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Social choice Maurice salles CREM, Université de Caen‐Basse‐Normandie CPNSS, London School of Economics Murat Sertel Center, Bilgi University, Istanbul 1. Introduction. Social choice in its modern guise is a young subject that can be dated back to the end of the 1940s and the beginning of the 1950s in the works of Black 1948 , Arrow 1950, 1951 and Guilbaud 1952 . This is nowadays considered as a rebirth, the first birth being generally attributed to Borda 1784 and Condorcet 1785 . However, as will be clear in this Chapter, one can find a number of precursors. Social choice is concerned with the selection of options on the basis of the opinions of individuals over these options. It should be noted that there is an analogy with the choice by an individual of, say, an object, in the presence of multiple criteria. However, I will restrict myself to the multi‐individual framework. The selection procedures have been studied either from a rather abstract point of view or from a more practical point of view. In the former case, one considers notions such as aggregation functions, social choice functions and their properties and in the latter case one considers voting rules, voting games etc. It is interesting to note, in particular in this handbook perspective, that this dichotomy has a historical origin, the abstract aspect being generally associated with the utilitarianism tradition from Bentham to Bergson, Samuelson and with welfare economics and the practical aspect being associated with questions related to elections, be it elections in small committees or in larger organizations. The precursors I will use this phrase for authors living before the 18th century dealt mostly with voting. On the other hand, the 18th century saw an upsurge of interests under both aspects, in particular with scholars living at the same time: Bentham on one side, and Borda and Condorcet on the other side. It is surprising that during the 19th century the interest in voting rather faded away in spite of the emergence of democratic societies. There has been, however, some work on proportional representation or the equivalent apportionment methods by European scholars rediscovering for a part what some founding fathers of American democracy previously did and the brilliant exception of C.L. Dodgson, better known as Lewis Carroll. The rebirth of social choice theory in the 20th century offers the same dichotomy. Although both Duncan Black and Kenneth Arrow are economists, the former obviously belongs to the voting tradition and the latter to the welfare economics tradition. Even though voting aspects are not absent in the founding book by Arrow, book entitled “Social choice and individual values”, a significant part of this book is devoted to discussions of the compensation tests debate between Hicks, Kaldor and Scitovsky and of the Bergson‐Samuelson social welfare functions. “Modern” must I say Arrovian? social choice theory has to be presented even in a handbook devoted to the history of economic thought. I will refrain from describing the most recent
2 aspects. However, a large part of this Chapter will be devoted to Arrow’s theorem and its descendants and to other major results that form the cornerstone of the domain. 2. The precursors. We have been very lucky that several major classics were collected in an excellent anthology by McLean and Urken 1995 . Furthermore, McLean and Urken wrote a remarkable introduction that will be one of my main sources as far as pre‐Arrovian social choice is concerned. Even if both Plato and Aristotle’s views regarding the goodness of the various political systems could have been discussed in this chapter, since it seems that, for them, the ideal would be some authoritarian regime Oligarchy for Plato and monarchy with some kind of benevolent dictator for Aristotle with as objective the maximization of the happiness of the state as a whole, a theme related to both Bentham and Arrow , I will follow the standard but quite recent usage to consider than Pliny the Younger was the first author to consider a social choice problem1. 2.1. Pliny the Younger. Pliny AD 61 or 62‐113 was born in a Roman high society family in Como. His uncle his mother’s brother was Pliny the Elder, a naturalist, natural philosopher, naval and army commander, and friend of the emperor Vespasian. Pliny the Elder died during the Vesuvius eruption that destroyed Pompeii and Herculaneum in August 79 while he was trying to rescue friends. The younger Pliny inherited from his uncle and the change of name indicates his adoption by will as a son his father, Lucius Caecilius had died while he was very young . Pliny started his career at the Roman bar at the age of eighteen. He moved through the regular offices in a senator’s career, even becoming consul in 100 under Emperor Trajan. On this occasion he delivered the speech of thanks known as the Panegyricus. But he is better known for his letters that are considered as a social document of his times and are praised for the quality of their prose. The letter that interests us is letter 14 in Book VIII Pliny, 1969 . Pliny’s letter was brought to the attention of social choice theorists by Farquharson 1969 in his exceptional book on strategic voting on Farquharson, see Dummett, 2005 . Farquharson ‘s book includes a 1752 translation by John, Earl of Orrery, of Pliny’s letter. According to Riker 1986 , the translator “…did not seem to understand the parliamentary issues involved and therefore did not see what happened at the end of the event.” There is fortunately a new translation by Betty Radice in Pliny 1969 . Pliny’s letter is to Titius Aristo, a preeminent jurist. After a long digression praising Titius Aristo, Pliny explains that “ T he case at issue concerned the freedmen of the consul Afranius Dexter, who had been found dead; it was not known whether he had killed himself or his servants were responsible, and, if the latter, whether they acted criminally or in obedience to their master.” Three kind of decisions were then suggested: acquittal, banishment or death penalty. Pliny also suggests that some people were in favour of banishment for the freedmen and death for the slaves, although there is no further mention of slaves. Pliny wished to have a plurality voting 1
It is striking to remark that, for Plato, the ruling elite should have accomplished a full study of mathematics. The Athenian democratic system is described in The Constitution of Athens, generally attributed to Aristotle. One can see why, for Aristotle, democracy and elections are antinomic. The crucial role played by selection by lot is the explanation (see Tangian, 2008).
3 system in this case: “ M y own proposal was that the three sentences should be reckoned as three, and that two should not join forces under a temporary truce”. He feared the formation of a coalition of senators in favour of death with those in favour of banishment, defeating those who were in favour of acquittal. He even imagined a run‐off with a vote on death against banishment. Although the letter does not explain this, we can infer from it that the banishment penalty would have defeated both acquittal and death penalty in pair‐wise majority voting, being what is now called a Condorcet winner. Pliny succeeded in his request to have a one round plurality vote. Then: “ T he proposer of the death sentence was convinced by the justice of my request…, dropped his own proposal, and supported that of banishment. He was afraid, no doubt, that if the sentences were taken separately which seemed likely if he did not act the acquittal would have a majority, for there were many more people in favour of this than either of the other two proposals. Then, when those who had been influenced by him found themselves abandoned by his crossing the floor and the proposal thrown over by its author, they dropped it too, and deserted after their leader. So the three sentences became two, and the second carried the day by elimination of the third which could not defeat both the others, and therefore chose to submit to one.” It seems clear in Radice’s translation that the senate voted for the banishment penalty. According to Riker 1986 Pliny proposed plurality rule because he thought that voters would vote sincerely. He tried to manipulate the Roman Senate by promoting a rule that would generate the decision he was in favour of: acquittal. However, and I share Riker’s view, he did not understand clearly that those in favour of the death penalty would manipulate the voting rule by voting strategically. The 1752 translation is unclear on this and a French translation by Annette Flaubert 2002 has a footnote in which it is said that acquittal carried the day! It remains that Pliny describes a situation that is manipulable by a coalition in the modern sense of Gibbard‐Satterthwaite: a group of individuals by voting strategically forces the voting rule to generate an outcome that the members of the group prefer to the outcome that would have prevailed if they had not voted strategically. 2.2. Ramon Lull. Ramon Lull ca. 1233‐1316 was born in Mallorca in a wealthy family. As a young man, he was a troubadour writing poetry and songs. At the age of thirty or so, while he was writing a song to a lady he loved, he saw Jesus‐Christ on the cross and from then on devoted his life to religion incidentally, abandoning his wife and his two children! . He believed to have three missions: writing books against the errors of the unbelievers, founding schools for teaching foreign languages, converting Jews and Moslems. He wrote about 290 books 260 reached us , some of them in Arabic on a variety of subjects, including, of course, religion, but also logic and mathematics, and astrology and alchemy. He is still highly considered in particular in Catalonia and Germany. As far as I know social choice theorists heard of Lull and also for that matter of Cusanus from a paper by McLean and London 1990 . McLean and London identified two sources: A novel entitled “Blanquerna” and another text whose title is “De Arte Eleccionis”. Since then a third text, “Artificium Electionis Personarum”, was called to our attention by scholars from Augsburg, in particular Pukelsheim see Hägele and Pukelsheim 2001 and 2008 . It is remarkable that
4 “Blanquerna” is considered as one of the first novels ever written in Catalan in Europe. In “Artificium Electionis Personarum” Arti , the first published among the three texts as well as in “Blanquerna” Blan and the later “De Arte Eleccionis” Arte , Lull recommends systems based on pair‐wise majority voting. In both Arti and Blan, all pair‐wise comparisons are done. There is however a difference, since in Blan, the ballot is organized in two stages. The set of voters and the set of possible elected persons are identical. At a first stage, voters have to reduce the size of these two sets. Lull considers a set of twenty voters to be reduced to seven. He describes a method to reach these seven: each voter is asked to select seven among nineteen I suppose this means that voters are not permitted to vote for themselves , and the seven collectively chosen are those who have the most votes. The next step, pair‐wise majority voting is among some also reduced set of candidates, but this set is not identical to the reduced set of voters as Lull in his example considers nine candidates why nine and from where are they coming, I do not know . The winner is the candidate who is victorious in most of the pair‐wise contests. This method is known today as Copeland method and more sophisticated versions are used in tournaments, in particular in sports2. McLean and Urken 1995 hesitate to provide a clear‐cut interpretation as they mention that Lull’s description could be Borda’s rule. Lull was conscious that the method could generate ties. He then proposed a tie‐breaking rule that, to say the least, is rather obscure: “The art recommends that these two or three or more should be judged according to art alone. It should be found out which of these best meets the four aforementioned conditions, for she will be the one who is worthy to be elected.” These four conditions are: which of them best loves and knows God, which of them best loves and knows the virtues, which of them knows and hates most strongly the vices, and which is the most suitable person. Since ties would happen, given an odd number of voters and strict preferences linear orderings , in case of a top cycle, the only way to break this would be to organize a deliberation among voters and proceed to a new ballot among tied candidates. I am not sure that this corresponds to what Lull had in mind. In “Arte”, the procedure is quite different even if it is still based on majority pair‐wise voting. It is based on successive eliminations. This rule is often known as the parliamentary procedure since it imitates the successive votes on bill proposals and amendments. It seems that Lull did not see that this method is highly agenda‐manipulable the outcome is strongly linked to the order in which the pair‐wise contests are organized and that it can select a candidate who is Pareto‐dominated that is a candidate could be elected even though all voters prefer another candidate. Saari 2008 gives a wonderful example of “electing Fred” even though Fred is Pareto‐dominated by three out of five other candidates. One of the only virtues of the rule is probably that it will not select a Condorcet loser a candidate that is beaten by all the other candidates in pair‐wise contests since the elected candidate won the last confrontation. In all Lull’s writings on voting, the ballot is supposed to take place within religious institutions In “Blan”, for instance, nuns have to select their abbess. According to Hägele and Pukelsheim 2001 , “…whether the electoral systems have actually ever been used is not known.” 2.3. Nicholas of Cusa. 2
Copeland was a mathematician at the University of Michigan. His paper entitled “A reasonable social welfare function” (1951) has never been published.
5 Nicholas of Cusa also known as Nicholas Cusanus, Nikolaus von Kues… , was born in Kues in 1401 in a wealthy family. Kues is a small town on the Moselle valley situated between Trier and Coblence. He studied at the universities of Heidelberg, Padua and Cologne. He had a very successful ecclesiastical career, becoming a bishop and a cardinal. He is generally considered as one of the greatest polymaths of the 15th century. He died in Todi Umbria in 1464. He participated in the Council of Basel in 1433‐34. It is during these years that he wrote his first major work “The Catholic Concordance” an English translation of “De Concordantia Catholica” was published in 1991 . In “The Catholic Concordance”, Cusanus devotes some paragraphs to the description of a voting method for the election of the Emperor of the Holy Roman Empire. He considers an example with ten candidates. Each voter attributes a digit, 1, 2 , 3… and 10 to the best candidate. Obviously, he assumes that the voter ranks the candidate without ties from the least preferred to the most preferred and give marks to candidates from 1 to 10 on the basis of this ranking. He writes: “ T he teller must add up the numbers by each name, and the candidate who has collected the highest total will be emperor.” This is clearly Borda’s rule as it is now known. Cusanus adds that “By this method innumerable malpractices can be avoided, and indeed no malpractice is possible. In fact, no method of election can be conceived which is more holy, just, honest, or free. For by this procedure, no other outcome is possible, if the electors act according to conscience, than the choice of that candidate adjudged best by the collective judgment of all present.” Cusanus does not mention the possibility of ties whose probability is not negligible when the set of voters is small . His phrase about electors acting according to conscience is somewhat similar to the phrase attributed to Borda but probably apocryphal that his method was for honest people. 2.4. Samuel von Pufendorf. Samuel von Pufendorf was born in 1632. He studied in Leipzig, Jena and Leiden and held professorships in Heidelberg 1661 and Lund 1670 . He left Lund to Stockolm to become a political‐jurisprudential councilor at the courts of Sweden. He then wrote a monumental history of Sweden. He spent the last years of his life in Brandenburg‐Prussia as appointed historian. He died in 1694. The book that presents an interest for social choice theorists and, incidentally, economists in general, is “De Jure Naturae and Gentium” “The Law of Nature and Nations” . It was published in 1672. Among the translations, the translation from Latin to French with many additional comments by Jean de Barbeyrac played an important role in the French Enlightenment, in particular it influenced both Diderot and Rousseau. Wulf Gaertner 2005 describes with more details Pufendorf’s contributions to voting and economics. In Book VII, Chapter II, section 18, one can read: “Thus those who fix a fine upon a man, at twenty units of value, may be united with those who fix it at ten units, against such as would acquit him altogether, and the defendant will be fined ten units, because this is agreeable to the majority of judges, in view of the fact that those in favour of the twenty, are included with those in favour of the ten.”
6 Pufendorf insists on the difference between quantitative options fines and qualitative options here acquittal . According to Gaertner, this can be viewed as single‐peaked preferences, with preference orderings from most preferred to least preferred being either twenty, ten, acquittal, or acquittal, ten, twenty, or ten, twenty, acquittal. Of course, in this case, the median option, here ten, is selected by the majority rule. Another possibility would be to consider this example as analogue to Pliny’s example: the judges in favour of a fine of twenty join those in favour of a fine of ten to defeat the acquittal option which is the option they ranked last. This possibility could justify a reference to Pliny’s letter in Pufendorf’s work as indicated by Gaertner. 3. The founding fathers. Even if Nicholas of Cusa proposed Borda’s rule more than three hundred years before Borda, and Ramon Lull’s description of elections are generally based on pair‐wise majority voting announcing, maybe, Condorcet, their works cannot be compared with those of Borda and Condorcet. Borda was a great applied scientist of his time and Condorcet’s contribution to human knowledge is still probably underestimated. What they left us on voting is not commensurate with what Cusanus or Lull left. I will add a third founding father, because of his influence on later thinkers: Bentham might be the father of the utilitarian social welfare function. 3.1. Jean‐Charles de Borda. Jean‐Charles de Borda was born in Dax in the South West of France in 1733 in a family of little nobility he was “Chevalier”, that is “knight” and not “Count” as written in Risse 2005 even though the pun “Count de Borda”/ “Borda count” opposed to the “Marquis de Condorcet” was amusing and clever . He studied at the Collège Royal Henry‐le Grand A College of Jesuits at that time in La Flèche, a small town at 40 kilometers from Le Mans the most famous pupil of this Collège is Descartes and La Flèche is also well known for being the city where David Hume settled while in France and where he wrote most of “A Treatise of Human Nature” He became a member of the military engineering corps against his father’s wishes who had preferred to see him as a magistrate. He worked on ballistics and could enter the “Académie Royale des Sciences”. He participated to the American war of independence as a French Navy officer but was taken prisoner by the British. Later, he worked on the metric system as chairman of the “Commission des Poids et Mesures” “committee of weights and measures” . He died in 1799. His work in social choice is rather limited: nine pages in “ Histoire de l’Académie Royale des Sciences, Année M.DCCLXXXI” that was published in 1784.In his “Mémoire sur les elections au scrutin”, Borda presents his system: the so‐called Borda count. Each voter ranks the candidates without ties and one point is attributed to the candidate ranked last in a voter’s ranking, two points are attributed to the candidate ranked just before the last one, etc. the top candidate obtaining a number of points that is equal to the number of candidates. Note that we could start from zero up to the number of candidates minus one, or even, as indicated by Borda, start from any number and add the same fixed number when we go from one rank to the rank that is just above it. The points obtained by a candidate are added and the winner s is are the candidate s who has have obtained the greatest number. But there is more in these nine pages. First, he gives an example where a plurality winner is a Condorcet loser. This demonstrates in his view that the plurality rule is flawed. On the other hand, there is no proof that, in non trivial cases, a Borda winner cannot be a Condorcet loser. However, he derives simple inequalities for the case when there is a Borda winner that coincides with a plurality
7 winner. A remark: as noted by McLean and Urken 1995 , there are typos in the French text. These typos are obviously not Borda’s mistakes but printers’ mistakes. 3.2 Jean Antoine‐Nicolas de Caritat, Marquis de Condorcet. Condorcet was born in 1743 in Ribemont near Saint‐Quentin North‐East of Paris . He was also educated by the Jesuits, first by a private Jesuit tutor and then at the “Collège des jésuites” de Reims. He studied mathematics in Paris and at 26 he entered the “Académie Royale des Sciences”. He was a friend of Turgot and as such interested in economics but also in politics and in the theory of elections. He was elected to the “Legislative Assembly” in 1791, becoming its president. In this position, he devoted most of his time to public education. Member of the “Convention”, he contested the Assembly’s right to judge the King, and then voted against capital punishment. He protested against the arrest of the “Girondins”. He wrote the famous “Esquisse d’un Tableau Historique des Progrès de l’Esprit Humain” while he was hiding. Finally, he is arrested. He died on April 7th , 1794 of poisoning or exhaustion Badinter and Badinter, 1988 . The 1785 “Essai” is an impressive piece of work. It has nearly 500 pages if we include the so‐ called “Discours Préliminaire”. If it is the most important book Condorcet devoted to elections, it is not the only one. Other important works include, among others, the “Lettres d’un Bourgeois de
New Heaven à un Citoyen de Virginie, sue l’Inutilité de Partager le Pouvoir Législatif entre Plusieurs Corps” and the “Essai sur la Constitution et les Fonctions des Assemblées Provinciales” see.Condorcet, 1986 . Most scholars have concentrated their attention to the “Preliminary Discourse” for several reasons, an obvious one being that some pages of the main text are covered by long probability calculations that are very similar, at first sight, to expressions we find today in works about the probability of pathologies for specific voting rules see, for instance, Gehrlein and Lepelley, 2011 . However, according to Bru and Crépel 1994 , one cannot eschew this main part of the “Essai”. In particular, according to them, how could we explain why some crucial parts gave raise to contradictory interpretations from some excellent social choice theorists? The basic theme of the 1785 “Essai” concerns the probability to take a correct decision. This is the now famous Condorcet’s Jury Theorem, where we have members of a jury for whom the probability to have the correct opinion is given by v and to be in error is given by e 1 v . If v is greater than 0.5, majorities are more likely to select the correct opinion is greater than v and this likelihood will increase with the number of voters. However, Condorcet was not certain that v e, and, since with e v, the result would be inversed, he was rather prudent. He wrote: “The assumption that e v is not absurd. For many important questions either complex or under the influence of prejudices or passions, it is likely that a poorly educated man will have an erroneous opinion. There are, consequently, a great number of points for which, the more we increase the number of voters, the more we can fear to obtain, with plurality, a decision in contradiction with truth so that a purely democratic constitution would be the worst of all for all these objects on which the people would not know the truth.” Condorcet then recommended that only enlightened men be attributed the prerogatives to make proposals of law. The popular assemblies would not been asked to vote on whether the law is useful or dangerous, but only if it is against justice or against the primary rights of men. A “pure” democracy could only be good for a very well educated people, so well educated that there had never been such a people, at least among the “great” people.
8 Condorcet is most known now for the example he presented pages lxJ of the “Preliminary Discourse” showing that majority rule could generate a cycle. This example is rather complicated, but Condorcet’s intention was not to uniquely present the paradox but to provide an analysis of the case he considered. There are 60 voters and three candidates A, B and C. The ranking are given by: 23 voters: ABC, meaning A ranked first, B, second and C third; 17 voters: BCA; 2 voters: BAC; 10 voters: CAB; 8 voters: CBA. One can see that a majority of voters 33 prefer A to B, a majority 35 prefer C to A and a majority 42 prefer B to C. In the main text of the “Essai”, Condorcet proposed a method to deal with the cycle problem. This method, rather obscure in Condorcet’s words, has been the object of a reconstruction by, among others, Young 1984 and Monjardet 1990 . Of course one can obtain a cycle very simply with three voters whose rankings are respectively ABC, BCA and CAB. Page clxxviJ of the “Discours”, Condorcet also alluded to Borda he does not name Borda, using the circumlocution “le Géomètre célèbre” “the famous Geometer” and gave an example showing that Borda’s rule could select another candidate than the Condorcet winner: 81 voters have the following rankings over three candidates A, B and C: 30 voters: ABC; 1 voter: ACB; 10 voters: CAB; 29 voters: BAC; 10 voters: BCA; 1 voter: CBA. Candidate A is a Condorcet winner he beats B only by 41 against 40 , but B is the Borda winner. That A is a better candidate than B seems obvious to Condorcet, not so to me. But this was the beginning of a long debate with still contemporary participants see, for instance Dummett 1984, 1997 , Emerson 2008 , Risse 2005 , Saari 1995, 2006 . Of course, again, a very simple example is possible, for instance with 19 voters: 10 voters: ABC; 9 voters: BCA.
9 Candidate A is a Condorcet winner and a plurality winner but B is the Borda winner. Page clxxix, Condorcet alluded to some kind of strategic voting indicating that Borda’s rule is not immune to this possible voters’ behavior. In the “Lettres d’un Bourgeois de New Heaven…”, Condorcet proposed that the Condorcet winner be selected if there is one, and if not proposed to select he candidate that won the most pair‐wise confrontation again, this is Copeland method that what suggested by Lull long before . For the selection of committees of k members to be selected in a set of 3k candidates, Condorcet recommended in 1792 that each voter partitions the set of candidates in three set of k candidates and ranked the three sets a set of k most‐preferred candidates, a set of k intermediately‐preferred candidates, and a set of k least preferred candidates . Each voter indicates his k most‐preferred candidates and, as a supplementary list, the k “intermediate” candidates. If at least k candidates obtain a majority, this is done by selection of the k candidates who have obtained the most votes. If not, one considers the supplementary lists. It is at the same time original and unorthodox, but unfortunately still to be formally studied. The works of Condorcet prompted a number of studies by Laplace, Lhuillier, Lacroix, Morales, Daunou see McLean and Urken, 1995 . 3.3 Jeremy Bentham. I will be brief on Bentham 1748‐1832 because his role on the development of social choice was rather indirect. Even though the paternity of utilitarianism cannot be attributed to Bentham, he did popularize it, in particular through the publication in 1789 of “An Introduction to the Principles of Morals and Legislation”. Basically, utilitarianism belongs to the sphere of individual, not social, ethics. However, from the utilitarian basic principles, in David Wiggins’ s words 2006 , “… T he new moral philosophy of Jeremy Bentham, James Mill, and James’s son, John Stuart, came to be linked with a stupendous programme of social and political reform”. The search for the “greatest happiness of the greatest number” is the utilitarian motto, generally attributed to Bentham. It is unclear whether this means that Bentham had in mind some kind of utilitarian social welfare function. However, one can read, at the very beginning of “An Introduction”: “The community is a fictitious body, composed of the individual persons who are considered as constituting as it were its members. The interest of the community then is, what? the sum of the interests of the several members who compose it.” One can take this as a, somewhat vague, definition of a utilitarian social welfare function and Bentham can then be viewed as an ancestor of Bergson and Samuelson. Arrow’s contribution was obviously prompted by Bergson’s paper as stated page 22 of “Social Choice and Individual Values”. Furthermore, in the same page, Arrow alludes to Bentham. 4. Social Choice during the 19th century. During the 19th century, in spite of the development of democratic institutions, the theory of social choice has been rather dormant, with a few exceptions. A basic question that had to be solved in America was the apportionment question: given a state with its population what is the correct number of representatives that would respect the principle of equality of the citizens over the various states? The equality here means that each
10 representative should represent the same number of persons. Although this looks simple, and maybe secondary, the difficulty arises from the fact that the number of representatives are integers. The problem of apportionment is identical to the problem of proportional representation. A number of methods have been devised, in particular by some of the founding fathers of American democracy such as Jefferson and Hamilton, and rediscovered later by Europeans generally in the context of proportional representation. The definite book on this topic is Balinski and Young 1982 . It includes many historical developments. Two scholars have to be mentioned. The first one is Charles Lutwidge Dodgson 1832‐98 , better known as Lewis Carroll, the author of “Alice Adventures in Wonderland” and the second one is E.J. Nanson 1850‐1936 . Dodgson was a logician at Oxford University. He wrote so‐called pamplhlets that are reproduced in Black 1958 . Through time, he proposed various methods. For instance, even though he did not know either Borda nor Condorcet so that he never mentioned them, he proposed to use Borda’s rule and a run‐off between the two top candidates. Later, he proposed to select the Condorcet winner if there is one, and if not, to have recourse to Borda’s rule, a rule that is generally associated with Duncan Black.. Nanson spent most of his life in Australia as a professor of mathematics at the University of Melbourne. It is quite impressive that he knew the French literature on voting Condorcet as well as Borda . He wanted to promote procedures that select the Condorcet winner when there is one. He eventually proposed a method based on the Borda’s rule, used in an iterative way. At each stage the candidates who failed to obtain an average Borda score are eliminated. Then, the procedure starts again on the basis of the modified rankings and is repeated until only one candidate remains who must be the Condorcet winner, if there is one . 5. Arrovian social choice. A standard view of the British economists at the end of the 19 th century and the beginning of the 20th century was that welfare, utility, satisfaction etc. had a money measure Pigou, 1932 . It seems clear that in this case one could use a utilitarian social welfare function where the social welfare is the addition of the individual welfares. Since individual utility was measured in monetary terms, the problems of scales, origin, comparability etc. were solved by definition. Maximizing social welfare amounted to finding a maximum for the utilitarian social welfare function, given that individual utility functions were fixed, the variables being the social states, whatever this term of social state covers. However at the same time under the influence of Walras and above all Pareto, economists wanted to get rid of the measurement of utility problem. The solution was to use ordinal utility or even the underlying preference relation. With ordinal utility functions, still numerical functions, the real numbers/utilities could only be meaningfully compared according to the relation . All the other mathematical properties defining the field of real numbers were rejected. A kind of corollary to the ordinalism thesis was that interpersonal comparisons had to be excluded too, even when these comparisons are limited to the relation , that is, one could not assert that the utility of individual i in state x is, say, greater than the utility of individual j in state y. On this basis, the only possible concept relative to the social goodness of a social state was Pareto optimality: a social state x is optimal if there is no other feasible social state y such that all individual utilities are greater for y than for x or in its strong version, such that all individual utilities are at least as great for y than for x and one is greater . In the 1930s, interpersonal comparisons are, however, reintroduced as virtual compensations by economists as famous as Hicks, Kaldor, Harrod and Scitovsky see the books
11 edited by Arrow and Scitovsky 1969 and Baumol and Wilson 2001 . The principle of compensation is that in a change of social states, say from x to y, individuals who gain in the changes could virtually compensate those who lose, making the change a Pareto improvement, that is, after compensation every individual has a utility greater due to the change. It is obvious that this procedure entails interpersonal comparisons. Furthermore, it has been shown that it was not immune to paradoxes. It is in this context that Bergson proposed the new notion of social welfare function in 1938. The form of the function has been modified by Samuelson 1947 and it is in this form proposed by Samuelson that the function is generally presented. In Samuelson’s version, the social welfare function, say f, associates a real number to a list of individual utilities u1,…,un of individuals 1,…,n for some social state belonging to a fixed set of social states. The individual utility functions are fixed. For instance, if we have a Cobb‐Douglas utility function defined over the positive orthant of a k‐dimensional Euclidean space for individual i, say, ui x 3/5 x11/k x21/k … xk1/k , the parameters 3/5 and 1/k are fixed, whatever the variables x1 … xk. are To impose to such a social welfare function a Paretian property is to assume that ∂f/∂ui 0 loosely speaking, social welfare increases or decreases when individual i’s utility increases or decreases , all things being equal . The purpose of the function is then to select some Pareto‐optimal social state through classical maximization., and this for public policy. However, it remains to know how and who will construct the function. In some sense, Arrow provided a reply to this question, unfortunately doubly negative, with his impossibility theorem. Arrow’s analysis definitely marks the birth of modern social choice theory. 5.1. Formal preliminaries. I will introduce the standard social choice framework. One considers a set X of social states or states of the world . These social states must be interpreted as extremely detailed descriptions of atemporal situations. In particular, a given social state will include descriptions of characteristics pertaining to agents. Preferences over X are binary relations that will be assumed to be complete. I will denote the preference relation for social states x and y, x y and it will mean x is at least as good as y. The asymmetric part strict preference , x is better than y will be denoted by x y and defined, given completeness, by not y x, and the symmetric part indifference will be denoted by x y and defined by x y and y x. The preference is said to be complete if for all x and y in X either x y or y x; x and y are always comparable. The set of agents will be denoted by N. Each individual agent has a preference that is a complete preorder over X. Her preference is a transitive binary relation. If she finds x as least as good as y and y at least as good as z, then she must find that x is at least as good as z. One should note that in this case both the strict preference relation and the indifference relation are also transitive. Individual i’’s preference will be denoted by i. In the following analysis, the social preference, denoted by S, will be assumed to be complete, and will satisfy some rationality condition. We will consider three different conditions. i
Transitivity.
For all x, y and z in X, x ii
S y and y
S z
x
S z.
S z
x
S z.
Quasi‐transitivity.
For all x, y and z in X, x
S y and y
12 iii
‐acyclicity.
There is no finite subset of X, x1,…,xk , for which x1
x x
S 2, 2
x
x
S 3,…, k‐1
x
x
S k and k
x
S 1.
A complete binary relation satisfying transitivity is a complete preorder. In this case the social states X is finite, it means that the social states can be ranked from a top element to a bottom element with possible ties preference has the same characteristics as the individual preferences. When the set of social When the set of social states X is finite, it means that the social states can be ranked from a top element to a bottom element with possible ties Let be the set of complete preorders over X. The agents’ preferences are given by a profile which is a function from the set of individuals N into . This is a kind of labeling operation. It assigns a complete preorder to each individual. When N is finite of size n, we have the usual list of individual preferences 1,…, n . We will consider two types of aggregation rules. They will be called respectively aggregation functions and social choice functions the definition of a social choice function will be given in the sub‐section devoted to Gibbard‐Satterthwaite theorem . An aggregation function is a function f : S which associates to each possible profile a social preference S over X. When the social preference S is a complete preorder, the aggregation function is the classical Arrovian social welfare function Arrow 1951 . In the double finite case the set of individuals and the set of social states being finite , given a complete ranking of the social states by each individual, a social welfare function gives a complete ranking of the social states at the social/collective level. Individual and social rationalities are identical. When S is complete and satisfies ‐acyclicity, the aggregation function, following Sen 1970 will be called a social decision function and, when it satisfies quasi‐transitivity, it will be called a QT‐social decision function. 5.2. Arrow’s impossibility theorem. The conditions introduced by Arrow do not concern a particular class of aggregation functions. Their definitions never necessitate a condition of transitivity or other collective rationality property. They are valid for all aggregation functions. Condition U Universality . A profile may include any individual complete preorder. This means that the individual preferences are not restricted; they are complete preorders, but no extra rationality conditions are postulated. For instance with 3 social states, there are 13 complete preorders. Each of these 13 is feasible. Condition I Independence of irrelevant alternatives . Consider two social states a and b and two profiles 1 and 2. If for each individual, the preference regarding a and b is the same in profile 1 and profile 2, then the social preference regarding a and b must be identical for both profiles. This means that the information used in the aggregation is myopic, and, given the definition in terms of preferences, ordinal. For instance, when there is a set of 10 social states, the fact that individual j ranks a first and b tenth will have the same effect as if she had ranked a first and b second. Also, one should remark that this condition uses two profiles; it is a multi‐profile condition. Since preferences could be represented by ordinal utility functions, this means that the individual utility functions are possibly different so that an Arrovian social welfare function is quite different from a Bergson‐Samuelson social welfare function.
13 Condition P. Pareto principle . Let a and b be two social states and be a profile in which every individual prefers a to b for all i N, a i b , then a is socially preferred to b a S b . This is simply a unanimity principle. As a consequence, if individuals can either prefer a to b or b to a so that there is a tiny diversity among the feasible profiles which is the case, of course, by Condition U , then it is impossible that the aggregation function f be a constant function. Are then excluded functions that would be based on a moral or religious code, independently of the individuals constituting the society. A dictator would be an individual i such that for any social states x and y, the aggregation function would generate x S y whenever x i y. One can see that a dictator imposes his strict preferences to the society in Arrow’s framework, he does not impose his indifferences . Condition D No‐dictatorship . There is no dictator. Arrow’s Impossibility Theorem. If N is finite and includes at least two individuals, and if there are at least three social states, there is no social welfare function satisfying conditions U, I, P and D. Dishonest comments were made about the impossibility of democracy. First the theorem is about transitivity whose violation entails that there are three social states. With only two, there is no problem. Furthermore, one can challenge the conditions. Condition U was in fact challenged in Black 1948 that is even before the publication of Arrow’s first paper. But it is Condition I that has been contested the most often, among others by Sen 1970 and by Saari 1995 . One can also consider a weakening of the condition of transitivity to quasi‐transitivity of the social preference. In this case, that is for QT‐social decision function, Gibbard showed that if the function satisfied Conditions U, I and P, we obtained an oligarchy. An oligarchy is a group of individuals who have the power of a dictator when they agree and such that each member has a veto power that is, loosely speaking, a power sufficient to preclude that the social preference be inverse of his preference . If the number of oligarchs is small, we are not so far from dictatorship. As was shown by Mas‐Colell and Sonnenschein, there is not much to gain from a further weakening to acyclicity of the asymmetric part of the social preference, that is in considering social decision functions. Independently of the beauty of Arrow’s result, it is most remarkable that it was developed in an elegant framework that will become the framework of the whole subject for the origin of this framework, see Suppes, 2005 . 5.3. Sen and the Paretian liberal paradox. In a six‐page article published in the Journal of Political Economy Sen, 1970a , Sen introduces a notion of individual rights within the Arrovian framework of social choice. These six pages had a fundamental importance on the development of studies on non‐welfaristic aspects of normative economics. At about the same time, Kolm 1972, 1997 introduced the notions of fairness, equity and social justice using rather standard microeconomics models for instance, so‐called Edgeworth boxes . Sen introduced two conditions of liberalism, or individual freedom, the second one being a weakening of the first that reveals sufficient to get the result.
14 Condition L Liberalism . For each individual i N, there are two social states ai and bi such that we get ai S bi whenever ai i bi and bi S ai whenever bi i ai. The second condition sates that at least two individuals enjoy liberalism as defined above. Condition ML Minimal Liberalism . There exist two individuals i and j, two social states a and b for i and two social states c and d for j such that a S b whenever a i b, b S a whenever b i a, c S d whenever c j d and d S c whenever d j c. Sen’s impossibility theorem. If there are at least two individuals and two social states, there is no social decision function satisfying conditions U, P and ML. One can note that there is no need of the finiteness of N and of condition I and that the result already holds for two social states. To understand the power of this result which, incidentally, has no real meaning in a voting context since this would mean that two individuals have a partial dictatorship power , one has to consider that social states are descriptions of the states of the world as detailed as one wishes with possible personal elements. I will give an example that is adapted from Salles 2010 . I have two individuals i and j. The social states a and b are identical except that in a individual i eats legs of lamb with garlic and in b without garlic. Individual i is a garlic addict and, accordingly, strongly prefers a to b. The social states c and d are also identical except that in c individual j puts some Guerlain’s L’instant Magic perfume before going to sleep and, in d, does not. Individual j has a passion for L’instant Magic and so strongly prefers c to d. Now imagine that j is i’s wife and that she hates garlic as much as her husband hates perfume in general and L’instant Magic in particular. On this basis, let us suppose that the two individuals’ preferences are the following:
d i a i b i c b j c j d j a. Since there is nothing, in my view, more personal than culinary tastes or tastes related to smells, the social states a and b perfectly fits Mill’s notion of personal sphere regarding individual i, and likewise for individual j, c and d Mill, 1859 . This illustration exemplifies the difficulty one can encounters with this notion of personal sphere in presence of what the economists call externalities. We will assume that our society is only composed of i and j. It is in fact very easy to consider a general profile with appropriate preferences for the other individuals. By Condition P, since both individuals prefer d to a, we have d S a, and since they both prefer b to c, b S c. Now, since i prefers a to b, by Condition ML, a S b. Since individual j prefers c to d, then by Condition ML, c S d. We have accordingly a cycle: a S b, b S c, c S d, d S a. A major by‐product of this paper is the tremendous development of the freedom of choice literature see for instance Dowding and van Hees, 2009 . 5.4. Gibbard‐Satterthwaite theorem. As previously noted the problem identified by Pliny was the problem of strategic voting. I also mentioned that Condorcet seemed to allude to this problem regarding Borda’s rule. According to
15 a gossip, most probably apocryphal, Borda would have responded that his method was for honest people. The general result on this topic was obtained independently by Gibbard 1973 and Satterthwaite 1975 . A fundamental contribution due to Dummett and Farquharson 1961 , more than ten years before, in spite of being published in Econometrica was rather unnoticed. We will assume that X, the set of social states, here, say, candidates, is finite. and, for reasons of simplicity, that individual preferences are given by linear orders, denoted by i. This means that individuals rank the candidates without ties. A profile will then be a list of individual rankings X. This means that rather than selecting a 1,…, n . A social choice function is a function f social preference as in the case of aggregation functions, a social choice function selects, given a profile, a candidate. We will say that individual i manipulates the social choice function f in profile 1,…, n if there is a profile ’ which is identical to profile except for the preference of individual i such that f ’ i f . For simplicity, imagine that the profile is a profile of sincere individual preferences. Individual i manipulates f if, by misrepresenting her preference lying , she can force the function to generate a result that she prefers to the result that would have been obtained otherwise. We will assume that the social choice function is surjective: for any candidate x, there is a profile such that x f . This basically means that there is no fictitious candidate. A consequence is, of course, that if there are at least two candidates, the function cannot be a constant function. A dictator for a social choice function f will be an individual i such that for all profile , f for all x f .
i x,
Gibbard‐Satterthwaite theorem. Suppose that there are at least two individuals and three candidates, that all linear orderings individual preferences are permissible and that f is surjective and non‐manipulable. Then there is a dictator. This theorem has been at the origin of a tremendous number of contributions in social choice theory, but also in public economics and is strongly related to implementation theory Jackson 2001 , Maskin and Sjöström 2002 . 5.5. Black, single‐peakedness and majority rule. In 1948, that is even slightly before Arrow, Duncan Black 1908‐1991 , a British economist, introduced the notion of single‐peaked preference in a paper published in the Journal of Political Economy. He studied the effects of this assumption on the outcomes generated by majority rule. He proved among other results what is now called the ‘median voter’ theorem. He used a kind of geometrical setting. Assume that the set of options candidates, social states or whatever is a closed interval a,b . Furthermore, assume that individuals have ordinal continuous utility functions ui that until it reaches a maximum, and then decreasing until it reaches b. In the following picture, we have the curves of five functions representing preferences.
16
Figure 1. Black’s single‐peakedness. In this figure, u1 reaches its maximum for option a, u2 for option x2 etc. Individual 3 is the median individual and the option selected by the majority rule is x3. In the figure, the functions are strictly concave, but the definition allows strict quasi‐concavity. When the space of options is no longer one‐dimensional, difficulties arise see Austen‐Smith and Banks 1999 , Schofield 2008 . I will now present Black’s analysis in a discrete setting. The discrete version of single‐ peakedness is due to Arrow 1951 . First let me define majority rule. i assume that individual preferences are given by complete preorders. Majority rule is an aggregation function such that for all distinct options x and y, x S y if and only if the number of individuals i for whom x i y is than the number of individuals for whom y i x, and y S x otherwise. The following definition of single‐peakedness is adapted from Sen 1966 . A set of complete preorders over X satisfies the condition of single‐peakedness over a,b,c X if either a b and b c, or there is an option among the three options, say, b, such that b a or b c. We will say that a set of complete preorders satisfies the condition of single‐peakedness if it satisfies the condition of single‐peakedness over all x,y,z X. The following figure is a geometrical representation of this condition over a,b,c
Figure 2. Black’s single‐peakedness over a,b,c Black’s theorem. Let us assume that there are at least two individuals and three options, and that all individual preferences belong to a set of complete preorders satisfying the condition of single‐peakedness. Let us assume further that the number of individuals who are not indifferent between x, y, z is odd for any x,y,z X. Then the majority rule is a social welfare function satisfying conditions I, P and D. Since the majority rule obviously satisfies condition I, P and D, this simply means that S is transitive. The specific condition on the number of non‐indifferent individuals can appear as problematic, but it is not so problematic since if we drop it we still get a QT‐social decision function S is then transitive . The literature of this sort, where individual preferences are restricted by some kind of super‐rationality is abundant and has been excellently surveyed in Gaertner 2001 . A special kind of restriction refers to the so‐called economic domains and is, of course, related to Black’s analysis since in the standard micreconomic framework, individual preferences are continuous, convex or strictly convex etc. see Le Breton and Weymark, 2010 . 5.6. Harsanyi and utilitarianism.
17 John Harsanyi 1920‐2000 , in two papers 1953, 1955, 1976 , considers social choice in risky environments. In his 1955 paper, he presents a major result that, in some sense, justifies utilitarianism, more precisely a weighted version of utilitarianism, from a rather technical perspective. The set of social states is supposed to be a set of lotteries, that is the probability distributions over a finite set of prizes. If all probabilities are permitted, this set is infinite and uncountable. Individuals have a preference over this set of lotteries given by a complete preorder and representable by a utility function. In the lotteries setting, rather than using ordinal utility functions, economists generally adopt a framework developed by von Neumann and Morgenstern 1953 for dealing for risky analysis in game theory, the so‐called von Neumann‐Morgenstern utility functions. In fact, given appropriate assumptions on the set of lotteries and the set of complete preorders over the set of lotteries some of these assumptions are topological assumptions , it can be shown that there exists a utility function representing the complete preorder and having the so‐called expected utility property. If we assume that a lottery x is given by k prizes x1,…,xk and a probability distribution p1,…,pk, p1 being the probability of receiving prize x1 etc., the utility function u is said to satisfy the expected utility property if:
u(x) =
∑
k j =1
p j u ( x j ) .
This means that the utility associated to the lottery x is the sum of the utilities associated to the prizes, weighted by the probabilities of receiving these prizes. An important consequence of this property is that the utility function is not only unique up to an increasing transformation, as in the case of ordinal utility functions, but unique up to a specific form of the increasing transformation: a positive affine transformation; if u is a von Neumann‐Morgenstern utility function representing a complete preorder , then v α u β, where α and β are real numbers and α 0, is also a von Neumann‐Morgenstern utility function representing this same . As a major consequence the differences of utilities can be compared according to the relation , which was not possible for ordinal utility functions. Such functions are said to be cardinal. In the following presentation which is essentially due to Weymark 1991 three conditions are introduced. Since the prizes are fixed, a lottery will be assimilated to the associated probability distribution p p1,…,pk . Condition P‐I Pareto indifference . Let p and q be two lotteries. If for all i N, p i q, then p
S q.
If all individuals are indifferent between two lotteries, so does the society. Condition S‐P Strong Pareto . Let p and q be two lotteries. If for all i N, p i q, and for some i N, p i q, then p S q. Condition I‐P Independent prospects . For each i N, there exist two lotteries pi and qi such that pi i qi and for all j i pi j qi. For instance, let x1 be a piece of cheesecake and x2 a piece of chocolate cake. Assume that x1 x2 and for all j i x1 x2, then i will prefer lottery 1,0,0,…,. to lottery 0,1,0,0…,0 , but all the other individuals are going to be indifferent between the two lotteries. In Harsanyi’s theorem, both individual utility functions and the social welfare function w are supposed to be von Neumann‐Morgenstern utility functions. The theorem is then the following this presentation is Weymark’s presentation .
18 Harsanyi‐Weymark’s Theorem. 1 If condition P‐I is satisfied, then there exist real numbers ai such that S is represented by w ∑ ai ui. 2 if condition S‐P is also satisfied, the real numbers ai are positive. 3 If condition I‐P is also satisfied, the ai are unique up to a positive factor of proportionality. In 1 , it is not said that the weights attached to the individual utility functions are positive. With a negative weight, an increase in individual utility would decrease the social welfare. The result in 3 can be considered as the theorem about weighted utilitarianism. A number of people have challenged the use of von Neumann Morgenstern utility functions. Diamond 1967 in particular has criticized the assumption that the social preference could satisfy the assumptions introduced by von Neumann and Morgenstern. Furthermore, a number of authors have criticized Harsanyi by arguing that his utilitarianism could not be associated with classical utilitarianism, mainly because of the von Neumann‐Morgenstern necessary framework see Sen 1976 , Roemer 1996 and contributions in Fleurbaey et al. 2008 . 6. Conclusion. In this conclusion, I would like to explain my choices regarding Section 5. First, I wanted to consider results that are well established and had an important descent. Second, and it is related to my first point, I have not described more recent major trends such as judgment aggregation see List and Puppe, 2009 , freedom of choice, or empirical social choice. Third, I did hesitate about the inclusion of Nash’s epoch‐making contribution to bargaining theory and of voting games, in particular Nakamura’s theorem 1979 . The real difficulty is that social choice has some rather fuzzy frontiers. The overlap with political economy, public economics, game theory, political philosophy, formal political science, welfare economics, normative economics, social ethics, and I probably forget some areas, is quite large, as exemplified by Sen’s book 2010 as far as social justice is concerned. To have a good view of the present state of the subject, I would recommend the Arrow, Sen and Suzumura Handbook 2002, 2010 , the Anand, Pattanaik and Puppe Handbook 2009 and the introductory text by Gaertner 2009 . References Anand, P., Pattanaik, P.K. and C. Puppe eds. 2009 , The Handbook of Rational and Social Choice, Oxford: Oxford University Press. Arrow, K.J. 1950 , ‘A difficulty in the concept of social welfare’, The Journal of Political Economy, 58, 328‐346. Arrow, K.J. 1951 , Social Choice and Individual Values, Second Edition, 1963, New York: Wiley. Arrow, K.J. and T. Scitovsky eds. 1969 , Readings in Welfare Economics, Homewood, IL: Irwin. Arrow, K.J., Sen, A.K. and K. Suzumura eds. 2002, 2010 , Handbook of Social Choice and Welfare, Volume 1 2002 , Volume 2 2010 , Amsterdam: Elsevier. Austen‐Smith, D. and J.S. Banks 1999 , Positive Political Theory I, Collective Preference , Ann Arbor: The University of Michigan Press. Badinter, E. and R. Badinter 1988 , Condorcet. Un Intellectuel en Politique, Paris : Fayard.
19 Balinski, M. and H.P. Young, Fair Representation. Meeting the Ideal of One Man, One Vote, New Haven: Yale University Press. Baumol, W.J. and C.A. Wilson eds. 2001 , Welfare Economics, Volume 1, Cheltenham: Edward Elgar. Bentham, J. 1789, 1970 , An Introduction to the Principles of Morals and Legislation, Oxford: Oxford University Press. Bergson, A. 1938 , ‘A reformulation of certain aspects of welfare economics’, Quarterly Journal
of Economics, 52, 310‐334, and in Arrow and Scitovsky 1969 , 7‐38. Black, D 1948 , ‘On the rationale of group decision making’, The Journal of Political Economy, 56, 23‐34. Black, D. 1958 , The Theory of Committees and Elections, Cambridge: Cambridge University Press, revised second edition edited by I. Mac Lean and al., Boston: Kluwer. Borda, J.‐Ch. de 1784 , ‘Mémoire sur les élections au scrutin’, in Histoire de l’Académie des Sciences pour 1781, Paris : Imprimerie Royale, pp. 657‐665. Black, D 1948 , ‘On the rationale of group decision making’, The Journal of Political Economy, 56, 23‐34. Bru, B. and P. Crépel eds. 1994 , Condorcet. Arithmétique Politique : Textes Rares et Inédits, Paris : Institut National d’Etudes Démographiques. Condorcet, J.A‐N.de Caritat, Marquis de 1785 , Essai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix, Paris : Imprimerie Royale. Condorcet, J. A.‐N. de Caritat, Marquis de 1986 , Sur les Elections et Autres Textes, Paris : Fayard. Dowding, K. and M. van Hees, ‘Freedom of choice’, in Anand, P., Pattanaik, P.K. and C. Puppe eds , The Handbook of Rational and Social Choice, Oxford: Oxford University Press, pp. 374‐ 392. Dummett, M. 1984 , Voting Procedures, Oxford : Oxford University Press. Dummett, M. 1997 , Principles of Electoral Reform, Oxford: Oxford University Press. Dummett, M. 2005 , ‘The work and life of Robin Farquharson’, Social Choice and Welfare, 25, 475‐483. Dummett, M. and R. Farquharson 1961 , ‘Stability in voting’, Econometrica, 29, 33‐43. Emerson, P. ed. 2007 , Designing an All‐Inclusive Democracy, Heidelberg: Springer. Farquharson, R. 1969 , Theory of Voting, Oxford: Blackwell. Fleurbaey, M., Salles, M. and J.A. Weymark eds. 2008 , Justice, Political Liberalism, and utilitarianism. Themes from Harsanyi and Rawls, Cambridge: Cambridge University Press. Gaertner, W. 2001 , Domain Conditions in Social Choice Theory, Cambridge: Cambridge University Press.
20 Gaertner, W. 2005 , ‘De jure naturae et gentium : Samuel von Pufendorf’s contribution to social choice theory and economics’, Social Choice and Welfare, 25, 231‐241. Gaertner, W. 2009 , A Primer in Social Choice Theory, Oxford: Oxford University Press. Gehrlein, W.V. and D. Lepelley 2011 , Voting Paradoxes and Group Coherence, Heidelberg: Springer. Gibbard, A. 1973 , ‘Manipulation of voting schemes: A general result’, Econometrica, 41, 587‐ 217. Guilbaud, G.‐Th. 1952 , ‘Les théories de l’intérêt général et le problème logique de l’agrégation’,
Economie Appliquée, 5, 501‐551. Hägele, G. and F. Pukelsheim 2001 , ‘Llull’s writings on electoral systems’, Studia Lulliana, 41, 3‐ 38. Hägele, G. and F. Pukelsheim 2008 , ‘The electoral systems of Nicholas of Cusa in the Catholic Concordance and beyond’, in Christianson, G, T.M. Izbicki and CM Bellito eds. , The Church, the Councils, & Reform: The Legacy of the Fifteenth Century, Washington: The Catholic University of America Press. Harsanyi, J.C. 1953 , ‘Cardinal utility in welfare economics and in the theory of risk‐taking’, Journal of Political Economy, 61, 434‐435. Harsanyi, J.C. 1955 , ‘Cardinal utility, individualistic ethics, and interpersonal comparisons of utility’, Journal of Political Economy, 63, 309‐321. Harsanyi, J.C. 1976 , Essays on Ethics, Social Behavior, and Scientific Explanation, Dordrecht: Reidel. Jackson, M.O. 2001 , ‘A crash course in implementation theory’, Social Choice and Welfare, 18, 655‐708. Kolm, S‐C 1972 , Justice et Equité, Paris : Editions du C.N.R.S.. Kolm, S.‐C 1997, Justice and Equity, Cambridge Mass. : MIT press. Le Breton, M. and J.A. Weymark 2010 , ‘Arrovian social choice theory on economic domains’, in Arrow, K.J., Sen, A.K. and K. Suzumura eds. , Handbook of Social Choice and Welfare, Volume 2, Amsterdam: Elsevier, pp.191‐299. List, C. and C. Puppe 2009 , ‘Judgment aggregation’, in Anand, P., Pattanaik, P.K., and C. Puppe eds. , The Handbook of Rational and Social Choice, Oxford: Oxford University Press, pp. 457‐ 482. Maskin, E. and T Sjöström 2002 , ‘Implementation theory’, in Arrow, K.J., Sen, A.K. and K. Suzumura eds. , Handbook of Social Choice and Welfare, Volume 1, Amsterdam: Elsevier, pp. 237‐288. McLean, I and J. London 1990 , ‘The Borda and Condorcet principles: three medieval applications’, Social Choice and Welfare, 7, 99‐108.
21 McLean, I. and A.B. Urken eds. 1995 , Classics of Social Choice, Ann Arbor: The University of Michigan Press. Mill, J.S. 1859 , On Liberty, London, in Gray, J. ed. 1991 , John Stuart Mill, On Liberty and Other Essays, Oxford: Oxford University Press. Monjardet, B. 1990 , ‘Sur diverses formes de la “règle de Condorcet” d’agrégation des preferences’, Mathématiques, Informatique et Sciences Humaines, 54, 33‐43. Nakamura, K. 1979 , ‘the vetoers in a simple game with ordinal preferences’, International
Journal of Game Theory, 8, 55‐61. Nicholas of Cusa 1433 1991 , The Catholic Concordance, edited and translated by P.E. Sigmund, Cambridge: Cambridge University Press. Pline 2002 , Lettres, Livres I à X, Paris : Flammarion. Pliny 1969 , Letters, Books VIII‐X, Panegyricus, translated by Betty Radice, Cambridge Mass. : Harvard University Press Loeb Classical Library . Pigou, A.C. 1932 , The Economics of Welfare, London: MacMillan. Riker, W.H. 1986 , The Art of Political Manipulation, New Haven: Yale University Press. Risse, M. 2005 , “Why the count de Borda cannot beat the Marquis de Condorcet”, Social Choice and Welfare, 25, 95‐113. Roemer, J.E. 1996 , Theories of Distributive Justice, Cambridge Mass. : Harvard University Press. Saari, D.G. 1995 , Basis Geometry of Voting, Heidelberg: Springer. Saari, D.G. 2006 , ‘Which is better: the Condorcet or Borda winner?’, Social Choice and Welfare, 26, 107‐129. Saari, D.G. 2008 , Disposing Dictators, Demystifying Voting Paradoxes: Social Choice Analysis, Cambridge: Cambridge University Press. Salles, M. 2010 , ‘On rights and social choice’, working paper, CREM, Université de Caen‐Basse‐ Normandie Samuelson, P.A. 1974 , Foundations of Economic Analysis, Cambridge Mass. : Harvard University Press. Satterthwaite, M.A. 1975 , ‘Strategy‐proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions’ Journal of Economic Theory, 10, 187‐217. Schofield, N. 2008 , The Spatial Model of Politics, Abingdon: Routledge. Sen, A.K. 1966 , ‘A possibility theorem on majority decisions’, Econometrica, 34, 491‐499. Sen, A.K. 1970 , Collective Choice and Social Welfare, San Francisco: Holden‐Day.
22 Sen, A.K. 1970a , ‘The impossibility of a Paretian liberal’, Journal of Political Economy, 78, 152‐ 157. Sen, A.K. 1976 , ‘Welfare inequalities and Rawlsian axiomatics’, Theory and Decision, 7, 243‐ 262. Sen, A.K. 2009 , The Idea of Justice, London: Allen Lane. Suppes, P. 2005 , ‘The pre‐history of Kenneth Arrow’s social choice and individual values’, Social Choice and Welfare, 25, 319‐326. Tangian, A. 2008 , A mathematical model of Athenian democracy, Social Choice and Welfare, 31, 537‐572. Von Neumann, J. and O. Morgenstern 1953 , Theory of Games and Economic Behavior, Princeton: Princeton University Press. Weymark, J.A. 1991 , ‘A reconsideration of the Harsanyi‐Sen debate on utilitarianism’, in Elster, J. and J.E. Roemer eds. , Interpersonal Comparisons of Well‐Being, Cambridge: Cambridge University Press, pp. 255‐320. Wiggins, D. 2006 , Ethics. Twelve Lectures on the Philosophy of Morality , Cambridge Mass. : Harvard University Press. Young, H P. 1988 , ‘Condorcet’s theory of voting’, American Political Science Review, 82, 1231‐ 1244.
