Majority Judgment

Majority Judgment: Measuring, Ranking, and Electing

Michel Balinski
Rida Laraki
Copyright Date: 2010
Published by: MIT Press
Pages: 432
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  • Book Info
    Majority Judgment
    Book Description:

    In Majority Judgment, Michel Balinski and Rida Laraki argue that the traditional theory of social choice offers no acceptable solution to the problems of how to elect, to judge, or to rank. They find that the traditional model--transforming the "preference lists" of individuals into a "preference list" of society--is fundamentally flawed in both theory and practice. Balinski and Laraki propose a more realistic model. It leads to an entirely new theory and method--majority judgment--proven superior to all known methods. It is at once meaningful, resists strategic manipulation, elicits honesty, and is not subject to the classical paradoxes encountered in practice, notably Condorcet's and Arrow's. They offer theoretical, practical, and experimental evidence--from national elections to figure skating competitions--to support their arguments.Drawing on insights from wine, sports, music, and other competitions, Balinski and Laraki argue that the question should not be how to transform many individual rankings into a single collective ranking, but rather, after defining a common language of grades to measure merit, how to transform the many individual evaluations of each competitor into a single collective evaluation of all competitors. The crux of the matter is a new model in which the traditional paradigm--to compare--is replaced by a new paradigm--to evaluate.

    eISBN: 978-0-262-29560-4
    Subjects: Economics, Political Science

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Preface
    (pp. ix-xvi)
  4. 1 Majority Judgment
    (pp. 1-20)

    Throughout the world, voters elect candidates, and judges rank competitors, goods, alternatives, cities, restaurants, universities, employees, and students. How? Schemes, devices, ormechanismsare invented to reach decisions. Each defines

    the specific form of the voters’ and judges’inputs, themessagesused to exert their wills, and

    the procedure by which the inputs or messages are amalgamated or transformed into a final decision, social choice, oroutput.

    In piano competitions, a judge’s input message is a grade assigned to each competitor—often in the range from 0 (low) to 25 (high)—and the output is the rank-ordering determined by the...

  5. 2 Voting in Practice
    (pp. 21-46)

    Voting in practice invokes issues that go well beyond the problem of how to elect one candidate among several or how to determine their order of finish. Many candidates are elected as the representatives of regions (constituencies, congressional districts, states, departments, provinces, or nations) to legislatures (Assemblées nationales, Diets, Houses of Representatives, Knessets, Parliaments, or Senates), or as the representatives of political parties, or of both regions and parties, to legislatures. A multitude of different systems are used; they raise different problems, invoke different information, ask for different inputs, and are resolved with different mechanisms. Nevertheless, several central problems are...

  6. 3 Traditional Social Choice
    (pp. 47-66)

    History reveals three rounds of precursors to the development of a full-blown discipline devoted to the study of voting and the problem of social choice.¹ Every revival of interest seems to have begun in ignorance of the previous work. Nevertheless, the basic model for voting has remained the same from the analyses of Ramon Llull in 1299 and Nicolaus Cusanus in 1433 to those of the Chevalier de Borda (1781) and the Marquis de Condorcet (1785), from the studies of Charles L. Dodgson (1873; 1874; 1876; 1884) and E.J. Nanson (1882) to those of Kenneth Arrow (1951), Duncan Black (1958),...

  7. 4 Electing versus Ranking in the Traditional Model
    (pp. 67-92)

    Condorcet himself defined a method of ranking that behaves continuously, always places the Condorcet-winner first (when he exists), and always agrees with the majority-rule-ranking (when it is transitive). Moreover, it almost satisfies IIA. These facts were largely ignored until H. Peyton Young read and analyzed Condorcet’s famousEssaiwith care (Condorcet 1785; Young 1988). This oversight was due in part to the first person to have looked into the history of the methods used for voting, Duncan Black. Black (1958) discussed Borda’s and Condorcet’s ideas at length, but much of his commentary—most notably that concerning Laplace’s and Galton’s contributions...

  8. 5 Strategy in the Traditional Model
    (pp. 93-110)

    Pierre-Simon, Marquis de Laplace was a mathematician and astronomer, a founder of the theory of probability (“perhaps the greatest and certainly the most famous physicist of his day” [Kuhn 1961, 196]). For him, “[T]he most important questions of life, are in effect, for the most part, problems of probability. One can even say, in all rigor, that almost all our knowledge is probable; and in the small number of things that we can know with certitude, in the mathematical sciences themselves, the principal means to arrive at the truth, induction and analogy, are based on probabilities, so that the whole...

  9. 6 Fallacies of the Traditional Model in Voting
    (pp. 111-128)

    Several centuries of work on the theory of social choice have produced very substantial contributions, notably, in identifying a host of important properties or criteria that should (or should not) be satisfied by a mechanism that amalgamates the beliefs, desires, or wills of individuals into a decision of society.

    Arrow’s paradox must be avoided: a method should satisfy independence of irrelevant alternatives, that is, the presence or absence of some candidate should not cause a change of winner between two others. Condorcet’s paradox must be avoided: a method should yield a transitive order of finish among the competitors. A method...

  10. 7 Judging in Practice
    (pp. 129-160)

    “We’re ranking everybody,” said the playwright Arthur Miller, “every minute of the day”: economists and peace-makers, mathematicians and physicists, novelists and journalists, students and professors, divers and skaters, beauty queens and muscle-men, cities and countries, hotels and restaurants, movies and theatrical performances, hospitals and universities, wines and cheeses. To rank these and many other competitors, accomplishments, endowments, performances, goods, or services is fraught with differences of opinion among the judges—or conflicting appreciations of their characteristic attributes—that must be reconciled into the verdict of a jury.

    Athletes compete for glory (and money) at Olympic games; chess and go players...

  11. 8 Common Language
    (pp. 161-174)

    Everywhere, in all pursuits, scientific and societal, scales are invented to measure, to understand, to classify, to evaluate, to rank, or to make decisions. This applies to every activity, attribute, candidate, or alternative, be it an immutable concept of the universe—temperature and its degrees—or an ephemeral fancy—the value of a painting and its price. These scales or measures constitute common languages of words that have absolute meanings, clearly understood by those who use them. Many domains of the physical world have natural units of measurement, imposed as it were, by the physics of the situation: time, mass,...

  12. 9 New Model
    (pp. 175-186)

    Over seven hundred years of effort and a host of impossibility theorems show that the “Arrovian model”, where many individual rankings are to be resolved into a single collective ranking, cannot be made to work: there is no satisfactory mechanism for doing what is wanted. Experience shows, on the other hand, that it is a relatively simple matter to invent grades, scores, levels, or measures to evaluate the performances of students, figure skaters, divers, and musicians, the qualities of wines and cheeses, and the intensities of seismic events, and so by inference to determine the relative merits of competitors in...

  13. 10 Strategy in Grading
    (pp. 187-198)

    The members of a jury assign grades. Asocial grading function defines a mechanism for transforming the individual grades of several or many judges into one final grade of the jury. The issues addressed in this chapter focus on the question, What strategies will judges use in the game of assigning grades? Later chapters consider other strategic games that judges may play, notably, how they may act and react to giving grades when these are also used to rank competitors.

    Experience clearly establishes the fact that assigning gradesisa game, because the players—the judges—may assign their grades strategically....

  14. 11 Meaningfulness
    (pp. 199-208)

    The languages used to grade students vary from nation to nation, the ranges of numbers used to evaluate flautists, pianists, and wines change from competition to competition, and the highest and lowest scores earned by Olympic competitors differ from one to another athletic discipline. When numbers are used, the differing languages are not related by a simple change in scale. For example, a school grade of 10 on a scale of [0, 20] in France has an entirely different meaning than a 50 on a scale of [0, 100] in the United States (indeed, it may be more accurate to...

  15. 12 Majority-Grade
    (pp. 209-218)

    The previous chapters have presented mounting evidence that argues for a jury to arrive at a final grade by using one of the order functions. A completely different set of arguments will single out one function that happens to be an order function. Sir Francis Galton pointed in the right direction a century ago:

    I wish to point out that the estimate to which least objection can be raised is the middlemost estimate, the number of votes that it is too high being exactly balanced by the number of votes that it is too low. Every other estimate is condemned...

  16. 13 Majority-Ranking
    (pp. 219-234)

    Some applications do not seek complete rank-orderings of the competitors: wine competitions come to mind, the aim being to give gold, silver, and bronze medals to certain percentages of the entries. In other applications, notably sports and elections, an ordered list from first to last and a clear winner are necessary.

    A candidate (or alternative) who receives a higher majority-grade than another is naturally ranked higher in the order of the candidates or alternatives than the other: grades imply orders. But if rank-orderings are the outputs, the strategic behavior of judges and voters may well change. Does this imply that...

  17. 14 Large Electorates
    (pp. 235-250)

    The majority-values of competitors or candidates—and thus the majority-ranking—may be simplified when a jury has many judges or an electorate many voters, or when the common language contains few grades in comparison with the number of judges. To see why, look again at the example of candidate A in the fifty-two-voter SCW Society election (see table 13.1): twenty-four2s, eleven1s, and seventeen0s:

    $(\overbrace {{2_{52}},...,{2_{12}},{2_{10}},{2_8},{2_6},}^{24}\overbrace {{1_4},{1_2},{1_1},{1_3},{1_5},{1_7},{1_9},{1_{11}},...,{1_{17}}}^{11},\overbrace {{0_{19}},...,{0_{51}}}^{17})$.

    Thekth majority-grades fork= 1,...,52, are indicated by the subscripts. The majority-value is obtained from the ordered set of beginning with the majority-grade, or the lower middlemost grade, and then taking...

  18. 15 Common Language: Voting
    (pp. 251-278)

    The majority judgment relies on a language of evaluation that is common to the judges of a jury or the voters of an electorate. Practice shows that in judging competitions—of wines, divers, skaters, gymnasts—the numbers used by the judges of juries are defined by rules and regulations and constitute a well-understood language of grades that becomes better understood through use. Much the same may be expected to happen in voting with the majority judgment in large electorates: use will increasingly impart meaning; over many elections the language will become common. And when the method is actually used for...

  19. 16 Objections to Majority Judgment
    (pp. 279-292)

    The majority judgment enjoys a host of excellent properties. It is important to discover and understand under what circumstances, if any, it may fail to satisfy other desirable properties. Shortly after the theory was first publicly presented at the 8th International Meeting of the Social Choice and Welfare Society in Istanbul on July 14, 2006, objections began to emerge. The same ones have been rediscovered repeatedly. As James Stephens once remarked, “Nothing is perfect. There are lumps in it.”

    The first type of “lump” is perceived because of deeply ingrained attitudes anchored in habit and tradition, which often ignore the...

  20. 17 Point-Summing Methods
    (pp. 293-314)

    Point-summing methodsare much used in practice: a set of numerical points is specified, a voter or judge assigns any point of the set to each candidate, the winner is the candidate whose sum of points or average is highest, and the candidates are ordered according to the sums of points they receive or their averages. Point-summing methods arenotsum-scoring methods. In the latter, scores or points are associated with places in a voter’s rank-order of the candidates; in the former, voters are free to assign any point of the allowable set of points to a candidate. Approval voting...

  21. 18 Approval Voting
    (pp. 315-338)

    A relatively recent novelty in voting mechanisms, approval voting, is championed by Steven J. Brams, Peter C. Fishburn (Brams and Fishburn 1983), and many others. It was first proposed formally by Robert J. Weber (1977).¹ It has been used to elect the officers of several important scientific societies, to elect national representatives in Russia, and in a referendum held in the state of Oregon where one proposition among five was to be chosen.Approval voting, which was discussed in the context of left-right spectra in chapter 6, allows each voter to cast as many votes as he wishes, but at...

  22. 19 Comparisons of Voting Methods
    (pp. 339-350)

    Previous chapters have compared various methods of voting with the majority judgment on the basis of the theoretical properties the methods satisfy or fail to satisfy. The experimental evidence given in this chapter depends entirely on the 2007 Orsay experiment. We believe that the participants by and large expressed their true opinions, for they at once had no incentive not to do so—participation itself in a nonbinding vote was an indication of a will to cooperate—and their input messages were consistent with the expressions of their official votes (as well as other ancillary evidence). Thus the fact that...

  23. 20 The Game of Voting
    (pp. 351-374)

    The analysis and comparison of methods, traditional and new, and thus the evaluation of which are best, depend oncontext. Different contexts are encountered in theory and in practice. Often, in debates or arguments, when in one context a method satisfies the criteria that are sought, willy-nilly it is attacked in terms of another context. To be coherent, arguments should compare methods in each of the several possible contexts separately. In the various contexts discussed so far, the majority judgment has been shown, we believe, to dominate the other methods in that it comes closest to meeting the important desirable...

  24. 21 Multicriteria Ranking
    (pp. 375-386)

    Each voter distinguishes one candidate for political office from another by some ill-defined mix of criteria that may touch upon party affiliation and party platform; honesty and moral outlook; voice, appearance, and charisma; foreign, economic, and social policies; and a host of other considerations. But there is no agreement among voters on which of these aspects are more or less important: each voter is left to integrate all the criteria he believes of importance to reach a final judgment on the merit of each candidate.

    Skaters, gymnasts, countries, pianists, wines, . . . , however, are routinely evaluated on the...

  25. 22 A Summing Up
    (pp. 387-394)

    Paradoxes and impossibility theorems have dominated thetheoryandanalysisof social choice and voting from Condorcet’s to Arrow’s, and on to all the many others that continue to be found down to the present day. Paradoxes and anomalies—most notably and most importantly, Condorcet’s and Arrow’s—have plagued therealityandpracticeof voting and judging across the years. Today the world shows signs of a growing awareness that perhaps the mechanisms used to elect and to rank—pure inventions of the human mind—are not electing the candidates the voters want nor designating the order of finish the...

  26. References
    (pp. 395-404)
  27. Name Index
    (pp. 405-408)
  28. Subject Index
    (pp. 409-414)