Section 8-4: Apportionment - Am I Represented?

Welcome back to Professor Baker's Math Class! This week, we're diving into the fascinating world of apportionment, specifically Section 8-4 from your textbook, focusing on Chapter 8: Voting and Social Choice. Remember, there will be no quiz this week, so use this time wisely to prepare your final presentations and make them engaging!

Our main question this week is: Am I represented?. We'll explore different methods for allocating representatives to states based on population, ensuring fair and proportional representation. This is a crucial aspect of democratic governance, and understanding these methods will give you a deeper insight into how our political system works.

Key Concepts

  • Ideal District Size (Standard Divisor): This is the foundation of apportionment. We calculate it using the formula:
    • $$ \text{Ideal district size} = \frac{\text{U.S. population}}{\text{House size}} $$
  • State's Quota: This represents each state's fair share of representatives, calculated as:
    • $$ \text{State's quota} = \frac{\text{State's population}}{\text{Ideal district size}} $$

Apportionment Methods

Several methods have been developed to address the challenge of allocating whole numbers of representatives when quotas often involve fractions. Let's explore a few:

  1. Hamilton's Method: This method involves the following steps:
    • Calculate the quota for each state.
    • Allocate each state the whole number part of its quota.
    • Rank the states by the fractional part of their quota and allocate the remaining seats to states with the largest fractions until all seats are filled.
  2. Jefferson's Method: This is an adjusted divisor method:
    • Start by using the ideal district size as the divisor.
    • Calculate each state's quota by dividing its population by the divisor.
    • Round down each quota to the nearest whole number.
    • If the sum of the rounded quotas is less than the total number of seats, decrease the divisor and repeat steps 2 and 3. If the sum is too large, increase the divisor and repeat. Iterate until the sum equals the number of seats.
  3. Adjusted Divisor Methods: These methods (Adams and Webster) adjusts the way the quota is rounded.
    • Adams Method: round each quota up
    • Websters Method: round to the nearest whole number, up if the fractional part is 0.5 or greater.
  4. The Huntington-Hill Method: This method is currently used for congressional apportionment. It uses the geometric mean to determine whether a quota is rounded up or down. The formula for the geometric mean is: $$ \text{Geometric mean} = \sqrt{n(n + 1)} $$ where $n$ is the whole number part of the quotient.

Important Considerations

  • The Alabama Paradox: Be aware of the potential for paradoxical situations where increasing the total number of seats can actually *decrease* the number of seats allocated to a particular state.
  • Staying Within Quota: Ideally, any apportionment method should ensure that the final allocation for each state is within one of its quota (either rounded up or down).

Understanding these methods is crucial for evaluating the fairness and effectiveness of our representative democracy. Good luck with your final presentations!