Voting theory: the origins

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That collective decisions made through voting may lead to some weird consequences is something probably well known since the first time one tribe started to use some kind of voting in order to reach an agreement. Contrarily to what happens in many other areas of human wisdom, we don’t have any deep reflections on the topic from the otherwise prolific Greek philosophers of the Classical Age, though surely they encountered many cases in which voting procedures could be somehow “improved”: it seems that they tended to consider that the problems with voting were more an effect of the poor rationality of individual voters (as when the Athenian great jury condemned Socrates to death by more votes than had been cast to declare him guilty of impiety), than of the voting system itself. Some passing remarks about the possibility of manipulating the result of voting by modifying the number or the order in which the proposals are voted were made by the Roman writer Pliny the Younger, but apparently with no hindsight about the possibility of experimenting with different voting rules. It was the process for the election of new Popes in the Catholic Church what provided the opportunity for a deeper reflection on voting procedures, more than a millennium after Pliny: till the 12th century, the fiction was kept that those elections, as well as the canons from the councils, were always approved by unanimity or near unanimity (acclamation or compromissum), but in 1179 Alexander III introduced the rule of two thirds, a kind of qualified majority, surely backed by a thoughtful discussion amongst the nascent representatives of Scholastic philosophy, of which, unfortunately, no records exist.

It was the Spanish intellectual Ramon Llull (latinized as Raymundus Lullius), described by Martin Gardner as ‘one of the most inspired madmen who ever lived’, who gained the honour of being the first author who elaborated something like a theoretical discussion about voting rules, in several texts from the end of the 13th century, including his Catalan book Blanquerna, the first novel written in a modern European language. In particular, Llull proposed a system in which there was a vote amongst each possible pair of candidates (if candidates are also electors, the two members of each pair are not allowed to vote), and the candidate wining the most pair-wise single contests is then elected (this is, of course, a procedure analogous to the traditional ‘league’ tournaments in many sports). The candidate that would be elected under such a voting procedure will later became known as a ‘Condorcet winner’, although ‘Lullius winner’ would be a much more suitable name). More than one century later, the churchman, philosopher and mathematician Nicholas Cusanus proposed a method equivalent to what will become known as ‘Borda count’: each elector orders all the n candidates according to his or her preferences (these procedures were intended to be applied to the election of abbots and abbesses by the monks or nuns of the monastery), and numbers them from n to 1; the winner is the candidate that gets more points when these numbers are summed up.

After this brief flash of relatively primitive reflection on voting, nothing of interest was produced within the next three centuries, until a group of philosophes, very probably without any knowledge of those forerunners, started to study the topic in the eve of the democratic revolutions of the Enlightenment. The two most important figures were the already mentioned Jean-Charles de Borda and the Marquis of Condorcet. It was Condorcet who, reflecting on the method of voting proposed by Borda, discover what became known as the ‘Condorcet paradox’: the fact that pairwise comparisons amongst candidates can lead to cycles (candidate A beating candidate B, B beating C, and C beating A). This may happen not only with the Borda count, but with most other procedures, and entails that collective preferences may not be transitive, at least if derived from individual preferences by means of voting. Another severe problem soon discovered in these voting rules is that they might be subject to manipulation, not by the people with power to decide who are electors and candidates, but by the electors themselves, in the sense that there may be cases in which a voter has a higher chance of reaching a better result by voting in contradiction with his or her real preferences: for example, in a Lullian-Condorcet contest between candidates A and B, one elector may vote for B even if preferring A, if that makes C (the elector’s favourite candidate) more likely the final winner; or, in a Cusan-Borda election, one voter may rank his second-best candidate in the last position, if that makes the victory of his favourite candidate more probable. It was again a Spaniard mathematician, José Isidoro Morales, who discovered this fact, to which Borda replied that his method was only attempted for ‘honest electors’.

Unfortunately, the Borda-Condorcet-Morales debate was mostly forgotten for almost another century, and some of their ideas had to be independently rediscovered in the 1870’s by the mathematician Charles Dodgson (better known by his pseudonym Lewis Carroll), who also worked on proportional representation. Another Victorian Englishman (but based in Australia) who worked on electoral rules, this time with knowledge of the Enlightened precursors, was Edward Nanson. But, again, neither of their contributions had any repercussion on the political or theoretical development of electoral rules in their time, in spite of being an epoch of growing introduction of more or less democratic systems in many Western countries; to use the words of the author whose historical survey is my guide on this topic: “the record shows that these electoral systems were mostly adopted by politicians who perceived partisan advantages in them, and not because of any mathematical arguments”.

Non-transitivity and manipulability are two members of the triad of ‘classical paradoxes’ that may affect voting rules, and that are studied in the field of economic theory known as social choice. This triad was completed by the American mathematician Edward Huntington in a paper of 1938, where he showed that procedures like the Borda count violate what he called the ‘principle of relevance’: introducing a new alternative or candidate which is not the favourite one of any elector may affect the choice of the winner. This is, of course, the principle that soon was to be baptized as ‘independence from irrelevant alternatives’ in what became the birth of the modern, systematic theory of voting, under the leadership of Kenneth Arrow, but this is another story.

References

McLean, I. (2015), “The strange history of social choice, and the contribution of the Public Choice Society to its fifth revival”, Public Choice, 163: 153-165.

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