Apportionment and Voting

1987 Philippine Constitution
Article VI- The Legislative Department
Section 5 (1)

The House of Representatives shall be composed of not more than two hundred and fifty members, unless otherwise fixed by law, who shall be elected from legislative districts apportioned among the provinces, cities, and the Metropolitan Manila area in accordance with the number of their respective inhabitants, and on the basis of a uniform and progressive ratio, and those who, as provided by law, shall be elected through a party-list system of registered national, regional, and sectoral parties or organizations.

1987 Philippine Constitution
Article VI- The Legislative Department
Section 5 (3)

Each legislative district shall comprise, as far as practicable, contiguous, compact and adjacent territory. Each city with a population of at least two hundred fifty thousand, or each province, shall have at least one representative.

Apportionment

✓ is a method of dividing a whole into various parts or among units
✓ has its roots in the US Constitution
✓ main question: How many voters must be represented by each member of the House of Representatives?

Two Competing Plans in 1790

✓ The Hamilton Plan (Alexander Hamilton)
✓ The Jefferson Plan (Thomas Jefferson)

Standard Divisor (D)

The number of votes represented by each representative

𝐷 = 𝑃/𝑛

where:
𝐷 − the standard divisor
𝑃 − the total population
𝑛 − the number of representatives

Standard Quota (Q)

The whole number part of the quotient of a population divided by the standard divisor

𝑄 = 𝑃/𝐷

where
𝑄 − the standard quota
𝑃 − the state’s (city or province) population
𝐷 − the standard divisor

The Hamilton Plan

✓ If the total standard quota of all subgroups does not equal the number of people to apportion (or the number of representatives), review the calculation of the standard quota and assign an additional representative to the subgroup with the largest decimal number.
✓ Continue this process until you have reached the desired number of representatives.

The Jefferson Plan

✓ Uses a modified standard divisor that yields the correct number of representatives by trial and error
✓ The modified standard divisor is always smaller than the standard divisor.

Average Constituency

The average constituencies of the subgroups are approximately the same

𝐶 = 𝑐/𝑛𝑟

where
𝐶 − the average constituency.
𝑐 − the state (city or province) population
𝑛𝑟 − the number of representatives from the state

Apportionment Principle

When adding a new representative to a subgroup, the representative is assigned to the subgroup in such a way as to give the smallest relative unfairness of apportionment.

R = A/C

Where
R – relative unfairness of apportionment
A – absolute unfairness of apportionment

𝐴 = |𝐶1 - 𝐶2|

C is the average constituency of the subgroup receiving the new representative

Huntington-Hill Apportionment Method

✓ Method of equal proportions
✓ Used by the US House of Representatives since 1940
✓ When there is a choice of adding one representative to a number of subgroup, the representative should be added to the subgroup with the greatest Huntington-Hill number

𝐻 = (𝑃𝐴)^2 / 𝑎 (𝑎 + 1)

where
𝑃𝐴= the population of subgroup 𝐴
𝑎 = the current number of representatives of subgroup 𝐴

Balinski-Young Impossibility Theorem

Any apportionment method will either violate the quota rule or produce paradoxes.

Alabama Paradox

Even when the population of subgroups do not change, one subgroup loses a representative.

Population Paradox

A subgroup loses a representative even when its population is increasing faster than that of the other subgroups.

New States Paradox

Some other subgroups lose or gain a representative even when a number of representatives was added to account for a new subgroup of a population.