91Ó°ÊÓ

In Exercises 1-2, the preference ballots for three candidates \((A, B\), and \(C)\) are shown. Fill in the number of votes in the first row of the given preference table. ABC BCA BCA CBA CBA ABC ABC BCA BCA CBA ABC ABC $$ \begin{array}{|l|l|l|l|} \hline \text { Number of Votes } & & & \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ BCA ABC ABC CBA

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 wil have 80 seats, divided among the four states according 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.

A university is composed of five schools. The enrollment in each school is given in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \begin{array}{c} \text { Human- } \\ \text { ities } \end{array} & \begin{array}{c} \text { Social } \\ \text { Science } \end{array} & \begin{array}{c} \text { Engi- } \\ \text { neering } \end{array} & \text { Business } & \begin{array}{c} \text { Educa- } \\ \text { tion } \end{array} \\ \hline \text { Enrollment } & 1050 & 1410 & 1830 & 2540 & 3580 \\ \hline \end{array} $$ There are 300 new computers to be apportioned among the five schools according to their respective enrollments. Use Hamilton's method to find each school's apportionment of computers.

A university is composed of five schools. The enrollment in each school is given in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \begin{array}{c} \text { Liberal } \\ \text { Arts } \end{array} & \begin{array}{c} \text { Educa- } \\ \text { tion } \end{array} & \text { Business } & \begin{array}{c} \text { Engi- } \\ \text { neering } \end{array} & \text { Sciences } \\ \hline \text { Enrollment } & 1180 & 1290 & 2140 & 2930 & 3320 \\ \hline \end{array} $$ There are 300 new computers to be apportioned among the five schools according to their respective enrollments. Use Hamilton's method to find each school's apportionment of computers.

A small country is composed of five states, \(A, B, C, D\), and \(E\). The population of each state is given in the following table. Congress will have 57 seats, divided among the five states according to their respective populations. Use Jefferson's method with \(d=32,920\) to apportion the 57 congressional seats. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Population } & 126,316 & 196,492 & 425,264 & 526,664 & 725,264 \\\ \hline \end{array} $$

The following preference table gives the results of a straw vote among three candidates, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 0} & \mathbf{8} & \mathbf{7} & \mathbf{4} \\ \hline \text { First Choice } & \text { C } & \text { B } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { C } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$ a. Using the plurality-with-elimination method, which candidate wins the straw vote? b. In the actual election, the four voters in the last column who voted \(A, C, B\), in that order, change their votes to \(\mathrm{C}, \mathrm{A}, \mathrm{B}\). Using the plurality-with-elimination method, which candidate wins the actual election? c. Is the monotonicity criterion satisfied? Explain your answer.

The travel club members are voting for the American city they will visit next semester: New York (N), San Francisco (S), or Chicago (C). Their votes are summarized in the following preference table. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 6} & \mathbf{8} & \mathbf{6} & \mathbf{4} \\ \hline \text { First Choice } & \text { S } & \text { N } & \text { N } & \text { C } \\ \hline \text { Second Choice } & \text { N } & \text { S } & \text { C } & \text { N } \\ \hline \text { Third Choice } & \text { C } & \text { C } & \text { S } & \text { S } \\ \hline \end{array} $$ Which city is selected using the plurality method?

What is the Alabama paradox?

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{8} & \mathbf{4} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { D } & \text { A } \\ \hline \text { Third Choice } & \text { C } & \text { C } & \text { C } \\ \hline \text { Fourth Choice } & \text { D } & \text { A } & \text { B } \\ \hline \end{array} $$ a. Using the Borda count 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. Suppose that candidate \(C\) drops out of the race. Using the Borda count method, who among the remaining candidates wins the election? Is the irrelevant alternatives criterion satisfied? Explain your answer.

What is the population paradox?