Apportionment: Divisor Methods Can Violate Quota

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451


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The Apportionment Problem involves translating claims based on "population" to an integer number of "seats" based on h the "house size" of a parliament. Thus, if party A got 32% of the vote in a parliamentary election where there were 160 seats in parliament, the party would be entitled to 51.2 seats. This number is sometimes called the exact quota or exact share for Party A. The quota rule suggests that a "reasonable" apportionment method would give party A either its exact quota (share) if it is an integer, or its exact quota rounded up or down to the nearest integer. The method of largest remainders or Hamilton's Method "obeys quota" while the so-called divisor methods, Jefferson (D'Hondt), Dean, Webster, Huntington-Hill (H-H,) and Adams don't. The examples below, due to Michel Balinski and H. P. Young, address the fact that divisor methods can violate quota and show the germ of why no apportionment can obey quota and be "population" monotone. (There are different versions of a method being population monotone but the intuitive idea is that the method cannot give a state (party) fewer seats when its population goes up or does relatively better in terms of growth compared to other states.) Table 1 shows the results of apportioning 35 seats to 4 parties (or states) based on the vote for the parties or populations of the states. If a method obeyed the "Quota Rule" then state D should receive 26 or 27 seats but since Huntington-Hill and Webster assigns 25 seats, in this example we see that these methods violate the "Quota Rule." Webster's method and H-H's method are population monotone. Table 2 has a similar set-up to Table 1 except that although the total population remains the same, the parties' (states') populations have shifted a bit. Note this results in the exact quota changing but the apportionment of the seats in a 35 seat house staying the same. Note that now, state D's having been given 25 seats does not violate the "Quota rule" because it got 25 seats based on an exact quota of 25.491.

Party (State) Vote (population) Exact quota (share) of house Webster or Huntington-Hlll number of seats
A 70,653 1.552 2
B 117,404 2.579 3
C 210,923 4.633 5
D 1,194,456 26.236 25
Total 1,593,436 35 35

Table 1

Party (State) Vote (population) Exact quota (share) of house Webster or Huntington-Hlll number of seats
A 86,228 1.894 2
B 113,908 2.502 3
C 232,778 5.113 5
D 1,160,522 25.491 25
Total 1,593,436 35 35

Table 2

Suppose we have a method which, when applied to Table 1, obeys the Quota Rule by giving Party D 26 seats, and we do this by taking a seat away from Party B. If now we look at Table 2 and the method gave the results shown, we would have a violation of the "population monotonicity " rule. Party B went down by 3% and D down by 2.8%, so it is now relatively larger than Party B but B gives up a seat to D. The spirit of this example is what Balinski and Young use to show their theorem that a method can't obey both the quota rule and a population montoniticity rule, that is, there is no "perfect" apportionment method.


Balinski, M. and H.P. Young, Fair Representation, Second edition, Brookings Institution Press, 2001.