Voting Systems that Combine Approval and Preference
Social choice theory, while postulating that voters have preferences over candidates, does not ask them to stipulate where, in their preference rankings, they would draw the line between acceptable and unacceptable candidates. Approval voting (AV) does as
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1 Introduction Social choice theory, while postulating that voters have preferences over candidates, does not ask them to stipulate where, in their preference rankings, they would draw the line between acceptable and unacceptable candidates. Approval voting (AV) does ask voters to draw such a line, but it ignores rankings above and below this line. Rankings and approval, though related, are fundamentally different kinds of information. They cannot necessarily be derived from one another. Both kinds of information are important in the determination of social choices. We propose a way of combining them in two hybrid voting systems, preference approval voting (PAV) and fallback voting (FV), that have several desirable properties. Approving of a subset of candidates is generally not difficult, whereas ranking all candidates on a ballot, especially if the list is long, may be arduous. PAV asks for both kinds of information, whereas FV asks voters to rank only those candidates they approve of, making it simpler than systems that elicit complete rankings. We describe, analyze, and compare each of these systems in tandem. In Sect. 2 we give definitions and assumptions. In Sect. 3 we describe PAV and analyze which candidates can and cannot win under this system. Although a PAV winner may not be a Condorcet winner or AV winner, PAV satisfies what we call the strongestmajority principle for voters. More specifically, if a majority-approved candidate is preferred by a majority to the AV winner and other majority-approved candidates, PAV “corrects” the AV result by electing the majority-preferred candidate. That is, PAV elects the majority-approved candidate who is most preferred. A majority-preferred candidate is likely to have a more coherent point of view than an AV winner, who may be the most popular candidate because he or she is
S.J. Brams (¬) Department of Politics, New York University, New York, NY 10012, USA e-mail: [email protected] S.J. Brams et al. (eds.), The Mathematics of Preference, Choice and Order: Essays in Honor 215 of Peter C. Fishburn, Studies in Choice and Welfare, c Springer-Verlag Berlin Heidelberg 2009
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S.J. Brams and M.R. Sanver
bland or inoffensive – a kind of lowest common denominator who tries to appease everybody. However, this problem does not seem to be a common one (Brams & Fishburn, 2005; Brams, 2008, Chap. 1). Sometimes not choosing such a candidate when two or more candidates receive majority approval makes PAV coherenceinducing for candidates by giving an advantage to candidates who are principled but, nevertheless, command broad support. In Sect. 4 we describe FV and compare its properties with those of PAV. Like PAV, FV tends to help those candidates who are relatively highly ranked by a majority of voters. Both systems may give different winners from nonranking systems (e.g., plurality voting and AV), ranking systems (e.g., the Borda count and single transferable vote, or STV), and each other. In Sect. 5 we show that PAV and FV are monotonic in two different senses: Voters,
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