and was brought up by his mother. The main in- ﬂ uence on his education was his mother’s brother, Pliny the Elder, a Roman naval commander and avid natural philosopher.
CHAPTER THREE THE MYSTIC I n the Athenian Assembly the choices to be made were usually of the form yes/no, for/against, guilty/innocent. The decision-making process usually worked all right, because deciding between two options presented no special problems; a simple majority vote does the trick. And whenever an ofﬁ cial had to be chosen from among more than two candidates, the selection was usually deferred to chance, or god, by drawing lots. With time it became apparent, however, that choices between more than two alternatives were inevitable, and electors were often hard put to agree on a winner. Reluctant to let the lot—or fortune, or god—decide on impor- tant issues, many institutions drew up their own rules as they went along. By and by, special methods were instituted to elect emperors, popes, and the doges of Venice. But house rules did not always meet with everybody’s approval. Pliny may have been the ﬁ rst to organize, and to manipulate, a three-way decision. It was only the earliest-known instance of problem- atic decision-making procedures, innumerably more followed. For exam- ple, thirteen centuries after Pliny, during the Papal Schism that lasted from 1378 until 1417, two, and at a certain time three, popes reigned over their ﬂ ock. The need for more sophisticated electoral schemes became apparent during the Middle Ages.
CHAPTER FOUR THE CARDINAL A fter Llull’s foray, no further progress was made in the theory of voting and elections for over a century. Then, in 1428, the German student Niko- laus Cusanus happened upon one of Llull’s documents in Paris. He found the text sufﬁ ciently interesting to make a transcript and take it home. But he did more than reproduce the text for his own use. A few years later he improved upon Llull’s method, thereby establishing himself as the second pioneer of the modern theory of elections.
CHAPTER FIVE THE OFFICER T he eighteenth century was a period of enlightenment throughout the Old and New World. France, the United States, and Poland granted them- selves constitutions. Nations were in upheaval as their citizens started de- manding equal justice for all, showing concern for human rights, and call- ing for a regulation of the social order. At the same time, demands for quality government arose and the question of how ofﬁ cials were to be elected to high positions became important again. In this atmosphere two eminent French thinkers appeared on the scene. One was a military ofﬁ - cer with numerous distinctions in land and sea battles. His name was Che- valier Jean-Charles de Borda. The other was the nobleman Marquis de Condorcet. The two men, outstanding scientists in Paris during the time of the French Revolution, did something amazing: they reinvented the election methods that Llull and Cusanus had proposed a few hundred years earlier. Actually, they did more than that: they provided the appro- priate mathematical underpinnings. At odds with each other on many sub- jects, they also engaged in a lively debate on the theory of voting and elections.