Paradoxes at the polling booth

Michael Dummett is an eminent philosopher who has had a long-standing interest in voting procedures. In this short and elegant book, he undertakes a study of alternative forms of voting with the practical aim of clarifying public understanding of the workings and consequences of alternative methods. He concentrates on the election of Parliament, and his comments are clearly directed to a British context or at least to systems similar to the British, which might also include the German and Italian. Systems with somewhat independent strong executives, such those of the United States and France, raise additional questions, but Dummett's analysis surely has pertinence even to these more complex systems.

British elections are determined by two principles: constituency representation with single-member districts, and choice of the candidate who has more votes than any other (plurality voting, sometimes misleadingly called "first past the post"). These principles of course also apply to elections for both houses of the US Congress and for all of the US state legislatures. Elsewhere, however, various forms of multi-member constituencies (in the extreme case of Israel, the whole country) and proportional representation are common.

The choice of voting procedures has been the object of some degree of analytic study beginning at least with the medieval Catalan logician and theologian, Ramon Lull, in the 13th century. It would probably be impossible to find a single analyst who would defend plurality voting. Its deficiencies are obvious and grave. Suppose that people vote sincerely. If there are several parties of more or less equal size, then it can easily happen that an extremist candidate, one much disliked by everyone not a supporter, will win. If voters vote strategically, then supporters of smaller parties will switch their votes to more likely winners to prevent a more disliked candidate from winning. Hence, there is a pressure for reducing the number of parties, as we observe in the US and the United Kingdom, so that viewpoints with lesser strength have no chance of being expressed. Both of these points show the value of having ballots which are more richly informative by showing not merely first choices but second, third, and more choices.

Dummett does a splendid job of explaining these and more complicated implications of different voting systems. The exposition is clear; quite complex arguments are expressed in terms available to a large number of in-telligent voters without sacrifice of accuracy. He is not dogmatic; he expresses his pref-erences clearly but presents the arguments for different systems fairly and does not hesitate to point out difficulties with his preferred systems.

He argues that there are really two different functions for the election of a legislative body: who best represents a given constituency, and how the seats in the legislature should be distributed among parties to represent the voting preferences in the nation as a whole. For each function, there are two questions, what do we want to achieve and how should we achieve it.

He argues strongly for the German system of dual voting: the legislative seats are filled partly from constituencies, partly on the basis of votes for party lists. Hence, he argues, the two functions can be carried out by different mechanisms, and there is no need to balance between them. (He does point to some problems with specific aspects of the German system.) He then discusses the criteria for election in a single-member constituency (these might also be applicable to the election of a chief executive in countries such as the United States and France). He agrees with Lull and with the 18th-century philosophe Condorcet that if a candidate would have a majority against any other candidate in a two-person election, that candidate should be elected. (If each ballot gives the full preferences of the voter over all candidates, the results of the hypothetical two-person elections can be easily calculated.) Notice that this "Condorcet leader" need not have more first choices than any other candidate; in fact, it is possible that a candidate who has no first choice votes at all is the Condorcet leader (the moderate candidate who is everyone's second choice, for example).

But, as Condorcet already noted, there may be no "Condorcet leader" in this sense. Dummett, in an interesting analysis, gives two different criteria in this case. They are based on the Borda score (suggested by Jean-Charles de Borda in 1770, though not published until 1784). If, for example, there are four candidates, give each first choice three marks, each second choice two marks, and so forth. Borda's own method was to choose the candidate with the highest total number of marks. However, it can easily happen that a Condorcet leader would not be elected.

One method suggested by Dummett (following, in part, the mathematician A. H. Copeland) is to choose the candidate who has a majority over the most other candidates. If there is more than one such, choose the one who has the highest Borda score. Another method is to choose the Condorcet leader if there is one, otherwise select the candidate with the highest Borda score. The merits of these proposals, including their effects on causing individuals to vote strategically rather than sincerely, are illustrated by very well-chosen examples.

In chapter ten, Dummett analyses the voting method called "alternative vote", or "preferential voting". It is an application of the well-known single transferable vote to the case when only one candidate is to be elected. By means of examples, he shows very well how arbitrary this system is, in particular how changes of preference in favour of a candidate can actually work to that candidate's disadvantage.

Subsequently, Dummett analyses alternative methods of voting in multi-member constituencies. He continues the analysis of single transferable vote systems and shows beautifully and clearly the arbitrary elements in the system, in particular how small changes can produce large and unintuitive alterations in outcome. He further shows how mean-ingless is the claim that single transferable vote systems waste no votes. He puts the protection of minorities as the main criterion in multi-member constituencies and shows how this can be achieved by a new method of his devising.

The subject of voting is inherently paradoxical, as illustrated by Condorcet's proof that there need not exist a Condorcet leader. Therefore, one can always find possible cases when any given system produces unexpected and undesirable results. It is also true that, for any electoral system, there will be circumstances in which it pays to vote insincerely. But Dummett has shown great clarity in taking account of these possibilities and yet forming judgements on possible systems.

I have only two observations of a somewhat critical nature. First, the sharp distinction he makes between representing a constituency and having the legislature reflect the national votes is not very clear to me. This difficulty is related to a second, that Dummett takes as given the existence of parties whose members have homogeneous preferences. The political parties of the US are notoriously heterogeneous in nature, and, from what I read of internal conflicts within the two major British parties, they are far from homogeneous. This raises the question of the extent to which individual preferences should be reflected in the outcomes. This issue is only significant in multi-member constituencies. List systems, such as that of Israel, in which the voter only votes for the party, impose an artificial homogeneity (and also deny intra-party democracy, because the ranking of the candidates of a given party is determined by the party leadership, not the voters).

This book is to be strongly recommended to anyone with a serious interest in the major issue of electoral reform.

Kenneth J. Arrow, Nobel laureate, is emeritus professor of economics and operations research, Stanford University, California.