Abstract

In an election, a population of individuals (the voters) express their preferences over the list of available alternatives (the candidates), and a voting rule is a mechanism that assigns chosen candidates to these preference profiles. A prominent example is the plurality rule which proceeds by counting, for each candidate, the number of voters who place her or him in the top position of their respective ordering. The candidates with the highest number of first-place positions are then elected. The formal analysis of voting rules can be traced back at least two centuries. Among the most fundamental early contributions are those of Bentham (1776), Borda (1781) and, about a century later, Dodgson (1873). We examine some proposals that appear in these classics, hoping to be able to shed further light on the proposed election methods and their properties. There are various possibilities of defining a voting rule. One consists of assigning a set of chosen alternatives (that is, elected candidates) to each profile of individual voters' preferences. A second option is to establish a social ordering of the candidates based on the preference orderings expressed by the voters. Either of these methods can be employed for the purposes of our project; we choose the latter for ease of exposition. That is, in Arrow's (1951; 1963; 2012) terms, we examine voting rules in the form of social welfare functions. The plurality rule and the Borda (1781) rule are special cases of the class of positionalist voting rules. These are election procedures that can be defined solely on the basis of the positions in which each candidate appears in each voter's preference ordering. We propose a general definition of positionalist rules that encompasses those that are currently available in the literature, and we illustrate how our definition distinguishes itself from these earlier approaches. See Bossert and Suzumura (2017a; in preparation) for details on positionalist voting rules. Furthermore, we argue that Bentham's (1776) principle of the greatest happiness of the greatest number can be given an ordinal interpretation, in which case it is equivalent to the plurality rule. We also examine a natural counterpart, the principle of the greatest unhappiness of the least number, a proposal that focuses on the voters whose bottom-ranked candidates are chosen. This alternative principle is tantamount to the inverse plurality rule. Greatest happiness and least unhappiness principles are discussed in Bossert and Suzumura (2017b). Finally, we reexamine Dodgson's (1873) method of marks, according to which each voter gets to allocate a vote budget among the candidates. Our focus is on identifying the number of votes that each voter should be given on the basis of two plausible and intuitively appealing criteria. This analysis of vote budgets can be found in Bossert and Suzumura (2017c). Authors: Walter Bossert (U Montreal) and Kotaro Suzumura (Weseda University)