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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.
Describe the head-to-head criterion.
What is the plurality-with-elimination method? Why is it advantageous to rank the candidates when using this method?
Is it possible to have election results using a particular voting method that satisfy all four fairness criteria? If so, does this contradict Arrow's Impossibility Theorem?
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. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Number of Votes } & 10 & 6 & 6 & 4 & 2 & 2 \\ \hline \text { First Choice } & \mathrm{M} & \mathrm{C} & \mathrm{D} & \mathrm{C} & \mathrm{D} & \mathrm{M} \\ \hline \text { Second Choice } & \mathrm{C} & \mathrm{M} & \mathrm{C} & \mathrm{D} & \mathrm{M} & \mathrm{D} \\ \hline \text { Third Choice } & \mathrm{D} & \mathrm{D} & \mathrm{M} & \mathrm{M} & \mathrm{C} & \mathrm{C} \\ \hline \end{array} $$ \(\text { Which type of play is selected using the plurality method? }\)
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