Social Choice has to do with the aggregation of individual preferences to determine an overall social preference or social choice. Over 200 years ago a French enlightenment philosopher by the name of the Marquis de la Condorcet noticed that there is no straightforward way to do this and discovered what has come to be called the Paradox of Voting. The Paradox of Voting states that, given the distribution of voter’s preferences, in certain cases there may be no solution which obeys certain common sense rules.
A number of people have worked on this problem over the years including the Rev. C. L. Dodgson better known as Lewis Carroll who wrote “Alice in Wonderland.” In the 1950s, Kenneth Arrow wrote a book entitled “Social Choice and Individual Values” in which he formally and mathematically proved that Social Choice is impossible if a possible solution has to satisfy a set of rational and ethical criteria. His work was essentially a formalization of Condorcet’s Paradox of Voting.
Various people have tried to water down Arrow’s assumptions in an attempt to find Social Choice solutions that are possible according to the watered down set of criteria. There’s also the possibility that Arrow’s criteria, as originally conceived, are somehow wrong.
Social Choice is an important field because, if in fact Social Choice is impossible, there is no solid theoretical ground for a satisfactory version of political democracy and also any possible version of economic democracy would likewise be invalidated.
To delve more deeply into the subject we note that Pliny the Younger in 105 AD, Lull in 1299 and Cusanus in 1431 wrote on the subject, but the Golden Age of Social Choice did not occur until the French Enlightenment. Borda was a member of the Paris Academy of Sciences founded in 1666. He was primarily an experimentalist and a naval engineer. In 1770 he delivered a paper to the Academy proposing the method of the rank-order count for electing members to the Academy. Not much attention was paid to the paper, and Borda probably had little to do with having the paper published. It was published in 1784. Borda managed to survive the Terror of the French Revolution a feat not accomplished, however, by the Marquis de la Condorcet. Borda’s method involved having each elector submit an ordered list with the most favored candidate listed first and the least favored listed last. Then each candidate is awarded a number of points from 1 to n, where n is the number of candidates. The most favored is awarded n points while the least favored is awarded 1 point. Thus the method is called the Borda count. Each candidate’s points are added up and the one with the most points wins. The French Academy adopted Borda’s method for elections to its membership in 1784. It remained in effect until 1800 when it was attacked by a new member of the French Academy, Napoleon Bonaparte.
In 1785 Condorcet published his Essai sur l’application de l’analyse a la probabilite des decisiones rendues a la pluralite des voix. Condorcet’s method for deciding a multi-candidate election is for voters to vote on candidates 2 at a time. This should determine an overall winner - the one that wins over every other candidate in every pairing. The problem is that this may not always be the case. For example, for 3 candidates, A, B and C, and 3 voters, 1, 2 and 3, if voter 1 votes ABC, voter 2 votes BCA and voter 3 votes CAB, then in the pairing A and B, A wins 2 votes to 1, in the pairing A and C, C wins 2 votes to 1 and in the pairing B and C, B wins 2 votes to 1. Therefore, A wins over B, B wins over C and C wins over A. There is no clear winner. This is known as the paradox of voting. Those familiar with my work know that my proposed solution to this problem is a tie among A, B and C.
In 1792, Condorcet was made chairman of a committee to draw up a constitution for France. He was also connected with the US and Polish constitution writers. According to “Classics of Social Choice,” by Iain McLean and Arnold B. Urken, he took an active part in the French Revolution tirelessly promoting voting schemes. As a Girondan, Condorcet found himself out of power after the Jacobin coup in 1793. His constitution was dumped in favor of one drawn up by Robespierre. Condorcet denounced the Jacobin constitution and was proscribed by the Jacobins. He hid in the house of a courageous landlady. While there he wrote his “Outline of an Historical Picture of the Progress of the Human Mind” which was published posthumously. This is a peon to the idea of Progress which was enshrined by the French Revolution. After the Jacobins passed a law passing a death sentence on anyone harboring a fugitive, Condorcet left his landlady’s house in order not to place her in danger. He was shortly discovered and died in prison. Thus was one of the greatest minds in social choice who might have implemented progressive voting methods in the French Constitution eliminated. He is buried in the Pantheon in Paris along with other French intellectuals such as Voltaire and Rousseau.
According to Duncan Black in “The Theory of Committees and Elections”:“The second half of the eighteenth century in France was one of the outstanding epochs of scientific thought. Science had felt its strength and its impulse and did not know what barriers it might not cross. ...
The hope had sprung up to carry the methods of rigorous and mathematical thought beyond the physical and into the realms of the human sciences. But after a brilliant start, its fate was to be misunderstood and forgotten.
The initial step was taken by Borda, who ... had already achieved distinction as a mathematician and had for the centre of his life the Academy of Sciences. It was no doubt elections to the Academy, membership of which was for him the most valuable of all privileges, and not the wider problems of politics that first directed his mind to the theory. In it he showed the same eye for the significant fact and for the simplifying assumption as in his other researches, and he broached the subject in a new way. His work has the robust good sense of the practical man.
It is not only the same strand of reasoning on which both Borda and Laplace rely and the same conclusion that they adopt, but it is also the same spirit that actuates the work of both of them. By a straight and untortuous line of reasoning they proceed from premiss to conclusion; and no one, whether mathematician or politician, can fail to feel the attraction of their theory. It combines a logical with a practical appeal and its virtues are much more visible than its defects: the real phenomena involved in elections are far more complex, tortuous and indecisive than it represents.
Condorcet, on the other hand, saw the complexities. He had a deep intuitive vision of the truth and tried to impart it, more, perhaps, by the use of symbols than by a strict Mathematics. ...”
Condorcet is better known, perhaps, for his essay on Progress which he wrote while hiding from Robespierre and the Terror in the winter of 1793-1794, the title of which is “Outline of an Historical Picture of the Progress of the Human Mind” (Esquisse d’un tableau historique des progres de l’espirit humain Paris: Masson, 1822). It was published posthumously in 1795. What strikes one right off the bat is how Condorcet, while hiding out from a barbarous regime that was trying to kill him and eventually did, could be so positive about the future of mankind as is evidenced by the following quote:“Such is the goal of the work which I have undertaken, of which the result will be to show by reason and by evidence that no limit has been set to the perfection of human faculties; that the perfectibility of man is really indefinite; that the progress of this perfectibility, henceforth independent of every power which might wish to stop it, has no limit other than the duration of the globe on which nature has cast us.”
I don’t agree with Condorcet’s assessment of the “perfectibility of man.” Shortly after writing this, Condorcet himself died a miserable death in prison at the hands of a barbarous regime. Since his time much more powerful weapons, including nuclear weapons, have been developed. Mankind has proven to be consistently barbarous, if not more so, in terms of wars that have raged since Condorcet’s death including the Napoleonic War, the American Civil War, World Wars 1 and 2, not to mention numerous more minor wars and genocides. Human nature has remained remarkably consistent and warlike over its 150 thousand years of existence. However, during that period and, in particular, since Condorcet’s death, there has been remarkable technological progress including electricity, air and space travel, radio, TV, telecommunications, computers, and the internet.
We have seen major belief systems in a better age such as “The Age of Aquarius” come and go while progress in the techniques of war relentlessly advances. Marxism predicted a “Golden Age” in which all the tensions of competition would be resolved, and people would operate in the admirable fashion “from each according to their abilities, to each according to their needs.” As we’ve seen this belief in a future utopia has been, for all intents and purposes, relegated to the ash heap of history while war, genocide and barbarism march steadily on. I think that the study of utopias, the study of how things would operate in an ideal society may have some useful purpose. However, not much practical progress has been made since the constitution writing phase of the Enlightenment. There is always at least one critical element missing from whatever systems have been proposed to date.
While I don’t agree with Condorcet on the perfectibility of man, I do agree with him that progress can be made in human institutions although few advances have been made in this area either. In terms of voting systems, there are some entities using proportional representation which is arguably better than single district, single representative winner take all systems. The UN arguably represents a worthwhile ongoing attempt at bringing nations together in a cooperative forum. Judicial systems in advanced countries have done much to settle disputes in a civilized manner. Major religions and charitable institutions have done much to ameliorate the plight of the “least among us.”
Still advances in the technology of war and weaponry not to mention the legalization of assault weapons putting incredible firepower in the hands of ordinary citizens in the US at least more than offsets the meager attempts of the human race to civilize itself. However, the appalling historical record is no reason not to try and develop, on a theoretical level, at least the tools that might result in progress for the human race if they were somehow adopted. I don’t think that social choice, even if proven “possible”, is a panacea that will result in everyone’s laying down their arms and coexisting peacefully. However, it might be part of the process of conflict resolution and the more equitable distribution of resources. Therefore, I believe it’s worth pursuing.
The Nineteenth Century
In the nineteenth century, work on social choice theory crossed the Channel (English or French according to your perspective) and became rooted among English philosophers and mathematicians. E.J. Nanson in 1882 read his paper, “Methods of Election,” to the Royal Society of Victoria. Born in England, Nanson immigrated to Australia in 1875 where he was a professor of Mathematics at Melbourne. Nanson developed a voting system which involved using an iterated Borda count, at each stage dropping the candidates with less than the average Borda count. This produces the sensible result in the voting paradox of dropping the candidate with the smallest majority. For instance, if A is preferred to B by a vote of 100 to 50, B is preferred to C by a vote of 90 to 60 and C is preferred to A by a vote of 80 to 70, then C would be dropped, and A would be the winner. In “The Theory of Committees and Elections,” by Duncan Black, he says: “The real defects in Nanson’s method of election are of a practical kind. It proceeds by a series of eliminations and, requiring a new set of marks at each round, works out very laboriously.” However, Black’s book was first published in 1958 shortly after the advent of computers and before the appreciation of their use especially in voting systems. I don’t think Nanson’s method would be insurmountable with the computing power available today.
The towering figure of social choice in the nineteenth century is the Rev. C. L. Dodgson better known as Lewis Carroll, the author of Alice in Wonderland among other children’s tales. Dodgson became a lecturer in mathematics and a fellow at Christ Church, Oxford, in 1856. He struck up a friendship with Alice Liddell, the daughter of H.G. Liddell, the Dean of Christ Church to whom he first told the stories that resulted in Alice in Wonderland and Through the Looking Glass. Dodgson worked primarily in isolation without researching extensively the work of other writers in the field. Dodgson’s work related to Proportional Representation (PR); The most distinctive feature of his scheme is “the giving to each candidate the power of transferring to any other candidate the votes given for him.” (Diary for 3 June 1884) Dodgson wrote three pamphlets, “A Discussion of the Various Methods of Procedure in Conducting Elections,” “Suggestions as to the Best Method of Taking Votes, Where More than Two Issues are to be Voted upon,” and “The Principles of Parliamentary Representation,” various letters to The St James Gazette, an evening newspaper and his Diary, all of which contain significant information regarding proportional representation.
Dodgson’s work was related to problems the Conservatives were having in the English Parliament. Electing members by majority vote in single member districts can produce a landslide for the party with a slight majority in each district while the minority although sizable can go totally unrepresented. The single member district with majority rule is exactly what exists in the US for electing members of Congress and state Assemblies as well. Dodgson saw that the survival of minorities in Parliament requires multimember districts and some method for apportioning seats among the voters. According to McLean and Urken, “The Principles is the earliest known work to discuss both the assignment of seats to each of a number of multimember districts (the apportionment problem) and the assignment of seats within each district to the parties (the PR problem).”
Jeremy Bentham, developed the utilitarian philosophy which states that society should try and promote the greatest happiness for the greatest number. John Stuart Mill (1812-1848) extended and modified Bentham’s work. Welfare economics builds on their work. Some of the work in welfare economics comes under the heading of social choice. In particular, if giving individuals, insofar as possible, what they want in terms of their stated individual choices, by combining those individual choices into a social choice, is considered to be promoting happiness, then social choice is indeed utilitarian. Whether giving an individual what he or she wants indeed promotes happiness is another question altogether.
The Twentieth Century
In 1951 Kenneth Arrow published Social Choice and Individual Values. Arrow assumes that each individual has a preference list: R1,...,Rn for alternative social states. A social welfare function is a process or rule for combining these individual preference lists to obtain a social ordering of alternative social states, R, where n is the total number of social states. By a social state Arrow means “...a complete description of the amount of each type of commodity in the hands of each individual, the amount of labor to be supplied by each individual, the amount of each productive resource invested in each type of productive activity, and the amounts of various types of collective activity...” The relationship xRy means x is preferred or indifferent to y. Alternatively, the alternatives could be political candidates. Arrow states that there are essentially two means of making decisions in a capitalist democracy: voting and the market mechanism. However, both political and economic decisions can be made by a social choice mechanism. Political decisions made in this way would represent an extension to the antiquated voting mechanisms currently used for political democracy and economic decisions made in this way would represent an economic democracy that would replace the market mechanism.
Arrow postulates five “rational and ethical” criteria which any social welfare function must meet. Therefore, his entire analysis is dependent on the fact that you will agree with him that any other way of combining individual choices to form a social decision is either irrational or unethical. This is one of the problems with Arrow’s analysis: there is not universal agreement with his five criteria. Arrow’s analysis “proves” that social choice is impossible. He does this in a very precise and mathematical manner. His conclusions have come to be known as “Arrow’s Impossibility Theorem.” Arrow’s Impossibility Theorem has been endorsed by those who champion a Madisonian or representative democracy as opposed to those who champion a Jeffersonian or participatory or direct democracy, and also by those who champion a capitalistic economic system since economic democracy has supposedly been proven impossible.
In the last 50 or so years much work has been done in the field of social choice which either validates Arrow’s conclusions or attempts to find a way around Arrow’s conclusions. The work represented by this web site is of the latter variety.
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