Soc Choice Welf DOI 10.1007/s00355-013-0745-z BOOK REVIEW

Michel Balinski and Rida Laraki: Majority judgment. Measuring, ranking, and electing MIT Press, 2010, ISBN 978-0-262-01513-4, xv, 414 pp Marc Fleurbaey

Received: 7 September 2012 / Accepted: 12 September 2012 © Springer-Verlag Berlin Heidelberg 2013

In the authors’ own concluding words, this book offers a ‘change in view [that] immensely improves the representation of reality, permits deeper understanding and analysis, and so leads to a vastly better mechanism for juries and electorates to rank and to elect.’ (p. 394) Certainly, this is not a boring book. In twenty-two short chapters, the reader is told a very interesting story about methods of voting and various ways of evaluating them. This book has two main messages. The best known, already, is that it advocates a new voting mechanism—or, rather, a class of mechanisms. It consists in asking voters not to pick a candidate, or to rank candidates, but to grade every candidate with a suitable grading system (e.g., from Excellent to To Reject). The aggregation of voters’ grades, for every candidate, is then made by taking the median of the distribution of voters’ grades for this candidate. This may produce ties, especially when the number of grades is small. Ties are then resolved by eliminating the median grade from the distribution and seeking the median grade in the remaining distribution. For large electorates, a good approximation of this tie-breaking mechanism consists in measuring the fractions of the electorate who give a greater grade and a lower grade than the median grade, and then to focus on the greatest of these two fraction numbers.1 The other message is that solving the election problem requires a new model, the traditional model of social choice being, according to the authors, radically flawed. The traditional model focuses on voters’ preference orderings and produces impossibility

1 Of two candidates with the same median grade, the one with more people wanting to give a higher grade than people wanting to give a lower grade is preferred to the one in the opposite configuration; of two candidates with same grade and more higher graders than lower graders, prefer the one with the greatest number of higher graders; of two candidates with same grade and more lower graders than higher graders, prefer the one with the lowest number of lower graders.

M. Fleurbaey (B) Princeton University, Princeton, NJ, USA e-mail: [email protected]

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results showing that it is hard to avoid cycles (Condorcet’s paradox), or sensitiveness to irrelevant candidates (Arrow’s theorem). The new model introduces a richer language of evaluation, allowing the collectivity of voters to grade every candidate, one by one, thereby avoiding sensitiveness to irrelevant candidates, and producing an ordering of candidates, avoiding the Condorcet paradox—in fact more than an ordering is delivered, as the grades of the candidates carry a message from the electorate: it is less honorable to be elected as Acceptable against a Reject candidate than to be elected as Very Good against a Good contender. Let us briefly summarize the contents of the book. Chapter 1 explains the new method and announces the two key messages just summarized. Chapter 2 describes many voting rules used in several countries, illustrating the dreadful properties of most of them, and showing that the Condorcet and Arrow problems are real, not just theoretical. Chapter 3 introduces the traditional social choice model and Arrow’s impossibility, with a nice analysis of three different ways in which the independence axiom can be formulated (including Nash’s independence) and brought to entail impossibility. Chapter 4 argues that electing a winner and ranking all candidates are different exercises in the traditional (preference-based) model. Similar axiomatic characterizations single out the Borda score for the selection of a winner and the Kemeny rule for the selection of a ranking (which always ranks the Condorcet winner first, unlike the Borda score). The authors conclude that the traditional model is inconsistent, because the winner should obviously be the top-ranked candidate. Chapter 5 reviews the manipulability of vote mechanisms and introduces the idea of looking at the median rather than the mean, attributing it to Galton, who realized that the median is less vulnerable to ‘cranky’ voters. Chapter 6 summarizes the ‘fallacies of the traditional model’. The authors argue in particular that grading candidates is cognitively easier and conveys more information on voters’ views than ranking. The former point relies on experiments in which voters refused to give full linear orders of candidates and other experiments in which voters were happy to grade candidates. It is however hard to understand why voters would have a greater difficulty to sort out candidates by rough categories than giving them a small number of qualitative grades. As grades contain more information than a ranking, it is hard to believe that it is less cognitively demanding to grade than to rank. The other main criticism of Chapter 6 against the traditional model is that the standard rules (Borda, Kemeny) are biased in favor of the center. However, the evidence consists mostly of observing that majority judgment is less favorable to the center. This could be taken as evidence that majority judgment is biased against the center—more on this later. Chapters 7–16 study majority judgment. Chapter 7 describes grading practices in a long list of activities (education, sports, arts, wine…) and Chapter 8 discusses the requirement that grades be formulated in a language common to all voters–judges. As an additional argument in favor of the median and against the summing approach, the authors argue that it is difficult to use grades having an interval value, because it requires that the proportion of candidates between two consecutive grades be always the same. This restriction appears excessive, however. It is standard to compute average grades (over different courses) for students, even when the distribution of students over grades is very far from being uniform in every course. That the difficulty of jumping to a higher grade may differ from one grade to another or differ across courses

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does not make the average grade over courses meaningless. The reason is simple: the grades may reflect a natural interval scale (e.g., the percentage of correct answers in a list of questions of similar difficulty). Requiring the grades to be adjusted to the distribution of performance, as the authors do, would make it impossible to assess if global performance changes over the years. Chapter 9 introduces the ‘new model’, in which the inputs are the grades given by every voter to every candidate. The output is either a grading, or a ranking as in the traditional model. The authors argue that grades are very different from voters’ utilities (which may depend on other things than the value of candidates). Chapter 10 shows that manipulation is minimized by ‘order functions’, which, for every candidate, pick his grade at a fixed percentile in the distribution of grades—the median used in majority judgment being a particular case. Chapter 11 provides additional arguments in favor of order functions (e.g., the ranking of candidates does not depend on a rescaling of grades that is common to all voters), and reformulates Arrow’s theorem in terms of allowing grades to be rescaled in different ways for different voters. Chapters 12 and 13 provide arguments in favor of taking the median, in the class of order functions. There are four main arguments: (1) respecting the majority (for any other grade than the median, there is a majority in favor of another specific grade); (2) minimizing the probability of finding a voter who could change the grade; (3) maximizing the average distance of the grade from each voter’s preferred grade; (4) making the grade independent of extreme grades (‘cranks’). Chapter 14 studies how to formulate majority judgment in a simple way for large electorates. Chapter 15 examines if, in experiments, voters use the grades in a homogeneous way across subsamples, which is supposed to test if they use a common language. It is hard to be convinced that this is a test of common language, as subgroups of voters can have a common language but nevertheless use it in systematically different ways, and conversely. Chapter 16 addresses objections to majority voting. There are two sorts of objections. The first rely on the observation that a candidate with the median grade may be unacceptable to a minority, whereas another candidate with a slightly lower median grade may be acceptable to all and preferred by every voter but one, as in the following table, in which it is assumed that voters are arranged by decreasing order of grades:

Candidates

Voter 1’s grade

…

Median grade

…

Voter n’s grade

A

50

…

50

50

0

…

0

B

100

…

100

49

49

…

49

The authors’ reply is that one must respect the majoritarian judgment, even when it goes against the quasi-unanimous preference. The other class of criticism refers to the fact that by abstaining from participating, a voter or a group of voters (e.g., those on the left of the above table) may turn the ranking of candidates in the direction of its preferences. Interestingly, a variant of the above table could be used to argue that majority judgment is biased against the center (we now assume that there are seven

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voters and that they are aligned from left to right). In the following table, the median grade of the left wing candidate (50) is greater than the centrist candidate’s median grade (49), who is nevertheless preferred by 5/7 of the electorate and receives a much greater minimal grade:

Candidates Left wing Centrist

Voter 1’s grade

…

…

Voter n’s grade

100

80

60

50

40

20

0

45

49

80

100

80

49

45

Chapters 17–19 compares with other voting methods, the point-summing methods (Ch. 17), approval voting (Ch. 18), and others (Ch. 19). Again the authors argue that summing points means nothing unless the distribution of points is uniform (interval measure). More convincing is the observation that points invite voters to reason in relative rather than absolute terms. They argue that this also occurs with approval voting. Therefore, even if, formally, approval voting is at the same time a point-summing method and a majoritarian judgment, due to the restriction to a yes-no grading, the authors argue that it drawn toward the point-summing perspective when voters reason in relative terms. Finally, Chapter 20 proposes an interesting strategic analysis of the voting game generated by majority judgment, and Chapter 21 applies the approach to multicriteria decision-making. Chapter 22 concludes as cited in the beginning of this review. So, is all this a paradigm shift? Formally and conceptually, it is not. The grading model is formally identical to the model of social choice with interpersonally comparable utilities. The only originality is, for some of the results, the formulation with a finite number of grades, which, as far as I know, has not been studied in the theory of social choice with utilities—although verbal variables have been considered in the theory of fuzzy social choice. The authors make great efforts to argue that grades in their model have little to do with utilities in social choice theory, but the fact that two notions receive different semantic interpretations in two approaches does not prevent the models from being formally equivalent. Similarly, positional dictatorship is a classical idea both in the theory of social choice with utilities and in the analysis of strategic voting. Many results of this book come (and this is acknowledged) from these parts of the literature. If this book is original, it is more in the insistence on the possibility of using grades in political voting. This is a very interesting idea, which deserves to be explored and tested. The book is very convincing in this respect and should have a great impact, as it radically questions the conventional wisdom that voting rules exclude all forms of interpersonal comparability. There are two limitations in the authors’ contribution to democratic theory. First, once one accepts that a common system of grades can be proposed to voters, it is of utmost importance to select the language of grades and study the framing effect of words and formulations of the question. Strangely enough, very little is said about this in the book. It is not even obvious that a consensus can be reached about the appropriate question and grading system to be used. Should the President of France

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be ‘excellent’, as suggested in the questionnaire of the authors’ experiment, or should he do the best for the country, or should he prepare the best future for our children, or should he serve best the interests of the voter, or should he be the most attentive to the welfare of the population…? Different conceptions of the common good may point in different directions. The choice of the question may therefore become a political and ideological quarrel by itself. The difficulty in finding the right question and grading does not mean that this is a flaw of the grading approach. One can argue that the traditional voting system is totally oblivious to the heterogeneity in voters’ motivations when they express their views with an information-deprived ballot. Adopting a grading system might be the occasion to align voters’ perceptions of what the vote is about and make the outcome much more meaningful. There is a lot to be explored in this respect. The second limitation is that for all their expertise in voting, the authors do not mention a key problem with the majoritarian approach, which is the oppression of minorities. It is obvious, for instance, that Condorcet cycles always involve a tyranny of the majority (i.e., a majority with low intensity preference imposing its will on a minority with high intensity preference). The objection illustrated in the first table above involves a tyranny of the majority as well. The example displayed in the table suggests a remedy. Instead of taking the median grade, one could take a greater percentile, so that no more than a small minority of voters would like to lower the grade. If one picked the grade that appears in the column on the right of the median grade, candidate B would be preferred over candidate A, by a wide difference in grades. While the authors discuss the extreme methods that pick either the lowest grade or the greatest grade (in relation to another issue, namely, consistency when concatenating electorates), they do not study qualified majority. A qualified-majority judgment would also contribute to reducing the bias of majority judgment against centrist candidates, as can be seen from the second table. In conclusion, this is a thought-provoking book that points in a direction that deserves to be explored further. There are several ways of enriching the information used and conveyed by the voting mechanism. The grading approach is not the only one, but it is appealing and promising. It may substantially improve the quality of the democratic process, and even, perhaps, positively affect the substance of political debates. That the authors are enthusiastic and excited about their proposal makes it very pleasant to readers who can bear the uncharitable light in which the classical literature is sometimes depicted. Marc Fleurbaey Princeton University

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