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Chapter 13: Problem 43

What is the plurality-with-elimination method? Why is it advantageous to rank the candidates when using this method?

Short Answer

Expert verified
The plurality-with-elimination method is a voting system where voters rank candidates in order of preference, initially only considering first-preference votes. If no candidate secures more than half of these votes, the candidate with the least number of first-preference votes is eliminated and their votes are reassigned based on the voters' second preferences. This process continues until a candidate secures over half of the votes. Ranking candidates is advantageous as it ensures every vote counts, boosting the democratic nature of the election process.

Step by step solution

01

Definition of the Plurality-With-Elimination Method

The plurality-with-elimination method, also known as the instant-runoff voting, is a voting system used in elections with more than two candidates. Voters rank the candidates in order of preference. Initially, only the first preference votes are tallied. If no candidate receives more than half of these votes, the candidate with the least number of first-preference votes is eliminated.
02

Procedure After Elimination

After a candidate is eliminated, their votes are transferred to the remaining candidates based on the second preferences marked on the eliminated candidate's ballots. This process continues, with the candidate with the fewest votes being eliminated and their votes being reassigned, until a candidate has more than half of the votes.
03

Advantage of Ranking Candidates

Ranking candidates can be advantageous in this system as it ensures that votes are not 'wasted'. If a voter's most preferred candidate is eliminated, their vote is not discarded but instead transferred to their next preferred candidate (as per their ranking). This makes certain that every vote contributes to the final result, ultimately leading to a more representative and democratic outcome.

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Most popular questions from this chapter

Three candidates, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\), are running for mayor. Election rules stipulate that the plurality method will determine the winner. In the event that the plurality method leads to a tie, the Borda count method will decide the winner. The election results are summarized in the following preference table. Under these rules, which candidate becomes the new mayor? $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2 , 0 0 0} & \mathbf{7 5 0 0} & \mathbf{4 5 0 0} \\ \hline \text { First Choice } & \text { C } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { B } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$

Explain why Hamilton's method satisfies the quota rule.

In Exercises 19-22, suppose that the pairwise comparison method is used to determine the winner in an election. If there are five candidates, how many comparisons must be made?

A small country has 24 seats in the congress, divided among the three states according to their respective populations. The table shows each state鈥檚 population, in thousands, before and after the country鈥檚 population increase. $$ \begin{array}{|l|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \begin{array}{l} \text { Original Population } \\ \text { (in thousands) } \end{array} & 530 & 990 & 2240 & 3760 \\ \hline \begin{array}{l} \text { New Population (in } \\ \text { thousands) } \end{array} & 680 & 1250 & 2570 & 4500 \\ \hline \end{array} $$ a. Use Hamilton鈥檚 method to apportion the 24 congressional seats using the original population. b. Find the percent increase, to the nearest tenth of a percent, in the population of each state. c. Use Hamilton鈥檚 method to apportion the 24 congressional seats using the new population. Does the population paradox occur? Explain your answer.

Four professors are running for chair of the Natural Science Division: Professors Darwin (D), Einstein (E), Freud (F), and Hawking (H). The votes of the professors in the natural science division are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & 30 & 22 & 20 & 12 & 2 \\ \hline \text { First Choice } & \text { D } & \text { E } & \text { F } & \text { H } & \text { H } \\ \hline \text { Second Choice } & \text { H } & \text { F } & \text { E } & \text { E } & \text { F } \\ \hline \text { Third Choice } & \text { F } & \text { H } & \text { H } & \text { F } & \text { D } \\ \hline \text { Fourth Choice } & \text { E } & \text { D } & \text { D } & \text { D } & \text { E } \\ \hline \end{array} $$ Who is declared the new division chair using the plurality method?
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