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Majority Judgment Vs. Majority Rule ∗

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Majority Judgment Vs. Majority Rule ∗ View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Liverpool Repository Majority Judgment vs. Majority Rule ∗ Michel Balinski CNRS, CREST, Ecole´ Polytechnique, France, [email protected] and Rida Laraki* CNRS, University of Paris Dauphine-PSL, France, [email protected] and University of Liverpool, United Kingdom June 6, 2019 Abstract The validity of majority rule in an election with but two candidates|and so also of Condorcet consistency| is challenged. Axioms based on evaluating candidates|paralleling those of K. O. May characterizing ma- jority rule for two candidates based on comparing candidates|lead to another method, majority judgment, that is unique in agreeing with the majority rule on pairs of \polarized" candidates. It is a practical method that accommodates any number of candidates, avoids both the Condorcet and Arrow paradoxes, and best resists strategic manipulation. Key words: majority, measuring, ranking, electing, Condorcet consistency, domination paradox, tyranny of majority, intensity problem, majority-gauge, strategy-proofness, polarization. 1 Introduction Elections measure. Voters express themselves, a rule amalgamates them, the candidates' scores|measures of support|determine their order of finish, and the winner. Traditional methods ask voters to express themselves by comparing them: • ticking one candidate at most (majority rule, first-past-the-post) or several candidates (approval voting), candidates' total ticks ranking them; • rank-ordering the candidates (Borda rule, alternative vote or preferential voting, Llull's rule, Dasgupta- Maskin method, . ), differently derived numerical scores ranking them. However, no traditional method measures the support of one candidate.1 Instead, the theory of voting (or of social choice) elevates to a basic distinguishing axiom the faith that in an election between two candidates majority rule is the only proper rule. Every traditional method of voting tries to generalize majority rule to more candidates and reduces to it when there are only two candidates. Yet majority rule decides unambiguously only when there are two candidates: it says nothing when there is only one and its generalizations to three or more are incoherent. Using the majority rule to choose one of two candidates is widely accepted as infallible: since infancy who in this world has not participated in raising their hand to reach a collective decision on two alternatives? A. de Tocqueville believed \It is the very essence of democratic governments that the dominance of the majority be absolute; for other than the majority, in democracies, there is nothing that resists" ([52], p. 379)2. Judging ∗The authors are deeply indebted to Jack Nagel for very helpful comments and suggestions, particularly with regard to Dahl's intensity problem. yOn February 4, 2019, my co-author Michel Balinski, passed away at 85 years old. He was not only a great researcher, but also a kind, considerate gentleman and an attentive and faithful friend. I will miss him. 1A significant number of elections, including U.S. congressional elections, have but one candidate. 2Our translation of: \Il est de l'essence m^emedes gouvernements d´emocratiques que l'empire de la majorit´ey soit absolu; car en dehors de la majorit´e,dans les d´emocraties, il n'y a rein qui r´esiste." from W. Sadurski's assertion|\The legitimating force of the majority rule is so pervasive that we often do not notice it and rarely do we question it: We usually take it for granted" ([49], p. 39)|that conviction seems ever firmer today. Students of social choice well nigh universally accept majority rule for choosing between two alternatives, and much of the literature takes Condorcet consistency|that a candidate who defeats each of the others separately in majority votes must be the winner (the Condorcet-winner)|to be either axiomatic or at least a most desirable property. Why this universal acceptance of majority rule for two candidates? First, the habit of centuries; second, K.O. May's [39] axiomatic characterization; third, the fact that it is strategy-proof or incentive compatible; fourth, the Condorcet jury theorem [20]. Regrettably, as will be shown, the majority rule can easily go wrong when voting on but two candidates. Moreover, asking voters to compare candidates when there are three or more inevitably invites the Condorcet and Arrow paradoxes. The first shows that it is possible for a method to yield a non-transitive order of the candidates, so no winner [20]. It occurs, e.g., in elections [35], in figure skating ([4] pp. 139-146, [8]), in wine- tasting [6]. The second shows that the presence or absence of a (often minor) candidate can change the final outcome among the others. It occurs frequently, sometimes with dramatic global consequences, e.g., the election of George W. Bush in 2000 because of the candidacy of Ralph Nader in Florida; the election of Nicolas Sarkozy in 2007 although all the evidence shows Fran¸coisBayrou, eliminated in the first-round, was the Condorcet-winner. A recent econometric study of French parliamentary (1978-2012) and local (2011, 2015) elections [46] shows that in 19.2% of the 577 elections the presence of a third minor candidate in run-offs causes a change in the winner and concludes \that a large fraction of voters are . `expressive' and vote for their favorite candidate at the cost of causing the defeat of their second-best choice" ([46], p. 42). How are these paradoxes to be avoided? Some implicitly accept one or the other (e.g., Borda's method, Condorcet's method). Others believe voters' preferences are governed by some inherent restrictive property. For example, Dasgupta and Maskin [23, 22] appeal to the idea that voters' preferences exhibit regularities|they are \single-peaked," meaning candidates may be ordered on (say) a left/right political spectrum so that any voter's preference peaks on some candidate and declines in both directions from her favorite; or they satisfy \limited agreement," meaning that for every three candidates there is one that no voter ranks in the middle. Another possibility is the \single crossing" restriction (e.g., [12, 47]): it posits that both candidates and voters may be aligned on a (say) left/right spectrum and the more a voter is to the right the more she will prefer a candidate to the right. Such restrictions could in theory reflect sincere patterns of preference in some situations; nevertheless in all such cases voters could well cast strategic ballots of a very different stripe. Moreover, all the sets of ballots that we have studied show that actual ballots are in no sense restricted. There are many examples. (1) An approval voting experiment was conducted in parallel with the first-round of the 2002 French presidential election that had 16 candidates. 2,587 voters participated and cast 813 different ballots: had the preferences been single-peaked there could have been at most 137 sincere ballots ([4] p.117). (2) A Social Choice and Welfare Society presidential election included an experiment that asked voters to give their preferences among the three candidates: they failed to satisfy any of the above restrictions [17, 48]. (3) A recent study of voting in Switzerland uses principal component analysis to show that at least three dimensions are necessary to map voters' preferences, so the one-dimensional single-peaked condition cannot be met [26]. Real ballots|true opinions or strategic choices|eschew ideological divides. All of this, we conclude, shows that the domain of voters' preferences is in real life unrestricted. An entirely different approach is needed. This has motivated the development of majority judgment [3, 4, 8] based on a different paradigm: instead of comparing candidates, voters are explicitly charged with a solemn task of expressing their opinions precisely by evaluating the merit of every candidate in an ordinal scale of measurement or language of grades. Thus, for example, the ballot in a presidential election could ask: \Having taken into account all relevant considerations, I judge, in conscience, that as President of France, each of the following candidates would be:" The language of grades constituting the possible answers may contain any number of grades3 though in elections with many voters five to seven have proven to be good choices. A scale that has been used is: Outstanding, Excellent, Very Good, Good, Fair, Poor, To Reject; 3Majority judgment theory [4] put no restrictions on the size or the nature of the scale, which can be numerical or verbal, finite or infinite. Methods based on a numerical scale (e.g. point-summing methods) are characterized in Chapter 17. 2 The method then specifies that majorities determine the electorate's evaluation of each candidate and the ranking between every pair of candidates|necessarily transitive|with the first-placed among them the winner. Majority judgment has been criticized by the proponents of the traditional paradigm in articles, reviews, and talks [13, 15, 24, 27, 28].. Most objections have been answered in chapter 16 of [4], see also [2]. New arguments to the no-show paradox charge are detailed in a companion paper [9]. The present article gives a completely new argument for majority judgment that responds to the objection that it does not agree with the majority rule on two candidates, i.e., that it is not Condorcet-consistent. This paper first gives a new description of majority judgment that shows it naturally emerges from a majority principle when candidates are assigned grades. Then, it shows that majority rule on two candidates is not incontestable, it is open to serious error because it admits the \domination paradox": it may elect a candidate whose grades are dominated by another candidate's grades. The infallibility of majority rule on two candidates has been challenged before.
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