4.6 Lowndes’ Method

William Lowndes (1782-1822) was a Congressman from South Carolina (a small state) who proposed a method of apportionment that was more favorable to smaller states. Unlike the methods of Hamilton, Jefferson, and Webster, Lowndes’s method has never been used to apportion Congress.

Lowndes believed that an additional representative was much more valuable to a small state than to a large one. If a state already has 20 or 30 representatives, getting one more doesn’t matter very much. But if it only has 2 or 3, one more is a big deal, and he felt that the additional representatives should go where they could make the most difference.

Like Hamilton’s method, Lowndes’s method follows the quota rule. In fact, it arrives at the same quotas as Hamilton and the rest, and like Hamilton and Jefferson, it drops the decimal parts. But in deciding where the remaining representatives should go, we divide the decimal part of each state’s quota by the whole number part (so that the same decimal part with a smaller whole number is worth more, because it matters more to that state).

# Math in Society 4.6 -- Lowndes' Method: who does a leftover seat matter MOST to?
# Hamilton ranks leftovers by the raw decimal part. Lowndes divides that decimal by
# the whole number already earned, so the same decimal counts for more when the
# state is small.
def compare(pops, seats, label):
    divisor = sum(pops.values()) / seats
    quota = {k: v / divisor for k, v in pops.items()}
    base = {k: int(q) for k, q in quota.items()}
    dec = {k: quota[k] - base[k] for k in pops}
    ratio = {k: dec[k] / base[k] if base[k] else float("inf") for k in pops}
    spare = seats - sum(base.values())

    ham_wins = sorted(pops, key=lambda k: (-dec[k], k))[:spare]
    low_wins = sorted(pops, key=lambda k: (-ratio[k], k))[:spare]

    print(f"{label}: {seats} seats, divisor {divisor:,.5f}, "
          f"{spare} leftover seat(s) to place")
    print(f"   {'':<12}{'quota':>10}{'lower':>7}{'decimal':>10}"
          f"{'dec/whole':>11}{'Hamilton':>10}{'Lowndes':>9}")
    for k in sorted(pops, key=lambda k: -pops[k]):
        print(f"   {k:<12}{quota[k]:>10.4f}{base[k]:>7}{dec[k]:>10.4f}"
              f"{ratio[k]:>11.4f}{base[k] + (k in ham_wins):>10}"
              f"{base[k] + (k in low_wins):>9}")
    if set(ham_wins) != set(low_wins):
        print(f"   the two methods disagree: Hamilton hands the spare seat(s) to "
              + ", ".join(ham_wins))
        print(f"   {'':<26}Lowndes hands them to " + ", ".join(low_wins))
    else:
        print("   both methods place the leftover seat(s) identically here")
    print()

delaware = {"Kent": 162310, "New Castle": 538479, "Sussex": 197145}
rhode_island = {"Bristol": 49875, "Kent": 166158, "Newport": 82888,
                "Providence": 626667, "Washington": 126979}
compare(delaware, 41, "Delaware")
compare(rhode_island, 75, "Rhode Island")     # EDIT: change either house size

print("New Castle's decimal (.5872) beats Kent's (.4111), so Hamilton gives it the")
print("seat. But .5872 spread over 24 seats New Castle already holds is worth far")
print("less per seat than .4111 spread over Kent's 7, so Lowndes gives it to Kent.")
print("Neither is 'right' -- they answer different questions about what a seat is for.")
print("Try it: run compare(rhode_island, 73, 'Rhode Island') and then 77. Count how")
print("often Providence, the largest county, wins a leftover under each method.")

Note

Lowndes’s Method

Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the divisor.
Divide each state’s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the quota.
Cut off all the decimal parts of all the quotas (but don’t forget what the decimals were). Add up the remaining whole numbers.
Assuming that the total from Step 3 was less than the total number of representatives, divide the decimal part of each state’s quota by the whole number part. Assign the remaining representatives, one each, to the states whose ratio of decimal part to whole part were largest, until the desired total is reached.

Example 1

We’ll do Delaware again. We begin in the same way as with Hamilton’s method:

County	Population	Quota	Initial
Kent	162,310	7.4111	7
New Castle	538,479	24.5872	24
Sussex	197,145	9.0017	9
Total	$897, 934$		$40$

Show solution

We need one more representative. To find out which county should get it, Lowndes says to divide each county’s decimal part by its whole number part, with the largest result getting the extra representative:

$\begin{matrix} Kent: & 0.4111 / 7 \approx 0.0587 \\ New Castle: & 0.5872 / 24 \approx 0.0245 \\ Sussex: & 0.0017 / 9 \approx 0.0002 \end{matrix}$

The largest of these is Kent’s, so Kent gets the $41 to the power th$ representative:

County	Population	Quota	Initial	Ratio	Final
Kent	162,310	7.4111	7	0.0587	8
New Castle	538,479	24.5872	24	0.0245	24
Sussex	197,145	9.0017	9	0.0002	9
Total	$897, 934$		$40$		$41$

Example 2

Rhode Island, again beginning in the same way as Hamilton:

County	Population	Quota	Initial
Bristol	49,875	3.5538	3
Kent	166,158	11.8395	11
Newport	82,888	5.9061	5
Providence	626,667	44.6528	44
Washington	126,979	9.0478	9
Total	$1, 052, 567$		$72$

Show solution

We divide each county’s quota’s decimal part by its whole number part to determine which three should get the remaining representatives:

$\begin{matrix} Bristol: & 0.5538 / 3 \approx 0.1846 \\ Kent: & 0.8395 / 11 \approx 0.0763 \\ Newport: & 0.9061 / 5 \approx 0.1812 \\ Providence: & 0.6528 / 44 \approx 0.0148 \\ Washington: & 0.0478 / 9 \approx 0.0053 \end{matrix}$

The three largest of these are Bristol, Newport, and Kent, so they get the remaining three representatives:

County	Population	Quota	Initial	Ratio	Final
Bristol	49,875	3.5538	3	0.1846	4
Kent	166,158	11.8395	11	0.0763	12
Newport	82,888	5.9061	5	0.1812	6
Providence	626,667	44.6528	44	0.0148	44
Washington	126,979	9.0478	9	0.0053	9
Total	$1, 052, 567$		$72$		$75$

As you can see, there is no “right answer” when it comes to choosing a method for apportionment. Each method has its virtues, and favors different sized states.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.