11.5 Fairness in Apportionment Methods
The Alabama, population, and new-states paradoxes, where adding seats or people costs a group representation, and how to limit the damage.
Contemporary Mathematics, 11.5 Fairness in Apportionment Methods
8 exercises
11.5 Exercise 1
The 1880 census recorded the population of Alabama as 1,513,401 and that of the U.S. as 62,979,766.
- Calculate the standard divisor and standard quota for the State of Alabama based on a house size of 299.
- Calculate the standard divisor and standard quota for the State of Alabama based on a house size of 300.
- Did the standard quota increase or decrease when the house size increased?
- Consider the H…
11.5 Exercise 3
Suppose that States A and B each have a population of 6, while State C has a population of 2.
- Use the Hamilton method to apportion 10 seats.
- Use the Hamilton method to apportion 11 seats.
- Does this example demonstrate the Alabama paradox? If so, how?
11.5 Exercise 5
Suppose that 18 respirators are to be apportioned to three hospitals based on their capacities. The Hamilton method is used to allocate the respirators in 2020, then to reallocate based on new capacities in 2021. The results are shown in the table below. How does this demonstrate the population paradox?
11.5 Exercise 7
The country of Costaguana has three states: Azuera with a population of 894,000; Punta Mala with a population of 696,000; and Esmeralda with a population of 215,000. There are 38 seats in the Congress of Costaguana. The apportionment of the seats is determined by Hamilton’s method to be: 19 for Azuera, 15 for Punta Mala, and 4 for Esmerelda. A census reveals that the population has grown and the s…
11.5 Exercise 9
The country of San Lorenzo has grown from two states to three. The house size of the congress has been increased by eight and the seats have been reapportioned to accommodate the new state of Oscuridad. The constitution mandates the use of the Hamilton method of apportionment. Use this information and the following table to answer the questions.
- What was the original house size?
- What is the new…
11.5 Exercise 11
The country of Gulliversia has two states: Lilliput with a population of 700,000 and Brobdingnag with a population of 937,000. The constitution of Gulliversia requires that the 90 congressional seats be apportioned by Hamilton’s method. Lilliput has received 38 seats while Brobdingnag has received 52 seats. Recently, the island of Houyhnhnmsland with a population of 170,000 has joined the union, b…
11.5 Exercise 13
Oklahoma was admitted as the 46th state on November 16, 1907. Before Oklahoma joined the union, the U.S. population was approximately 75,030,000 and the House of Representatives had 386 seats. The new state had a population of approximately 970,000. Use this information to estimate the original standard divisor to the nearest hundred, the new population, the new house size, and the number of seats…
11.5 Exercise 15
The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents and The Villages with 10,746 residents. A council with 100 members supervises the municipality’s operations. The council votes to annex an unincorporated subdivision called The Lakes with a population of 630. They plan to increase the size of the council to maintain the ratio of seats to residents suc…
Adapted from Contemporary Mathematics © 2023 Rice University. Licensed under CC BY 4.0. Source: Contemporary Mathematics. Imported at revision fa7c1a1.