In previous posts, we talked about current apportionment and concerns about it, proposed the Wyoming Rule to address the concerns and even provided several arguments against the proposed rule. Before we can decide if the Wyoming Rule is the way to go, we need to understand something more basic, how representatives have been divided up. Over time, different methods have been created to solve this problem. Each method tries to be fair, but each defines “fair” a little differently. There have been four different methods used so far:
- Hamilton/Vinton Method
- Jefferson Method
- Webster Method
- Huntington-Hill Method (the one we use today)
The Core Problem: Dividing Whole Seats Fairly
Imagine you have:
- 3 states
- 20 total seats
- Different populations in each state
You calculate how many seats each state should get based on population. But you might end up with numbers like:
- 3.4 seats
- 1.7 seats
- 4.9 seats
That adds up to 20 seats—but you can’t assign decimals in real life.
So the question becomes:
How do we round these numbers into whole seats without being unfair?
Different methods answer that question in different ways.
The Hamilton Method (Also Called the Vinton Method)
Basic Idea
Give each state what it clearly deserves first, then hand out the remaining seats based on which is closest to getting another one.
How It Works
- Calculate each state’s number of seats
- Give each state the whole number part first
- Count how many seats are left
- Give the remaining seats to states with the largest decimal remainders
Example
| State | Population |
| State A | 1,500 |
| State B | 9,100 |
| State C | 9,400 |
| Total | 20,000 |
Find the Standard Divisor
- Calculate the average number of people per seat: Standard Divisor = 20,000 total population / 20 seats = 1,000
- Test the Standard Quota
- Divide each population by the divisor and round down to the nearest whole number.
- State A: 1,500/1,000=1.5→ 1 seat
- State B: 9,100/1,000=9.1→ 9 seats
- State C: 9,400/1,000=9.4→ 9 seats
- Total Seats: 1+9+9=19
- Problem: We have 20 seats to fill, but we only assigned 19
- State A has a remainder of 0.5
- State B has a remainder of 0.1
- State C has a remainder of 0.4
- Since State A has the largest remainder (0.5), it receives the 20th seat.
The Problem
The Hamilton method has a strange issue called the Alabama Paradox which occurs when adding more total congressional seats can cause a state to lose a seat. That feels wrong. If the House gets bigger, no state expects to lose representation. Because of this, the method was abandoned at the federal level.
The Jefferson Method
Basic Idea
Favor larger states slightly by adjusting how we round numbers.
How It Works
Instead of normal rounding:
- You always round down
- Then adjust a divisor (a scaling number) until all seats are assigned
This process tends to:
- Give bigger states a small advantage
- Reduce the chances of paradoxes like the Hamilton method
Example
- Test the Standard Quota
- Divide each population by the standard divisor and round down (this is called the lower quota).
- State A: 1,500/1,000=1.5→ 1 seat
- State B: 9,100/1,000=9.1→ 9 seats
- State C: 9,400/1,000=9.4→ 9 seats
- Total Seats: 1+9+9=19
To give out more seats, we need to make the divisor smaller. Through trial and error, let’s try a modified divisor of 940 instead of 1,000.
- Test the revised quota
- State A: 1,500/949=1.5→ 1 seat
- State B: 9,100/949=9.5→ 9 seats
- State C: 9,400/949=10→ 10 seats
- Total Seats: 1+9+10=20
- By lowering the divisor from 1,000 to 940, State C’s quotient bumped up just enough to reach 10, allowing it to claim the 20th seat. This method tends to favor larger states because their larger populations are more likely to “absorb” the decrease in the divisor and cross the next whole-number threshold.
The Problem
- Smaller states often lose out
- Representation becomes slightly less equal per person
In other words, this method values stability over strict fairness.
The Webster Method
Basic Idea
Try to treat large and small states equally by using standard rounding.
How It Works
- Calculate each state’s seat number
- Round normally:
- .5 and above → round up
- below .5 → round down
- Adjust the divisor until the total number of seats is correct
Example
- Divide each population by the standard divisor and round either up or down
- State A: 1,500/1,000=1.5→ 2 seats
- State B: 9,100/1,000=9.1→ 9 seats
- State C: 9,400/1,000=9.4→ 9 seats
- Total Seats: 2+9+9=20
- With this example, the smallest state gained an extra seat based on the rounding
The Problem
- Can still produce small inconsistencies
- Not perfect at avoiding all paradoxes
Still, many people see Webster’s method as a “middle ground” approach.
The Huntington-Hill Method (Used Today)
Basic Idea
Focus on equal representation per person, not just fair rounding.
How It Works
Instead of simple rounding, it uses a formula based on geometric means to decide when to round up or down. Without getting too technical:
- It compares how representation changes when a state gains or loses a seat
- It tries to keep the population per representative as equal as possible across all states
Example
Get ready for some “math”, buckle-up.
- Using the same numbers from prior examples, including the same standard divisor (20,000/20=1,000)
- State A: 1,500/1,000=1.5→Lower (n) = 1. Upper (n+1) = 2
- Threshold = Square root 1×2 = 1.414
- Result: 1.5 is greater than 1.414→ 2 seats
- State B: 9,100/1,000=9.1→ Lower (n) = 9. Upper (n+1) = 10
- Threshold = Square root 9×10 = 9.487
- Result: 9.1 is less than 9.487→ 9 seats
- State C: 9,400/1,000=9.4→ Lower (n) = 9. Upper (n+1) = 10
- Threshold = Square root 9×10 = 9.487
- Result: 9.4 is less than 9.487→ 9 seats
- Total Seats: 2+9+9=20
- In this small-scale example, Webster and Huntington-Hill yielded the same result; however, on a national scale with 435 seats, the Huntington-Hill method’s lower threshold for small numbers makes it slightly easier for small states to “keep” a second or third seat compared to Webster.
The Problem
- Harder to explain and understand
- Less intuitive than other methods
- Feels more like a formula than a rule
This complexity is one reason apportionment is not widely discussed.
Why This Matters for the Bigger Question
If you remember from earlier posts, my starting belief is that, the House may be too small, and the Wyoming Rule might help fix representation problems. However, even if we change the size of the House, we still need a method to divide the seats if the math isn’t exact. This means the Wyoming Rule doesn’t replace apportionment methods it becomes part of the equation.