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Chapter 13: Problem 3

MTV's Real World is considering three cities for its new season: Amsterdam (A), Rio de Janeiro (R), or Vancouver (V). Programming executives and the show's production team vote to decide where the new season will be taped. The winning city is to be determined by the plurality method. The preference table for the election is shown at the top of the next column. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2} & \mathbf{9} & \mathbf{4} & \mathbf{4} \\ \hline \text { First Choice } & \text { A } & \text { V } & \text { V } & \text { R } \\ \hline \text { Second Choice } & \text { R } & \text { R } & \text { A } & \text { A } \\ \hline \text { Third Choice } & \text { V } & \text { A } & \text { R } & \text { V } \\ \hline \end{array} $$ a. Which city is favored over all others using a head-tohead comparison? b. Which city wins the vote using the plurality method? c. Is the head-to-head criterion satisfied? Explain your answer.

Short Answer

Expert verified
a. Amsterdam is favored over all others using a head-tohead comparison. b. Amsterdam also wins the vote using the plurality method. c. Yes, the head-to-head criterion is satisfied as the city that won in plurality method also won the head-to-head comparisons.

Step by step solution

01

Tally each first choice

By counting the number of votes for each city as the first choice, obtain the first choice vote count. There are 12 votes for Amsterdam (A), 9 votes for Vancouver (V), and 4 votes for Rio de Janeiro (R).
02

Apply Plurality Method

In the Plurality voting method, the city with the majority of first choice votes should win. With 12 votes, Amsterdam (A), receives the most votes, thus it would be the winner in the Plurality Method.
03

Head-to-Head Comparison

Consider all possible head-to-head contests among the three cities. There will be three comparisons: A vs R, A vs V, and R vs V. In a head-to-head competition between A and R, A gets 12 + 4 = 16 votes while R gets 9 + 4 = 13 votes. Thus, A wins this matchup. For A vs V, A gets 12 + 4 = 16 votes and V gets 9 + 4 = 13 votes. A wins again. Having won twice, Amsterdam (A) would be the head-to-head winner.
04

Verify Head-to-Head Criterion

The head-to-head criterion is satisfied if the city that wins in a head-to-head comparison also wins in the plurality voting. This criterion is satisfied, since Amsterdam (A) is the winner in both the plurality method and head-to-head comparisons.

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Most popular questions from this chapter

The table shows the 1790 United States census. In 1793, at the direction of President George Washington, 105 seats in the House of Representatives were to be divided among the 15 states according to their 1790 populations. Use this information to solve Exercises 23-26. $$ \begin{aligned} &1790 \text { UNITED STATES CENSUS }\\\ &\begin{array}{|l|r|l|r|} \hline \text { Connecticut } & 236,841 & \text { New York } & 331,589 \\ \hline \text { Delaware } & 55,540 & \text { North Carolina } & 353,523 \\ \hline \text { Georgia } & 70,835 & \text { Pennsylvania } & 432,879 \\ \hline \text { Kentucky } & 68,705 & \text { Rhode Island } & 68,446 \\ \hline \text { Maryland } & 278,514 & \text { South Carolina } & 206,236 \\ \hline \text { Massachusetts } & 475,327 & \text { Vermont } & 85,533 \\ \hline \text { New Hampshire } & 141,822 & \text { Virginia } & 630,560 \\ \hline \text { New Jersey } & 179,570 & & \\ \hline \end{array} \end{aligned} $$ Use Hamilton's method to find each state's apportionment of congressional seats.

Make Sense? In Exercises 49-52, determine whether each statement makes sense or does not make sense, and explain your reasoning. A candidate has a majority of the vote, yet lost the election using the plurality method.

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