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Chapter 13: Problem 34

Explain why Hamilton's method satisfies the quota rule.

Short Answer

Expert verified
In Hamilton's method, apportionment strictly follows the quotas. The first portion of the allocation adheres to the lower quotas and the distribution of leftover seats follows the decimal parts of the quotas. Therefore, the final apportionment is either equal to or very close to the quota, hence, satisfying the quota rule.

Step by step solution

01

Understanding Hamilton's Method

Hamilton's Method is an apportionment method which involves calculating a quota for distribution based on population. It assigns seats strictly based on the quotas and disregards fractions.
02

Applying Hamilton's Method

First, a standard divisor is calculated by dividing the total population by the total number of available seats. The standard quota for each state, party, or group is then calculated by dividing its population by the standard divisor. The lower quota (whole part of the standard quota) is the initial apportion. The leftover seats are distributed to the states, parties, or groups with the highest decimal parts in their quotas.
03

Explaining the Quota Rule

Quota Rule involves assigning an allocation that is as close as possible to the 'deserved' allocation or quota. The deserved allocation or quota is calculated by dividing the total items to allocate by the total number of participants.
04

Proving Hamilton's method satisfies the quota rule

As we see in Hamilton's method, the first part of the apportionment strictly follows the lower quotas. And the distribution of residual seats also strictly follows the decimal part of the quotas. Therefore, the final apportionment is either the floor or the ceiling of the quota, thus satisfying the quota rule.

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Most popular questions from this chapter

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.

What is the plurality-with-elimination method? Why is it advantageous to rank the candidates when using this 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? $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2 , 0 0 0} & \mathbf{7 5 0 0} & \mathbf{4 5 0 0} \\ \hline \text { First Choice } & \text { C } & \text { A } & \text { A } \\ \hline \text { Second Choice } & \text { B } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { A } & \text { C } & \text { B } \\ \hline \end{array} $$

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. $$ \begin{array}{|l|l|l|l|} \hline \text { Number of Votes } & \mathbf{7} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ 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 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鈥檚 Impossibility Theorem? Explain your answer.
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