Problems of Voting

Abstract

It is now over 200 years since Condorcet discovered a serious paradox in the standard method of voting under substantially any of the rules of order. Most Americans are most familiar with the rules developed by General Robert’s, but almost all sets of rules used in genuinely democratic bodies are similar. A proposal is made and then modifications, called amendments are added. There is then a series of binary votes so arranged that the final outcome is a modified version of the original proposal. The modification can be extreme. Unfortunately, Condorcet discovered that some amendments or proposals, which had been defeated in the earlier part of the voting series, might be able to beat the ultimate outcome. A colleague of Condorcet in the academy proposed another voting method in which all alternatives are listed by each voter and then given a weight depending upon their location in the list. It was immediately pointed out that this method provided strategic opportunities for the voter to cheat by misspecifying his preferences. Borda replied that he designed his system for honest men, but since politics is not full of honest men this was regarded as a weak defense. The problem was largely ignored for the next two centuries. Some mathematicians seem to have been interested and one of them Dodson, better known as Lewis Carrol put in a good deal of work on the matter, but without reaching any solution. People advocating democracy, or for that matter opposing it, seem not to have known about this problem or in any event they did not discuss it. In the 1930s, a mathematician interested in field sports and engaged in scoring sports meets discovered that the method used for scoring in which the winner in a given sport was given five points, the second three points, and the third one point, and these points were aggregated in order to determine which university team won had a serious paradox (see Huntingdon, 1938). Specifically, whether A or B won the meet could be determined by whether C, a weak team was physically present or not. He published this result in a mathematical journal and it was ignored for some time. It was not until the late 40s that Arrow used this as part of his proof that voting methods all have severe difficulties.