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Chapter 4: Problem 67

For an arbitrary state \(X\), let \(q\) represent its standard quota and \(s\) represent the number of seats apportioned to \(X\) under some unspecified apportionment method. Interpret in words the meaning of each of the following mathematical statements. (a) \(s-q \geq 1\) (b) \(q-s \geq 1\) (c) \(|s-q| \leq 0.5\) (d) \(0.5<|s-q|<1\)

Short Answer

Expert verified
Statement a implies the state is over-represented by at least one seat, b implies the state is under-represented by at least one seat, c means the state's representation is almost fair (deviating by at most half a seat), and d indicates the state's representation deviates from the ideal representation by more than half but less than one seat.

Step by step solution

01

Interpretation of Statement a

The inequality \(s-q \geq 1\) implies that the number of seats apportioned to the state \(X\), represented by \(s\), exceeds its standard quota (\(q\)) by at least one. Therefore, the state is over-represented by at least one seat.
02

Interpretation of Statement b

The inequality \(q-s \geq 1\) implies that the standard quota (\(q\)) for the state \(X\) is at least one seat greater than the number of seats actually apportioned (\(s\)). Thus, the state is under-represented by at least one seat.
03

Interpretation of Statement c

The inequality \(|s-q| \leq 0.5\) indicates that the absolute difference between the number of seats apportioned (\(s\)) and the standard quota (\(q\)) is not more than 0.5. It means the representation of the state differs from the ideal representation by less or equal to half a seat. This indicates a nearly fair representation of the state.
04

Interpretation of Statement d

The inequality \(0.5<|s-q|<1\) means that the absolute difference between the seats apportioned (\(s\)) and the standard quota (\(q\)) is more than 0.5, but less than 1. This shows that the actual representation of the state deviates from the ideal representation by more than half a seat but less than one full seat.

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Key Concepts

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

Standard Quota
The term "standard quota" refers to the ideal number of seats a state should have based on its population size in a proportional representation system. To calculate this, you divide the state's population by a standard divisor, which usually represents the total population divided by the total number of available seats in the legislative body.

For instance, if a country's total population is 300 million and the legislative body has 100 seats, the standard divisor would be 3 million. A state with a population of 6 million would therefore have a standard quota of two seats (6 million divided by 3 million).

This concept provides a baseline for a fair representation: theoretically, every state gets as many seats as its standard quota dictates. However, since seats are whole numbers and quotas are often decimal numbers, the actual number of seats (s) may differ, requiring additional apportionment methods to allocate the fractional seat portions appropriately.
Over-Representation
Over-representation occurs when a state receives more seats than its standard quota suggests. In mathematical terms, this is indicated when \(s-q \geq 1\), meaning the allotted seats (\

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Most popular questions from this chapter

The Hamilton-Jefferson hybrid methed. The HamiltonJefferson hybrid method starts by giving each state its lower quota (as per Hamilton's method) and then apportioning the surplus seats using Jefferson's method. (a) Use the Hamilton-Jefferson hybrid method to apportion \(M=22\) seats among four states according to the following populations: \(A\) (population 18,000\(), B\) (population 18,179\(), C\) (population 40.950 ), and \(D\) (population 122,871\()\). (b) Explain why the Hamilton-Jefferson hybrid method can produce apportionments that are different from both Hamilton and Jefferson apportionments (c) Explain why the Hamilton-Jefferson hybrid method can violate the quota rule.

If the standard quota of state \(X\) is 35.41 , then which of the following are possible apportionments to state \(X\) under Webster's method? (a) 35 or 36 (b) 34,35,36 or 37 (c) 33 or 34 (d) 37 or 38 (e) All of the above are possible

If the standard quota of state \(Y\) is 78.24 , then which of the following apportionments to state \(Y\) is (or are) possible under Hamilton's method? (a) 78.2 or 78.3 (b) 78 or 79 (c) 78 only (d) 79 only (e) any positive integer less than 79

Explain why Webster's method cannot produce (a) the Alabama paradox (b) the new-states paradox

Explain why Adams's method cannot produce (a) the Alabama paradox (b) the new-states paradox
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