91Ó°ÊÓ

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{1 2} & \mathbf{1 0} & \mathbf{6} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { C } & \text { C } & \text { A } & \text { B } \\ \hline \text { Fourth Choice } & \text { D } & \text { D } & \text { D } & \text { A } \\ \hline \end{array} $$ a. Using the plurality-with-elimination method, who is the winner? b. The six voters on the right all move candidate A from last place on their preference lists to first place on their preference lists. Construct a new preference table for the election. Using this table and the plurality-with- elimination method, who is the winner? Is the monotonicity criterion satisfied? Explain your answer.

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?

The preference table shows the results of an election among three candidates, A, B, and C. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{7} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Using the plurality method, who is the winner? b. Is the majority criterion satisfied? Explain your answer. c. Is the head-to-head criterion satisfied? Explain your answer. d. The two voters on the right both move candidate A from last place on their preference lists to first place on their preference lists. Construct a new preference table for the election. Using the table and the plurality method, who is the winner? e. Suppose that candidate \(\mathrm{C}\) drops out, but the winner is still chosen by the plurality method. Is the irrelevant alternatives criterion satisfied? Explain your answer. f. Do your results from parts (b) through (e) contradict Arrow's Impossibility Theorem? Explain your answer.

Describe the monotonicity criterion.

The table shows the 1790 United States census. In 1793, at th direction of President George Washington, 105 seats in t House of Representatives were to be divided among the 15 stat according to their 1790 populations. Use this information to sol 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.

The programmers at the Theater Channel need to select a live musical to introduce their new network. The five choices are Cabaret \((C)\), The Producers \((P)\), Rent \((R)\), Sweeney Todd (S), or West Side Story \((W)\). The 22 programmers rank their choices, summarized in the following preference table. Use the table to solve.$ $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{5} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { C } & \text { S } & \text { C } & \text { W } & \text { W } & \text { P } \\ \hline \text { Second Choice } & \text { R } & \text { R } & \text { P } & \text { P } & \text { R } & \text { S } \\ \hline \text { Third Choice } & \text { P } & \text { W } & \text { R } & \text { R } & \text { S } & \text { C } \\ \hline \text { Fourth Choice } & \text { W } & \text { P } & \text { S } & \text { S } & \text { C } & \text { R } \\ \hline \text { Fifth Choice } & \text { S } & \text { C } & \text { W } & \text { C } & \text { P } & \text { W } \\ \hline \end{array} $$ Determine which musical is selected using the pairwise comparison method.

Explain why Hamilton's method satisfies the quota rule.

Three candidates, A, B, and 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? $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2 , 0 0 0} & \mathbf{7 5 0 0} & \mathbf{4 5 0 0} \\ \hline \text { First Choice } & \text { C } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { B } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$

Describe the difference between the modified divisor, \(d\), in terms of the standard divisor using Jefferson's method and Adams's method.

Three candidates, A, B, and \(\mathrm{C}\), are running for mayor. Election rules stipulate that the pairwise comparison method will determine the winner. In the event that the pairwise comparison 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? $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{6 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} \\ \hline \text { First Choice } & \text { A } & \text { C } & \text { B } & \text { A } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { B } & \text { A } & \text { B } & \text { A } \\ \hline \end{array} $$