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Chapter 13: Problem 39

How are modified quotas rounded using Webster's method?

Short Answer

Expert verified
Modified quotas are rounded using standard rounding rules in Webster's method. First compute the initial quotas for each entity, then establish their modified quotas by finding the geometric mean of the quotas directly above and below. Finally, round these quotas normally to allocate seats.

Step by step solution

01

Quote calculation

Start by calculating the initial quotas for each party or each state. You can do this by dividing the total population of each by the standard divisor, which is calculated by dividing the total population by the total number of seats available.
02

Compute Modified Quotas

The next step is to calculate the modified quotas by implementing the Webster's method. This is done by taking the geometric mean of the quotas lower and higher than the initial quota.
03

Round the Quotas

The final step is to round the modified quotas. With Webster's method, modified quotas are usually rounded using ordinary rounding rules (round down if decimal is less than 0.5, round up if decimal is 0.5 or more). Assign seats based on the resulting rounded quotas.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quota Calculation
Understanding quota calculation is a key step in Webster's method. It begins by determining the initial quotas for distributing seats among parties or states. This is accomplished by finding a standard divisor. To calculate this divisor, divide the total population by the number of seats available. This gives you an average that helps in determining how many seats each group deserves based on their population size.

Next, calculate each group's initial quota by dividing their total population by this standard divisor. The result is a non-integer number that indicates the proportion of seats a group should ideally receive. This figure shows how close or above each group's need is compared to others. Quota calculation ensures a fair distribution based on relative size."
Geometric Mean
Webster's method employs the geometric mean to adjust the quotas before finalizing seat distribution. To find this, we use the initial quota calculation.

The geometric mean helps with deciding how to round each quota. For each group's quota, you need both the lower and higher whole number quotas surrounding it. The lower quota is simply the floor of the quota, while the upper quota is the ceiling.
  • Calculate the geometric mean by multiplying the two whole number quotas, then take the square root of the product.
  • This mean serves as the balanced point for deciding if the actual quota is closer to its floor or ceiling.
By using the geometric mean, Webster's method ensures more equitable decisions, particularly in balancing between needs that are close to each other or could be unfairly skewed by simple arithmetic rounding."
Seat Allocation
Once the geometric means are calculated, Webster's method focuses on seat allocation through a rounding step. The concept here is to correctly round modified quotas using standard rounding rules鈥攔ound up if the decimal part is 0.5 or more, round down if less.

Using results from the geometric mean helps to decide the best way to round the quotas, ensuring a balanced and fair approach.
  • Rounding involves comparing each initial quota's decimal part against the geometric mean.
  • Allocation of seats is prioritized by how quotas are rounded, ensuring that numbers push higher or lower in alignment with fair distribution.
This results in seat distribution that reflects both size and need, minimizing significant deviations from fairness that could arise through other methods."

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Most popular questions from this chapter

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 programmers at the Theater Channel need to select a live musical to introduce their new network. The five choices are Cabaret \((C)\), The Producers \((P)\), Rent \((R)\), Sweeney Todd (S), or West Side Story \((W)\). The 22 programmers rank their choices, summarized in the following preference table. Use the table to solve.$ $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{5} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { C } & \text { S } & \text { C } & \text { W } & \text { W } & \text { P } \\ \hline \text { Second Choice } & \text { R } & \text { R } & \text { P } & \text { P } & \text { R } & \text { S } \\ \hline \text { Third Choice } & \text { P } & \text { W } & \text { R } & \text { R } & \text { S } & \text { C } \\ \hline \text { Fourth Choice } & \text { W } & \text { P } & \text { S } & \text { S } & \text { C } & \text { R } \\ \hline \text { Fifth Choice } & \text { S } & \text { C } & \text { W } & \text { C } & \text { P } & \text { W } \\ \hline \end{array} $$ Determine which musical is selected using the pairwise comparison method.

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 4} & \mathbf{1 2} & \mathbf{1 0} & \mathbf{6} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { B } & \text { C } \\ \hline \text { Third Choice } & \text { C } & \text { C } & \text { A } & \text { B } \\ \hline \text { Fourth Choice } & \text { D } & \text { D } & \text { D } & \text { A } \\ \hline \end{array} $$ a. Using the plurality-with-elimination method, who is the winner? b. The six voters on the right all 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 this table and the plurality-with- elimination method, who is the winner? Is the monotonicity criterion satisfied? Explain your answer.

What is the plurality-with-elimination method? Why is it advantageous to rank the candidates when using this method?

What is the population paradox?
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