10.2: Huntington-Hill Method

The Huntington-Hill Method attempts to minimize the percent differences (unfairness) of how many people each representative will represent.

Huntington-Hill Method

If states X and Y have already been allotted x and y representatives respectively, then state X should be given an additional representative over state Y if:

\[\begin{align}\frac{(\text{population of Y})^2}{y\times (y+1)}<\frac{(\text{population of X})^2}{x\times (x+1)}\end{align}\]

\[\begin{align}\frac{(\text{population of X})^2}{x\times (x+1)}\end{align}\] is called State X's Huntington-Hill Number.

Huntington-Hill Method Procedure

1. Each state gets one representative to begin with. Now calculate the Huntington-Hill number for each state with a Current Representation value of 1.
2. Create a table of Huntington-Hill numbers for each state.
3. Give the next representative to the state with the highest Huntington-Hill number.
4. Recalculate the Huntington-Hill number for that state with the next Current Representation value.
5. Repeat steps 3 and 4 until all the representatives have been given out.
6. State the results in a complete sentence.

Example 1

In a hypothetical world, State X has 750 people, State Y has 300 people, and State Z has 390 people. Use Huntington-Hill Method to apportion 9 representatives.

Solution

Step 1) State X, Y, and Z all get one representative. Note: The denominator will be 1(2).

State X has a population of 750 and 1 representative so the calculation for the Huntington-Hill number for State X is: \[\begin{align}\frac{750^2}{1\times 2}\end{align}\]

You can calculate the Huntington-Hill number for State Y and State Z by replacing the 750 with 300 or 390 respectively.

Step 2) Create a table where X, Y, and Z have one representative.

\(\begin{array}{lrrccc}
\text { Current Representation } & \text { State X } & \text{ State Y } & \text{ State Z } \\
\hline 1 & 281,250 & 45,000 & 76,050\\
\end{array}\)

Step 3) Give the 4th representative to X because it is the highest Huntington-Hill number. Recalculate State X's Huntington-Hill number for Current Representation of 2.

Step 4) Recalculate the Huntington-Hill number for X with 2 representatives. Note: The denominator will now be 2(3)

\(\begin{array}{lrrccc}
\text { Current Representation } & \text { State X } & \text{ State Y } & \text{ State Z } \\
\hline1 & 281,250 & 45,000 & 76,050\\2 & 93,750&&\\
\end{array}\)

Step 5) Repeat steps 3 and 4 until all 9 representatives have been assigned.

Give the 5th representative to X because it is the highest Huntington-Hill number. Recalculate State X's Huntington-Hill number for Current Representation of 3.

\(\begin{array}{lrrccc}
\text { Current Representation } & \text { State X } & \text{ State Y } & \text{ State Z } \\
\hline1 & 281,250 & 45,000 & 76,050\\2 & 93,750&&\\3&46,875&&\\
\end{array}\)

Give the 6th representative to Z and recalculate with Current Representation of 2.

\(\begin{array}{lrrccc}
\text { Current Representation } & \text { State X } & \text{ State Y } & \text{ State Z } \\
\hline1 & 281,250 & 45,000 & 76,050\\2 & 93,750&&25,350\\3&46,875&&\\
\end{array}\)

Give the 7th representative to X

\(\begin{array}{lrrccc}
\text { Current Representation } & \text { State X } & \text{ State Y } & \text{ State Z } \\
\hline1 & 281,250 & 45,000 & 76,050\\2 & 93,750&&25,350\\3&46,875&&\\4&28,125&&\\
\end{array}\)

Give the 8th representative to Y

\(\begin{array}{lrrccc}
\text { Current Representation } & \text { State X } & \text{ State Y } & \text{ State Z } \\
\hline1 & 281,250 & 45,000 & 76,050\\2 & 93,750&15,000&25,350\\3&46,875&&\\4&28,125&&\\
\end{array}\)

Give the 9th representative to X and there is no need to recalculate because that was the last representative.

Step 6) The representatives were assigned in order to State X, Y, Z, X, X, Z, X, Y, X. The final allocation is State X has 5 representatives, State Y has 2 representatives, and State Z has 2 representatives.

Example 2

Three Universities will be sending representatives from the debate team to Washington D.C. for a special hearing. University P has 103 students on the debate team, University Q has 396 students, and University R has 247 students. Use Huntington-Hill Method to apportion 11 representatives.

Solution

The final table for Huntington-Hill numbers is below. Each University gets one representative to begin with. The number in the parentheses is the order to give out the representatives 4 through 11.

\(\begin{array}{lrrccc}
\text { Current Representation } & \text { University P } & \text{ University Q } & \text{ University R } \\
\hline1&5,304.5(10)&78,408.0(4)&30,504.5(5)\\2&1,768.2&26,136.0(6)&10,168.2(8)\\3&&13,068.0(7)&5,084.1\\4&&7,840.8(9)&\\5&&5,227.2(11)&\\
\end{array}\)

The delegates were assigned in order to University P, Q, R, Q, R, Q, Q, R, Q, P, Q. The final allocation is University P has 2 representatives, University Q has 6 representatives, and University R has 3 representatives.