voting paradoxes and how to deal with themby hannu nurmi

Voting paradoxes and how to deal with them by Hannu NurmiReview by: Steven J. BramsSocial Choice and Welfare, Vol. 18, No. 4 (2001), pp. 835-838Published by: SpringerStable URL: http://www.jstor.org/stable/41106430 .Accessed: 15/06/2014 11:59

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Springer is collaborating with JSTOR to digitize, preserve and extend access to Social Choice and Welfare.

http://www.jstor.org

This content downloaded from 188.72.127.150 on Sun, 15 Jun 2014 11:59:49 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=springerhttp://www.jstor.org/stable/41106430?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

Soc Choice Welfare (2001) 18: 835-838 Z

7~T7Z Social Choice

Welfare Springer- Verlag 2001

Book review

Hannu Nurmi: Voting paradoxes and how to deal with them. Springer-Verlag, Berlin, Heidelberg New York 1999. X, 153 p, 54,65 ISBN 3-540-66236-7

Hannu Nurmi has undertaken the daunting task of not only collecting voting and social-choice paradoxes that have been discovered over the past two hundred years but also categorizing and interrelating them. He has also taken the further step of indicating how some might be ameliorated, if not resolved.

Nurmi analyzes almost twenty paradoxes, including some from outside social choice theory (e.g., that originated in statistics) that, nonetheless, are pertinent to issues of voting or representation. I think his list is well chosen and nearly exhaustive. Starting with the well-known paradoxes of Borda and Condorcet, discovered in the late 18th century, Nurmi discusses paradoxes formulated in the 19th and early 20th centuries (the Alabama and Ostrogorski paradoxes, respectively). Most of the paradoxes Nurmi assembles, however, were rigorously investigated only in the second half of the 20th century, including several found and analyzed in the 1990s.

What makes Nurmi's book much more than just a laundry list are the the- oretical concepts and tools he introduces to probe the paradoxes and draw out their problematic features for democratic choice. Some tools will be new to many readers, such as Saari's (1994) innovative geometric representations of voters and outcomes in three-candidate elections.

Nurmi's book is largely self-contained, so newcomers to the field and students with a modest background in logic or other analytical methods should be able to read it without much difficulty. While Nurmi proves, or outlines proofs, of several theorems, his book focuses mostly on examples (in 64 tables!) that enable one to obtain a good understanding of the paradoxes from studying these tables.

All these examples, however, create their own difficulties in distinguishing the forest from the trees. Relentlessly, Nurmi shows connections between the paradoxes, classifying them into four categories in the end (incompatibility, monotonicity, choice set variance, and representation). Although these are reasonable categories, I thought some of the chapter titles, such as "compound majority paradoxes," were more descriptive. But classifying paradoxes, especially where there is overlap in the categories (which Nurmi fully recog- nizes), is certainly bound to be controversial.

Even more controversial is what constitute voting paradoxes. Nurmi says they "contain something counterintuitive or self-contradictory" (p. 2). In fact, most paradoxes that Nurmi has selected fit this rather loose definition - if not


http://www.jstor.org/page/info/about/policies/terms.jsp

836 Book review

now, then when they were first discovered. But some, in my opinion, do not fit, or seem contrived.

Take the referendum paradox, which is one of the compound majority paradoxes. This paradox occurs when a parliament reverses the majority decision of a "consultative referendum," because a majority of parliamentary members have majorities in their districts that oppose the majority outcome of the referendum. For example, if there are three equal-size districts of 100 voters each, and there are 51-49 "no" majorities in two of the districts, it would take only a 53-47 "yes" majority in the third district to make the entire electorate favor yes over no by 151 (49 + 49 + 53) to 149 (51 + 51 + 47) votes.

Nurmi admits that "there is not anything paradoxical in this this" (p. 76), but he then proceeds to discuss extreme examples of this paradox in which large majorities of the electorate can be reversed by a parliament. Such reversals, however, require districts that vote very differently from one another. The relevant empirical question is how often, in reality, do big differences like this occur.

To Nurmi's credit, he discusses the underlying issue that such a discrepancy raises - namely, when should voters place their trust in the undifferentiated electorate, speaking with a single voice, and when should they rely on representatives, presumably reflecting the views of their constituents? Amalgamating the votes of representatives may produce outcomes diametrically opposed to those that majorities of the electorate, as a whole, support, which has generated a good deal of debate about what is the proper forum for making social choices.

Nurmi analyzes paradoxes that are decidedly less obvious than the referendum paradox. For example, the paradox of multiple elections occurs when the combination of propositions that wins in a referendum may, when there are as few as three propositions, be supported by absolutely no voters. This paradox actually occurred in the 1990 California general election, in which there were 28 propositions on the ballot and none of the 1.8 million voters in Los Angeles county supported the winning side on all 28 (Brams et al. 1997, 1998).

To mention a voting paradox of a very different character, the no-show paradox occurs when some voters, by not showing up at the polls and voting for their preferred candidates, actually benefit these candidates more than if they had voted. Because Nurmi demonstrates that Schwartz's paradox, which seems to me somewhat contrived, is essentially equivalent to the no-show paradox, I question whether it should be distinguished as a new paradox.

To stem the proliferation of voting paradoxes, I would propose the fol- lowing reality test for their admission into the pantheon: that one can find an empirical instance in which the paradox actually occurred. This is not to deny the importance of pure thought experiments, with no grounding in reality, in which some remarkable contradiction appears. Rather, I worry that Nurmi's 64 tables could become 128 in the next edition of his book, completely over- whelming the reader, if not Nurmi himself!

We continually need to simplify and generalize these paradoxes, which is of course what social choice theory, beginning with Arrow's theorem (para-



Book review 837

dox?), is all about. At the same time, I would not want to expunge from the literature the simple, and sometimes striking, amomalies that Nurmi analyzes in favor of only axiomatic treatments. The crux of democracy is fair and democratic elections, and it would be a mistake not to make voting proce- dures, and the problems they give rise to, the starting point for the study and teaching of social choice.

Nurmi's catalogue, while quite comprehensive, is not complete. Hillinger's paradox (Hillinger 1971; see also Miller 1975), which recently was shown to have occurred in Chilean elections (Chakravarty and Hojman 1999), is not included; neither are van Deemen's (1993) paradoxes of proportional- representation list systems, one of which may have occurred in the Nether- lands in 1989. Brams and Fishbura (1984) showed that under additional- member systems, increasing the representation of underrepresented parties may create an incentive for these parties deliberately to lose district elections in order to increase their representation. They illustrated this kind of non- monotonicity with data from the 1983 British general election, under the supposition that the election had been conducted under an additional-member system.

These omissions are minor ones in an already full catalogue. Some ques- tionable paradoxes notwithstanding, Nurmi's survey is the first I know that brings together so many paradoxes and also treats their normative implica- tions for the design of better electoral and representation systems.

This is not to say that I am as enamored, as Nurmi seems to be in his final chapter, of probabilistic voting as a solution to some of the paradoxes. I think it would be totally unacceptable to most voters for election winners, especially if they are Condorcet candidates (as most are), to be dumped by a probabilistic mechanism that decides, despite the odds, that they are not its choice.

For me, approval voting is a simple and practicable solution to several serious problems of single-winner, multiple-candidate elections, especially compared with plurality voting or plurality voting with a runoff. (I put aside a comparison with ranking systems, like the Borda count, which I think will be non-starters, at least in public elections in the United States, for the foresee- able future.) While approval voting lessens the need to be strategic, however, it is certainly not immune to paradoxes. Indeed, there are paradoxes related to strategic voting, which is an area that Nurmi leaves mostly unexplored (but see Riker 1986, for several empirical examples), that could benefit from the attention that Nurmi gives to sincere-voting paradoxes.

No voting system is perfect, as Nurmi reminds us throughout. His book will certainly contribute to helping both the practical designer and the social- choice theorist weigh trade-offs among different systems and, more funda- mentally, understand why these trade-offs must be made.

References

Brams S, Fishburn PC (1984) Proportional representation in variable-size legislatures. Soc Choice Welfare 1: 397-410



838 Book review

Brams S, Kilgour DM, Zwicker WS (1997) Voting on referenda: The separability problem and possible solutions. Electoral Stud 16: 359-377

Brams S, Kilgour DM, Zwicker WS (1998) The paradox of multiple elections. Soc Choice Welfare 15: 211-236

Chavravarty SP, Hojman DE (1999) Voting, collective action, and liberalisation in Latin America: The rise and fall of the Hillinger paradox. Pubi Choice 101: 215- 233

Hillinger C (1971) Voting on issues and platforms. Behav Sci 16: 564-566 Miller NR (1975) Logrollling and the Arrow paradox: A note. Pubi Choice 21: 107-

110 Riker WH (1986) The art of political manipulation. Yale University Press, New Haven Saari DG (1994) Geometry of voting. Springer, Berlin Heidelberg, New York Van Deemen A (1993) Paradoxes of voting in list systems of proportional representa-

tion. Electoral Stud 12: 234-241

Steven J. Brams

Department of Politics New York University New York, NY 10003, USA



Article Contentsp. [835]p. 836p. 837p. 838

Issue Table of ContentsSocial Choice and Welfare, Vol. 18, No. 4 (2001), pp. A3-A11, 619-838Front MatterAn introduction to strategy-proof social choice functions [pp. 619-653]A crash course in implementation theory [pp. 655-708]The probability of ties with scoring methods: Some results [pp. 709-735]Optimal decision rules for fixed-size committees in polychotomous choice situations [pp. 737-746]The political viability of a negative income tax [pp. 747-757]Core concepts for share vectors [pp. 759-784]Strategyproof single unit award rules [pp. 785-798]Horizontal inequity comparisons [pp. 799-816]Nondictatorially independent pairs and Pareto [pp. 817-822]Using elections to represent preferences [pp. 823-831]Erratum: Social Welfare Functions which preserve distances [pp. 833-833]Book reviewReview: untitled [pp. 835-838]

Back Matter

voting paradoxes and how to deal with themby hannu nurmi

Documents