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Mathematics LibreTexts

2.18: Exploration

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    36261
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    1. In the election shown below under the Plurality method, explain why voters in the third column might be inclined to vote insincerely. How could it affect the outcome of the election?

    \(\begin{array}{|c|c|c|c|}
    \hline \textbf { Number of voters } & \mathbf{9 6} & \mathbf{9 0} & \mathbf{1 0} \\
    \hline \textbf { 1st choice } & \mathrm{A} & \mathrm{B} & \mathrm{C} \\
    \hline \textbf { 2nd choice } & \mathrm{B} & \mathrm{A} & \mathrm{B} \\
    \hline \textbf { 3rd choice } & \mathrm{C} & \mathrm{C} & \mathrm{A} \\
    \hline
    \end{array}\)

    1. In the election shown below under the Borda Count method, explain why voters in the second column might be inclined to vote insincerely. How could it affect the outcome of the election?

    \(\begin{array}{|c|c|c|}
    \hline \textbf { Number of voters } & \mathbf{2 0} & \mathbf{1 8} \\
    \hline \textbf { 1st choice } & \mathrm{A} & \mathrm{B} \\
    \hline \textbf { 2nd choice } & \mathrm{B} & \mathrm{A} \\
    \hline \textbf { 3rd choice } & \mathrm{C} & \mathrm{C} \\
    \hline
    \end{array}\)

    1. Compare and contrast the motives of the insincere voters in the two questions above.
    1. Consider a two party election with preferences shown below. Suppose a third candidate, C, entered the race, and a segment of voters sincerely voted for that third candidate, producing the preference schedule from #17 above. Explain how other voters might perceive candidate C.

    \(\begin{array}{|c|c|c|}
    \hline \textbf { Number of voters } & \mathbf{9 6} & \mathbf{1 0 0} \\
    \hline \textbf { 1st choice } & \mathrm{A} & \mathrm{B} \\
    \hline \textbf { 2nd choice } & \mathrm{B} & \mathrm{A} \\
    \hline
    \end{array}\)

    1. In question 18, we showed that the outcome of Borda Count can be manipulated if a group of individuals change their vote.
      1. Show that it is possible for a single voter to change the outcome under Borda Count if there are four candidates.
      2. Show that it is not possible for a single voter to change the outcome under Borda Count if there are three candidates.
    1. Show that when there is a Condorcet winner in an election, it is impossible for a single voter to manipulate the vote to help a different candidate become a Condorcet winner.
    1. The Pareto criterion is another fairness criterion that states: If every voter prefers choice A to choice B, then B should not be the winner. Explain why plurality, instant runoff, Borda count, and Copeland’s method all satisfy the Pareto condition.
    1. Sequential Pairwise voting is a method not commonly used for political elections, but sometimes used for shopping and games of pool. In this method, the choices are assigned an order of comparison, called an agenda. The first two choices are compared. The winner is then compared to the next choice on the agenda, and this continues until all choices have been compared against the winner of the previous comparison.
      1. Using the preference schedule below, apply Sequential Pairwise voting to determine the winner, using the agenda: A, B, C, D.

    \(\begin{array}{|c|c|c|c|}
    \hline \textbf { Number of voters } & \mathbf{1 0} & \mathbf{1 5} & \mathbf{1 2} \\
    \hline \textbf { 1st choice } & \mathrm{C} & \mathrm{A} & \mathrm{B} \\
    \hline \textbf { 2nd choice } & \mathrm{A} & \mathrm{B} & \mathrm{D} \\
    \hline \textbf { 3rd choice } & \mathrm{B} & \mathrm{D} & \mathrm{C} \\
    \hline \textbf { 4th choice } & \mathrm{D} & \mathrm{C} & \mathrm{A} \\
    \hline
    \end{array}\)

      1. Show that Sequential Pairwise voting can violate the Pareto criterion.
      2. Show that Sequential Pairwise voting can violate the Majority criterion.
    1. The Coombs method is a variation of instant runoff voting. In Coombs method, the choice with the most last place votes is eliminated. Apply Coombs method to the preference schedules from questions 5 and 6.
    1. Copeland’s Method is designed to identify a Condorcet Candidate if there is one, and is considered a Condorcet Method. There are many Condorcet Methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist. Copeland’s method does not have a tie-breaking procedure built-in. Research the Schulze method, another Condorcet method that is used by the Wikimedia foundation that runs Wikipedia, and give some examples of how it works.
    1. The plurality method is used in most U.S. elections. Some people feel that Ross Perot in 1992 and Ralph Nader in 2000 changed what the outcome of the election would have been if they had not run. Research the outcomes of these elections and explain how each candidate could have affected the outcome of the elections (for the 2000 election, you may wish to focus on the count in Florida). Describe how an alternative voting method could have avoided this issue.
    1. Instant Runoff Voting and Approval voting have supporters advocating that they be adopted in the United States and elsewhere to decide elections. Research comparisons between the two methods describing the advantages and disadvantages of each in practice. Summarize the comparisons, and form your own opinion about whether either method should be adopted.
    1. In a primary system, a first vote is held with multiple candidates. In some states, each political party has its own primary. In Washington State, there is a "top two" primary, where all candidates are on the ballot and the top two candidates advance to the general election, regardless of party. Compare and contrast the top two primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc.
    1. In a primary system, a first vote is held with multiple candidates. In some many states, where voters must declare a party to vote in the primary election, and they are only able to choose between candidates for their declared party. The top candidate from each party then advances to the general election. Compare and contrast this primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc.
    1. Sometimes in a voting scenario it is desirable to rank the candidates, either to establish preference order between a set of choices, or because the election requires multiple winners. For example, a hiring committee may have 30 candidates apply, and need to select 6 to interview, so the voting by the committee would need to produce the top 6 candidates. Describe how Plurality, Instant Runoff Voting, Borda Count, and Copeland’s Method could be extended to produce a ranked list of candidates.

    2.18: Exploration is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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