Ramon Lull
Ramon Lull (c. 1233-1316) was born in Mallorca into a wealthy family. As a young man, he was a troubadour writing poetry and songs. At the age of about 30, 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 - 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 and 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, astrology and alchemy.Social choice theorists first heard of Lull (also, for that matter, of Nicholas 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 Friedrich Pukelsheim (see Hagele and Pukelsheim 2001, 2008). It is remarkable that Blanquerna is considered as one of the first novels ever written (in Catalan) in Europe. In Artificium Electionis Personarum, the first published among the three texts as well as in Blanquerna and the later De Arte Eleccionis, Lull recommends systems based on pair-wise (majority) voting. In both works all pair-wise comparisons are done. There is however a difference, since in Blanquerna, 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 20 voters to be reduced to seven. He describes a method to reach these seven: each voter is asked to select seven among 19 (this probably 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, it is impossible to know). The winner is the candidate who is victorious in most of the pair-wise contests. This method is known today as the Copeland method and more sophisticated versions are used in tournaments, in particular in sports. (Copeland was a mathematician at the University of Michigan. His paper, “A reasonable social welfare function” 1951, has never been published.) 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 [the reference is to a nun] will be the one who is worthy to be elected” (McLean and Urken 1995: 72).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. It is not clear whether this corresponds to what Lull had in mind.
In De Arte Eleccionis, the procedure is quite different even though 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). Donald Saari (2008) gives an 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.