Hamilton's Method: Videos & Practice Problems
Hamilton's Method is an apportionment procedure for distributing a fixed number of items fairly among groups based on their populations. The process begins with the standard divisor, found by dividing total population by total items: \(D=\frac{\text{total population}}{\text{number of items}}\). Each group’s standard quota is then its population divided by the standard divisor: \(q=\frac{\text{group population}}{D}\).
Next, assign every group its lower quota by rounding each standard quota down to the whole number part. Add these lower quotas and compare the total to the number of items available to determine any leftovers. The remaining items are assigned one at a time to the groups with the largest decimal parts in their standard quotas. This makes Hamilton’s Method a structured way to move from quotas to a final apportionment while keeping the total number of distributed items correct.
Hamilton's Method
A workplace has three divisions led by Angelica, Eliza, and Peggy. They want to distribute 20 new ergonomic office chairs fairly across the three divisions, based on the size of each. Use Hamilton’s Method to apportion the new chairs.
Angelica - 11; Eliza - 7; Peggy - 2
Angelica - 11; Eliza - 6; Peggy - 3
Angelica - 11; Eliza - 6; Peggy - 2
Angelica - 12; Eliza - 6; Peggy - 2
Hamilton's Method Example 1
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Hamilton's Method is an apportionment procedure used to fairly distribute a fixed number of items, such as seats or resources, among groups based on their populations. The process starts by calculating the standard divisor, which is the total population divided by the total number of items to be apportioned. Mathematically, this is represented as . Next, each group's standard quota is found by dividing its population by the standard divisor: . Each group is initially assigned its lower quota, which is the integer part of its standard quota. After summing these lower quotas, if there are leftover items, they are distributed one by one to the groups with the largest fractional parts of their quotas. This ensures the total number of items is correctly apportioned while maintaining fairness based on population sizes.
In Hamilton's Method, the standard divisor is a key value used to determine each group's share of the total items. It is calculated by dividing the total population by the total number of items to be apportioned. The formula is . This divisor represents the average population per item and serves as a benchmark to calculate each group's standard quota. By dividing each group's population by this divisor, you find how many items that group should ideally receive before rounding. This step ensures the apportionment is proportional to population sizes.
In Hamilton's Method, a group's standard quota is the exact fractional number of items it should receive based on its population. It is calculated by dividing the group's population by the standard divisor: . The lower quota is the integer part of this standard quota, found by rounding down the standard quota to the nearest whole number. The lower quota represents the minimum guaranteed number of items assigned to the group initially. After assigning all lower quotas, any leftover items are distributed based on the largest fractional parts of the standard quotas. This distinction helps maintain fairness while ensuring the total number of items is correctly allocated.
After assigning each group its lower quota in Hamilton's Method, there may be leftover items because the sum of the lower quotas is often less than the total number of items to be apportioned. These leftover items are distributed one at a time to the groups with the largest fractional parts of their standard quotas. The fractional part is the decimal portion remaining after subtracting the lower quota from the standard quota. By giving leftover items to groups with the largest fractional parts, Hamilton's Method ensures a fair and proportional distribution that respects the original population ratios while maintaining the total number of items.
Hamilton's Method is advantageous because it is straightforward and ensures that the total number of items is exactly distributed while maintaining proportionality based on population. It respects the lower quota rule, meaning no group receives fewer items than its lower quota. However, a potential drawback is the possibility of the "Alabama Paradox," where increasing the total number of items can cause a group to lose an item, which seems counterintuitive. Despite this, Hamilton's Method remains a popular and intuitive approach for fair apportionment in many contexts.
