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Chapter 13: Problem 19
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?
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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?
The theater society members are voting for the kind of play they will perform next semester: a comedy (C), a drama (D), or a musical (M). Their votes are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Number of Votes } & 10 & 6 & 6 & 4 & 2 & 2 \\ \hline \text { First Choice } & \mathrm{M} & \mathrm{C} & \mathrm{D} & \mathrm{C} & \mathrm{D} & \mathrm{M} \\ \hline \text { Second Choice } & \mathrm{C} & \mathrm{M} & \mathrm{C} & \mathrm{D} & \mathrm{M} & \mathrm{D} \\ \hline \text { Third Choice } & \mathrm{D} & \mathrm{D} & \mathrm{M} & \mathrm{M} & \mathrm{C} & \mathrm{C} \\ \hline \end{array} $$ \(\text { Which type of play is selected using the plurality method? }\)
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? $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2 , 0 0 0} & \mathbf{7 5 0 0} & \mathbf{4 5 0 0} \\ \hline \text { First Choice } & \text { C } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { B } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$
Throughout this Exercise Set, in computing standard divisors, standard quotas, and modified quotas, round to the nearest hundredth when necessary. A small country is comprised of four states, \(A, B, C\), and \(D\). The population of each state, in thousands, is given in the following table. Use this information to solve Exercises $1-4 . $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 138 & 266 & 534 & 662 & 1600 \\ \hline \end{array} $$ According to the country's constitution, the congress will have 80 seats, divided among the four states according to their respective populations. a. Find the standard divisor, in thousands. How many people are there for each seat in congress? b. Find each state's standard quota. c. Find each state's lower quota and upper quota.
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