Question 14.63

image 33. A country with a parliamentary government has two parties that capture 100% of the vote between them. Each party is awarded seats in proportion to the number of votes received.

  1. Explain why the Webster (Sainte-Laguë) and Hamilton (Hare) methods will always give the same apportionment in this two-party situation.
  2. Explain how to use the result of part (a) to show that the Alabama and population paradoxes cannot occur when the Hamilton method is used to apportion seats between two parties or states.
  3. Explain why the result of part (a) implies that the Webster method satisfies the quota condition when the seats are apportioned between two parties or states.
  4. Will the Jefferson and Hill-Huntington methods also yield the same apportionments as the Hamilton method?

33.

(a) One quota will be rounded up and the other down to obtain the Webster apportionment. The quota that is rounded up will have a fractional part greater than 0.5 and will be greater than the fractional part of the quota that is rounded down. The Hamilton method will give the party whose quota has the larger fractional part an additional seat. Thus, the apportionments will be identical.

(b) These paradoxes never occur with the Webster method, which gives the same apportionment in this case.

(c) The Hamilton method, which always satisfies the quota condition, gives the same apportionment.

A-34

Seats Whigs Tories Liberals Centrists
Priority Seat # Priority Seat # Priority Seat # Priority Seat #
1 5,525,381 1 3,470,152 3 3,864,226 2 201,203
2 1,841,794 4 1,145,717 6 1,288,075 5 67,068
3 1,105,076 7 694,030 10 772,845 9 40,241
4 789,340 8 495,736 14 552,032 12 28,743
5 613,931 11 385,572 17 429,358 15 22,356
6 502,307 13 315,468 21 351,293 19 18,291
7 425,029 16 266,935 24 297,248 22 15,477
8 368,359 18 231,343 28 257,615 26 13,414
9 325,022 20 204,127 227,307 29 11,835
10 290,810 23 182,640 203,380 10,590
11 263,113 25 165,245 184,011 9,581
12 240,234 27 150,876 168,010 8,748
13 221,015 30 138,806 154,569 8,048
14 204,644 128,524 143,119 7,452
15 190,530 119,660 133,249 6,938

The Whigs get 13 of the first 30 seats, the Liberals get 9, the Tories get 8, and the Centrists get none of the first 30 seats.

(d) No, each of these methods is capable of producing a different apportionment. For example, suppose that one party receives 99.9% of the vote and the other receives 0.1%. If the house size is 100, the Jefferson method would apportion all 100 seats to the dominant party. (So would the Hamilton and Webster methods.) The Hill-Huntington method would apportion the dominant party 99 seats, and 1 seat to the other party. If the dominant party received 99.4% and the other party received 0.6%, the Jefferson method would still apportion all of the seats to the dominant party, but the Hamilton and Webster methods would give 1 seat to the other party.