EXAMPLE 9 Apportioning Seats in Parliament
A country has four political parties. its parliament has 100 members, and seats are apportioned by the Hare method after each election so that the number of seats that each party is awarded is as close as possible to being proportional to the number of votes that the party receives.
An election is held, but the parties are unable to form a government, so there is a repeat election. Table 14.9 shows the results of the two elections. The three major parties—Whigs, Tories, and Liberals—all received more votes in the second election, but the Centrists received fewer. The quotas for each party, shown in Table 14.10, were determined by dividing each party’s votes by the standard divisors
for the first election, and 132,517.70 for the second election.
| Party | First Election | Repeat Election |
|---|---|---|
| Whigs | 5,525,381 | 5,657,564 |
| Tories | 3,470,152 | 3,507,464 |
| Liberals | 3,864,226 | 3,885,693 |
| Centrists | 201,203 | 201,049 |
| Totals | 13,060,962 | 13,251,770 |
| Party | First Election | Repeat Election |
|---|---|---|
| Whigs | 42.3045 | 42.6929 |
| Tories | 26.5689 | 26.4679 |
| Liberals | 29.5861 | 29.3221 |
| Centrists | 1.5405 | 1.5171 |
The lower quotas for the results of the first election were 42, 26, 29, and 1, with a sum of 98; thus, the Tories and the Liberals, with the largest fractions, get extra seats. The apportionment after the first election was Whigs, 42; Tories, 27; Liberals, 30; and Centrists, 1.
For the repeat election, the lower quotas were the same, but now the largest fractions belong to the Whigs and the Centrists. Therefore, the new apportionment is Whigs, 43; Tories, 26; Liberals, 29; and Centrists, 2.
The Centrists have gained a seat, although they received fewer votes in the repeat election, while the Tories and the Liberals each lost a seat, even though their vote totals increased in the repeat election. This is an instance of the population paradox. This has a disturbing implication: A group of voters might have unintentionally caused the Centrist Party to gain a seat by switching their votes to the Liberal Party.