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Chapter 13: Problem 38

In allocating congressional seats, how does Hamilton's method choose some states over others for preferential treatment? Explain how this is avoided in Jefferson's and Adams's methods.

Short Answer

Expert verified
Hamilton's method can give preferential treatment to states with largest fractions of an additional seat. Jefferson's and Adams's methods avoid this by employing modified divisors that cause all states to have their quotas rounded down (Jefferson's) or rounded up (Adams's), ensuring an equitable distribution of seats.

Step by step solution

01

Understanding Hamilton's Method

Hamilton's Method, also known as the Method of Largest Remainders, allocates seats based on a quota calculated from the total population. In this method, each state first receives an integer part of the quota, which represents the minimum number of seats they're entitled to. If there are remaining seats, the ones with the largest fractions of an additional seat get the extra seats first. This can create a problem of preferential treatment towards states with additional fractions.
02

Understanding Jefferson's Method

Jefferson's Method mitigates this by using a modified divisor which will cause all states to have their quotas rounded down. In this way, preferential treatment is avoided because the additional seats are distributed roundly and no state receives more seats than its calculated quotas.
03

Understanding Adams's Method

Adams's Method, on the other hand, involves modified divisors which will cause all states to have their quotas rounded up. This ensures again that no state receives more seats than its calculated quotas, thus avoiding any possibility of preferential treatment.

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Most popular questions from this chapter

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. \(\begin{array}{llll}\text { } \mathrm{ABC} & \mathrm{BCA} & \mathrm{BCA} & \mathrm{CBA} \\ \mathrm{CBA} & \mathrm{ABC} & \mathrm{ABC} & \mathrm{BCA} \\\ \mathrm{BCA} & \mathrm{CBA} & \mathrm{ABC} & \mathrm{ABC} \\ \mathrm{BCA} & \mathrm{ABC} & \mathrm{ABC} & \mathrm{CBA}\end{array}\) $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & & & \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$

The mathematics department has 30 teaching assistants to be divided among three courses, according to their respective enrollments. The table shows the courses and the number of students enrolled in each course.$$ \begin{array}{|l|c|c|c|c|} \hline \text { Course } & \begin{array}{c} \text { College } \\ \text { Algebra } \end{array} & \text { Statistics } & \begin{array}{c} \text { Liberal Arts } \\ \text { Math } \end{array} & \text { Total } \\ \hline \text { Enrollment } & 978 & 500 & 322 & 1800 \\ \hline \end{array} $$a. Apportion the teaching assistants using Hamilton鈥檚 method. b. Use Hamilton鈥檚 method to determine if the Alabama paradox occurs if the number of teaching assistants is increased from 30 to 31. Explain your answer.

Members of the Student Activity Committee at a college are considering three film directors to speak at a campus arts festival: Ron Howard (H), Spike Lee (L), and Steven Spielberg \((S)\). Committee members vote for their preferred speaker. The winner is to be selected by the pairwise comparison method. The preference table for the election is shown. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 0} & \mathbf{8} & \mathbf{5} \\ \hline \text { First Choice } & \text { H } & \text { L } & \text { S } \\ \hline \text { Second Choice } & \text { S } & \text { S } & \text { L } \\ \hline \text { Third Choice } & \text { L } & \text { H } & \text { H } \\ \hline \end{array} $$ a. Using the pairwise comparison method, who is selected as the speaker? b. Prior to the announcement of the speaker, Ron Howard informs the committee that he will not be able to participate due to other commitments. Construct a new preference table for the election with H eliminated. Using the new table and the pairwise comparison method, who is selected as the speaker? c. Is the irrelevant alternatives criterion satisfied? Explain your answer.

An HMO has 70 doctors to be apportioned among six clinics. The HMO decides to apportion the doctors based on the average weekly patient load for each clinic, given in the following table. Use Jefferson's method to apportion the 70 doctors. (Hint: A modified divisor between 39 and 40 will work.)$$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Clinic } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \\ \hline \begin{array}{l} \text { Average Weekly } \\ \text { Patient Load } \end{array} & 316 & 598 & 396 & 692 & 426 & 486 \\ \hline \end{array} $$

The preference table gives the results of a straw vote among three candidates, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & 14 & 12 & 10 & 6 \\ \hline \text { First Choice } & \text { C } & \text { B } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { A } & \text { C } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$ a. Using the plurality-with-elimination method, which candidate wins the straw vote? b. In the actual election, the six voters in the last column who voted \(A, C, B\), in that order, change their votes to \(C\), A, B. Using the plurality-with- elimination method, which candidate wins the actual election? c. Is the monotonicity criterion satisfied? Explain your answer.
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