Abstract
The majority judgement is a method of election. It is the consequence of a new theory of social choice where voters judge candidates instead of ranking them. The theory is developed elsewhere (Balinski and Laraki 2007, 2010). This article describes and analyzes electoral experiments conducted in parallel with the last two French presidential elections to: (1) show that the majority judgement is a practical method, (2) describe it and establish its salient properties, and (3) illustrate how in practice the well known electoral mechanisms all fail to meet important criteria.
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- 1.
Laplace only used this model to deduce Borda’s method via probabilistic arguments. He then rejected Borda’s method because of its evident manipulability.
- 2.
This, of course, assumes that the vast majority of Nader’s votes would have gone to Gore.
- 3.
In their last 11 predictions (late February to the election), the Sofres polls showed Jospin winning seven times, Chirac two times, a tie two times.
- 4.
By Tns – Sofres – Unilog Groupe Logica CMG, April 22, 2007.
- 5.
A candidate’s Borda-score is the sum of the votes he or she receives in all pair by pair votes. Equivalently, with n candidates, a voter gives n − 1 Borda-points to the first candidate on his/her list, n − 2 to the second, down to 0 to the last. The sum of a candidate’s Borda-points is the candidate’s Borda-score.
- 6.
The question in French: “Pour présider la France, ayant pris tous les éléments en compte, je juge en conscience que ce candidat serait:” The grades in French: “Très bien, Bien, Assez bien, Passable, Insuffisant, à Rejeter.” The names of the candidates are given in the official order, the result of a random draw.
- 7.
The question in French: “Pour présider la France, ayant pris tous les éléments en compte, je juge en conscience que ce candidat serait:” The grades in French: “Très bien, Bien, Assez bien, Passable, Insuffisant, à Rejeter.” The names of the candidates are given in the official order, the result of a random draw.
- 8.
A collection of television interviews of participants prepared by Raphaël Hitier, a journalist of I-Télé, attests to these facts.
- 9.
With twelve candidates and six grades, there are 612 = 2, 176, 782, 336 possible messages.
- 10.
by TNS Sofres – Unilog Groupe Logica CMG, April 22, 2007, the same poll cited earlier.
- 11.
These elect the candidate who is ranked first by a majority. If there is no such candidate, then candidates are eliminated, one by one, their votes “transferred” to the next on the lists, until a candidate is ranked first by a majority. The choice of who to eliminate may differ. One mechanism eliminates the candidate ranked first least often; another eliminates the candidate ranked last most often. In the experiment the first elected Sarkozy, the second elected Bayrou.
- 12.
in the 1st precinct, 601 in the 2nd, 573 in the 3rd.
- 13.
No grade was assigned to each of the candidates in the following percentages: Nihous 7.2%, Schrivardi 5.8%, Laguiller 5.3%, Villiers 4.3%, Buffet 4.3%, Voynet 4.3%, Bové 4.2% Besancenot 3.2%, Bayrou 2.9%, Le Pen 2.7%, Royal 1.8%, Sarkozy 1.7%.
- 14.
The information in Table 2.10 does not suffice.
- 15.
The majority-grades and the majority-ranking of the candidates after Sarkozy is the same as for the three precincts except that Besancenot obtains a Poor−, and de Villiers is placed 9th and Nihous 10th.
- 16.
A Tnes-Sofres poll of March 14–15, 2007 showed 72% of Royal voters (respectively, 75% of Sarkozy voters) giving their votes to Bayrou in a second round against Sarkozy (respectively, against Royal).
- 17.
An extensive investigation, Balinski and Laraki (2010), uses many of the standard statistical tests to confirm this finding.
- 18.
Royal’s scores are consistently though slightly overestimated. This probably reflects changes in opinions in the 2 weeks that separated the two rounds of voting (due, in particular, to the televised debate between the two candidates).
- 19.
The voter’s preferences in grading are said to be “single-peaked.”
- 20.
In an entirely different context, a related technical result is proved in Moulin (1980).
- 21.
In the context of the traditional model, this is the Gibbard-Satterthwaite theorem.
- 22.
See Balinski and Laraki (2010). In point-summing methods, voters assign points from an interval to candidates and they are ranked according to the sum of their points.
- 23.
- 24.
The Institute for Operations Research and the Management Sciences, a scientific society. A large majority of the members are US citizens, but many members are citizens of other nations.
- 25.
The idea to experiment approval voting on a large scale in parallel with a presidential election actually goes back to 1995, when Balinski and Laurant Mann prepared a basic plan, but were too late to realize it. For a detailed account of the 2002 experiment, see Balinski et al. (2003).
- 26.
st, 5th, 6th, 7th, and 12th.
- 27.
This is standard practice. The 2007 ballot for the election of the officers of the Society for Social Choice and Welfare gives similarly neutral instructions: “You can vote for any number of candidates by ticking the appropriate boxes.”
- 28.
With 16 candidates there are 216 = 65, 536 possible messages. With the majority judgment, there are 616 or some 2.8 trillion possible messages.
- 29.
The crosses would have to be consecutive with regard to the alignment: there are 16 such messages with one cross, 15 with two, 14 with three, …, 1 with sixteen and 1 with none.
- 30.
For a different analysis of this experiment, see Laslier and Van Der Straeten (2004).
- 31.
The analyses are confined to the more important candidates.
- 32.
One ballot contained both. This permits analyses of potential interest. On the other hand, the participants expressed themselves twice simultaneously, which may have induced interdependencies.
- 33.
Three precincts in Illkirch (Alsace), two in Louvigny (Basse-Normandie), and one in Cigné (Mayenne).
- 34.
Applying this behavior to the majority judgment ballots of the Orsay experiment to simulate an approval vote gives the following percentages of ballots with circles: Bayrou 51.1%, Royal 44.8%, Sarkozy 44.1%, Besancenot 16.8%, Voynet 14.5%, Buffet 11.6%, Villiers 9.9%, Bové 9.0%, Laguiller 9.0%, Le Pen 8.7%, Nihous 3.2%, and Schivardi 2.6%.
- 35.
The previous Danish number scale had ten integers: 0 through 13 without 1, 2, 4, and 12. The information concerning the Danish grading systems was found in http://en.wikipedia.org/wiki/GPA, December 5, 2007.
- 36.
This analysis results from a theoretical argument developed in Balinski and Laraki (2010).
- 37.
Condorcet’s was, for a very short time, used to rank figure skaters, doubled – in case of an intransitivity – by Borda’s rule (see Balinski and Laraki 2010; in fact, the exact rule has been proposed and defended Dasgupta and Maskin 2004). Borda’s method was adopted in about 1,784 to elect members of France’s Academy of Sciences until a newly elected member, Napoléon Bonaparte, insisted it to be discarded in 1,800, presumably because it is highly manipulable, as Laplace had argued. It violates IIA, it ignores intensities, in Laplace’s words it gives “a big advantage to candidates of mediocre merit.” Arguments for it, alone or in convolutions, continue to be made to the present day Saari (2001).
- 38.
With these strategies, voters cannot manipulate the two-past-the-post method.
- 39.
In the experiment with three candidates, for example, Royal had 656 wins, Bayrou 0 wins, Sarkozy 9,261 wins, and there were 83 ties: the sum is 10,000 (and similarly for the other methods in both experiments). However, to Condorcet must be added 129 Condorcet-cycles in the experiment with three, and 114 Condorcet-cycles in the experiment with 12. Ties with the majority judgment means ties in the majority-gauges
- 40.
Our emphasis.
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Acknowledgments
We are deeply indebted to Cheng Wan whose final project as an undergraduate at the École Polytechnique (April–July, 2008) was devoted to statistical analyses of the various methods based on the 2007 Orsay experiment. The experience itself could not have been realized without the generous support of Orsay’s Mayor, Mrs. Marie-Hélène Aubry, the staff of the Mayor’s office, and our friends and colleagues who sacrificed their Sunday (a beautiful spring day) to urging voters to participate and explaining the idea: Pierre Brochot, Stéphanie Brochot Laraki, David Chavalarias, Sophie Chemarin, Clémence Christin, Maximilien Laye, Jean-Philippe Nicolai, Matias Nuñez, Vianney Perchet, Jérôme Renault, Claudia Saavedra, Gilles Stoltz, Tristan Tomala, Marie-Anne Valfort, and Guillaume Vigeral. Thanks to them, the experiment was successful and its expense limited to the costs of ballots, envelopes, and posters.
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Balinski, M., Laraki, R. (2011). Election by Majority Judgment: Experimental Evidence. In: Dolez, B., Grofman, B., Laurent, A. (eds) In Situ and Laboratory Experiments on Electoral Law Reform. Studies in Public Choice, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7539-3_2
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