The new book by Michel Balinski and Rida Laraki is a veritable encyclopedia of voting methods, even though its declared aim is to advance just one of them: Majority Judgement (MJ). The authors undertake a sweepingly comprehensive effort; not only they introduce and discuss the advantages and drawbacks of their favorite procedure, but also introduce and discuss — for the argument and comparison’s sake — the advantages and drawbacks of the competing ones. The authors set out on a formidable task:

The intent of this book is to show why the majority judgment is superior to any known method of voting and to any known method of judging competitions.

The arguments in favor of the MJ are theoretical, empirical and experimental. The authors indeed do a thorough study of the voting and grading situations. Chapter 2 is devoted to the social choice methods employed in several countries. (They actually tested their method at the polling stations in the 2007 French presidential election.) In Chapter 7 they gather information on grading used in various competitions: wine contests, figure skating, music contests, and the like.

Their theoretical investigation is even more thorough. Chapters 3-7 cover traditional voting methods and Chapters 17 and 18 the newer approval voting and its extension by point-summing methods. One thrust of the investigation is to discern the fallacies that — according to the authors — are built into the traditional methods. They prove 5 variants of Arrow’s Impossibility Theorem (three in Chapter 3 and two more in Chapter 11) and Gibbard-Satterthwaite’s theorem as well (Chapter 5). The conclusion? (p. 127)

The traditional model of social choice has been tried in theory and in practice and does not work. By Richard Feynman’s definition of science, it must be eliminated.

But was is wrong with the traditional model? In the traditional model the voters (or judges) rank the candidates in the order of their individual preferences. And this is the chief culprit; voters’ preferences are relative: it’s like voters are talking different languages; a strong preference of one may not be the same as a strong preference of another. This relativism leads to incompatibility and — when it comes to combining preference schedules of various voters — paradoxes.

The remedy suggested by Balinski and Laraki is three-pronged. First of all, in any election there needs to be a common language. It could be numeric or verbal, like the Mankoski pain scale (p. 163): Pain free, Minor annoyance, Annoying enough to be distracting, Can be ignored if you are really involved in your work, Can’t be ignored for more than 30 minutes, Can’t be ignored for any length of time, Makes it difficult to concentrate, Physical activity is severely limited, Unable to speak, Unconscious. (Compare this with a common request to evaluate pain on a scale from 1 to 10.)

Second, the voters grade the candidates using the common language. A voter grades the candidates independent of each other; several candidates may be assigned the same grade.

The third deviation from the traditional model concerns the manner of aggregation of individual grading schedules. A candidate is assigned the grade which is the median of the grades he/she received from all voters. (In the case of an even number of voters, Balinski and Laraki choose the lower median as the only one that satisifies a property of consensus.)

The approach is entirely axiomatic in the manner of the original proof in K. Arrow’s thesis. Theorem 9.1 shows that certain aggregation functions lead to the grading methods that are impartial, unanimous, monotonic, and — most importantly — independent of irrelevant alternatives in grading. I am not going to define all the terms but only note that independence of irrelevant alternatives is the major stumbling block in Arrow’s Impossibility Theorem.

Then in Chapter 12 the authors establish several uniquely appealing properties of the median aggregation referred in the book as the middlemost which follows the claim by the distinguished statistician Sir Francis Galton to the effect that “the middlemost estimate expresses the vox populi” (p. 101.) Again without going into the terminological details, the authors prove that the middlemost functions are the unique aggregation functions that

- assign a final grade of
*r* when the majority of judges assign *r* (Theorem 12.1);
- minimize the probability of cheating (Theorem 12.2);
- maximizes the social welfare (Theorem 12.3);
- counter crankiness (Theorem 12.4).

The proposed method, however, is not perfect; Chapter 16 covers some of its drawbacks. The major one is known as the “no-show” paradox. In essence, it’s this:

A jury decides that candidate X is ranked higher than another candidate Y . The jury is augmented by one additional judge who assigns a higher grade to X than to Y . Result: Y is ranked ahead of X. Suggested moral: the judge would have done better not to participate.

The authors counter with a lemma (p. 287):

If X with the majority-grade α is the winner against Y with majority-grade β, and a new judge assigns α or higher grade to X and strictly lower than α grade to Y, or symmetrically, assigns β or lower to Y and strictly higher than β to X, then X remains the winner.

The authors argue that if, for example, the extra judge gives both candidates a low grade, he is of poor opinion of both and is likely not to care who is the loser. In case he gives high grades to both, he may not care who is the winner. The lemma handles other possibilities and shows — according to the authors — that the paradox is not of much importance.

Finally, compared to other methods MJ is more likely to end in a tie; for this reason, the tie breaking rules become an essential part of MJ. Ties are broken by removing common grades and applying the aggregation function anew. The process may be repeated if necessary. (In rare cases throwing a die may be the only alternative.) The proposed mechanism is rather storage intensive as the tie breaking requires to preserve grading of given by every voter. This led the authors to propose a version that only stores three numbers for every candidate: the median itself and the percentages of grades that are worse and better than that. However, examples show that — in case of a tie — the tie breaking procedure may not lead to the same result as would be implied by the shortened version.

In my view, the method may be problematic when there are several candidates in that it requires each voter to have an opinion of every candidate. There is no clear strategy as to how blank spaces in individual preference schedules need to be treated. Replacing them with the least possible grade may change the final outcome.

I hope that my description of the MJ method is at least adequate, but most certainly a short review could not do justice to the book. It is fantastically rich in relevant information: methods, techniques, viewpoints… It’s a true treasure trove on the theme of social choice. The book is well written, the volume well published; it’s a pleasure to read and a pleasure to hold.

Alex Bogomolny is a freelance mathematician and educational web developer. He regularly works on his website Interactive Mathematics Miscellany and Puzzles and blogs at Cut The Knot Math.