Social Choice during the Nineteenth Century
During the nineteenth century, in spite of the development of democratic institutions, the theory of social choice has been rather dormant, save for a few exceptions.
A basic question that had to be solved in the USA 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 across the various states? The equality here means that each representative should represent the same number of people.
Although this looks simple, and maybe secondary, the difficulty arises from the fact that the numbers 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, especially Thomas Jefferson and Alexander Hamilton, and rediscovered later by Europeans generally in the context of proportional representation. The definitive book on this topic is Michel Balinski and Peyton Young (1982). It includes many historical developments.Two scholars have to be mentioned. The first is C.L. Dodgson (Lewis Carroll) (18321898) and the second is Edward John Nanson (1850-1936). Dodgson was a logician at Oxford University. He wrote some pamphlets that are reproduced in Black (1958). Through time, he proposed various voting methods. For instance, even though he apparently knew neither Borda nor Condorcet, he proposed the use of Borda’s rule and a run-off between the two top candidates. Later, he proposed selection of 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. 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 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).