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Math 110: An Introduction to the

Mathematics of Voting

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What’s The Problem?

CHAPTER

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1: Math 110: An Introduction to the Mathematics of Voting

The Plurality Method

The simplest and perhaps most common (?) method of deciding an election between candidates is the familiar plurality method.

DEFINITION 1.1 (Plurality Method). In the plurality method, each voter selects one

candidate on the ballot. The winner is the candidate with the most votes. Note that the winner does not need to have a majority of the votes. (In the rare case of a tie, some other method must be used to select the winner. We will largely ignore the possibility of ties in what follows.)

EXAMPLE 1.1. In a 3 candidate election, candidate A gets 12 votes, candidate B gets 20 votes,

Voting theory is the mathematical description of the process by which democratic societies resolve the di↵erent and conﬂicting views of the group’s members into a single choice for the group. Each vote is an expression of that voter’s preference about the outcome of an election. Why do we need a mathematical theory about something so simple as voting? How hard is it be to ﬁnd a simple, fair, and consistent procedure for determining the outcome of an election?

In the United States with its two party system we have become used to elections that involve only two major candidates. Suppose that there is an election between

Professors Critchlow and Eck for chairmanship of the Department of Mathematics and Computer Science. In this case the situation is as simple as you might imagine.

How should the winner be determined?1

However, many elections involve making a decision from among more than two candidates or choices. For example, think about how might this class decide whether to have the second exam on November 5(W), 7(F), or 10(M)?

In my lifetime, at least two presidential elections have been greatly a↵ected by third party candidates. In 1968 Richard Nixon (R) received 43.4% of the vote, Hubert

Humphrey (D) received 42.7%, and George Wallace (American Independent Party) received 13.5%. If Wallace had not run the results could easily have been di↵erent.

More recently in the 2000 election Gore (D) received 48.38%, Bush (R) 47.87% and

Nader (Green Party) 2.74% and Bush won. (Why?) The perception at the time was that most of those who voted for Nader would have voted for Gore if Nader had not run. (Why?) This likely would have changed the outcome of the election. For similar thoughts about this year’s election, at least how it was viewed in late July, see the article entitled “The Power of the Protest Vote” at http://campaignstops.blogs. nytimes.com/2008/07/29/the-power-of-the-protest-vote/index.html in The New

York Times.

In the material that follows we consider various attempts to decide an election fairly when there are three or more choices.

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You probably said that the candidate with the majority of the votes should be elected. (Recall that a majority means greater than 50% (half) of the votes.)

and candidate C 18 gets votes (total of 50 voters). Candidate B has a plurality of the votes but, in this case, not a majority of the votes. Nonetheless, Candidate B is the winner.

Assuming that there are no ties, the plurality method is easy to carry out. However there are problems with this method when there are more than two candidates, especially when two of the candidates are similar and split what potentially would be a majority of the vote. The next section examines this more closely.

Z Mindscape 1.1. Note: You should try these exercises as you encounter them in the reading.

This particular exercise is the one place where ties are considered.

(a) Devise a fair method to determine the winner of a three-way election when the two candidates with the most votes are tied.

(b) Now devise a method to determine the winner of a three-way election when all three candidates receive the same number of votes.