Elections Tidbit (09/13/2003)
Mathematics and Computing Department
York College (CUNY)
Jamaica, NY 11451
Email: email@example.com (for additions, suggestions, and corrections)
web page: www.york.cuny.edu/~malk
A pillar of American Democracy is the regular elections that are held for public officials at many levels of government. Elections or voting is also used in a wide variety of other settings: union officials, club officers, student government prom king and queen, food choice for the club picnic, and hunk of the month. The methods of election used vary considerably. Thus, for President of the United States we use a system based on the Electoral College while for Mayor of New York City we have a system where an initial election is held, and if no candidate gets a plurality of at least 40 percent of the vote, a run-off election is held at a later time between the two candidates who had the largest number of votes. But elections are not sufficient to guarantee that one is living in a democratic environment. There were elections in Stalinist Russia and in Nazi Germany. By far the most common election method is to have the voters pick a single choice from the list of candidates and the winner is that person having the largest number of first place votes (plurality voting or "first past the post." However, in a democracy, which of the the different methods for conducting elections is best and which is fairest, and why are so many different methods employed?
In constructing a mathematical model for an election system the key components are:
2. Alternatives to choose among.
3. Ballots for the voters to express their preferences among the alternatives.
4. A method of selecting a winner or winners based on the ballots.
Furthermore, in judging the success of the election system one needs to look at various choices for criteria for what one wishes to achieve. One's goal may be maximizing voter turnout, fairness, stability, minimization of cost, or other objectives.
To give some of the flavor of these issues, let us consider the issue of ballots first. For concreteness and simplicity we will consider only elections where a single individual is to be selected by the group of voters. Historically, the ballot nearly always used in American elections consisted of asking a voter to select one candidate from a list of candidates. This type of ballot will be referred to as the standard ballot. In recent years there has been considerable discussion of whether or not this type of ballot allows voters to express their preferences (opinions) in as full a way as possible. For example, instead of employing the standard ballot one can require each voter to rank all of the candidates on a preferential ballot where a voter is not allowed to be indifferent between choices or alternatives. (We will deal with objections to using this procedure later.) With this in mind one can use the symbolism below to represent a typical example of how a voter could express his/her views on the candidates A, B, C, and D:
The interpretation of this ballot, called the ordinal or preferential ballot, is that D is preferred strictly to C, B, and A, while C is strictly preferred to B and A, and B is preferred to A.
Note that in this environment we are not allowing a voter to express the view that under no circumstance would he/she be happy to have candidate A and B serves for office. All that is permitted is to allow the voter to compare his/her opinions of the candidates with respect to each other. Furthermore, the voter is not "allowed" to truncate his/her ballot and vote for only a subset of the candidates. (Of course, one can generalize what is discussed here by allowing truncation of ballots and/or indifference between candidates. One reason that is suggested why truncation should be allowed is that when there is a large list of candidates (e.g. the California Governor recall election) it is not realistic that voters will know much about all the candidates who are running.) Clearly, there are many, many approaches to ballet design. One comparatively recent new ballot innovation is the idea of an approval ballot. The idea is to have each voter indicate those candidates they approve to hold office. The candidate with the largest number of approval votes would win.
Suppose we have decided on a type of ballot to use, and suppose this type of ballot is the preferential ballot without ties allowed and no truncations allowed.
Given the use of a ballot of this kind, who do you think should win the election below, in which 55 ballots where cast:
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