*Felix P. Muga II teaches mathematics at the Ateneo de Manila University and is a Fellow of the Center for People Empowerment in Governance (CenPEG). His homepage is at http://www.math.admu.edu.ph/~fpmuga. This article was published in the Opinion Section, Yellow Pad Column of BusinessWorld, July 23, 2007 edition, page S1/4*.

In his article, “Law, Mathematics and the Party-List System” (Philippine Daily Inquirer, July 15, 2007, page 15), former Supreme Court (SC) Chief Justice Artemio Panganiban enumerates four inviolable parameters that govern the seat allocation method of the Philippine party-list system.

First, the twenty percent allocation—the combined number of all party-list congressmen shall not exceed twenty percent of the total membership of the House of Representatives, including those elected under the party list;

Second, the two percent threshold—only those parties garnering a minimum of two percent of the total valid votes cast for the party-list system are “qualified” to have a seat in the House of Representatives (section 11 of the Party-List Law, RA 7941);

Third, the three-seat limit—each qualified party, regardless of the number of votes it obtained, is entitled to a maximum of three seats, that is, one “qualifying” and two additional seats (section 11 of RA 7941); and

Fourth, proportional representation—the additional seats which a party is entitled to shall be computed “in proportion to their total number of votes” (sections 2 and 11 of RA 7941).

Let us call them the .2TOTAL, the .02THRESH, the 3-SEATCAP, and the PR parameters, respectively.

He failed to mention, however, whether the four inviolable parameters are consistent with each other. A consistent system of parameters is necessary in constructing a formula that will give a correct solution.

Consider a parameter that specifies “a number n cannot exceed 3” and another parameter that produces “the number n = 5.” There is no number that is common to the two parameters. Hence, there is no solution and therefore, the parameters are inconsistent.

I believe that the system determined by the four parameters is inconsistent. Hence, no formula can be formulated to give a correct solution.

**.2TOTAL parameter rephrased**

The 1987 Constitution mandates that “the party-list representatives shall constitute twenty per centum of the total number of representatives including those under the party list ….”

The .02TOTAL parameter did not faithfully follow this formulation. I believe the said parameter was rephrased so that the .02TOTAL and 3-SEATCAP parameters would be consistent.

**PR parameter violated**

But the High Court fails to make the 3-SEATCAP consistent with the PR parameter (or the principle of proportional representation) which asserts that the qualified party’s share of the total seats is equal to its share of the total votes of all parties qualified to receive a seat.

Mathematically, this means

**no. of seats of a qualified party no. of votes it obtained ——————————————- = ——————————————- total no. of seats total votes of all qualified parties **

Thus, by the PR parameter, the (ideal) number of seats of a qualified party is given by:

(ideal)no. of seats = its%share of the total votes x total no. of seats

where the total votes means the total votes of all parties qualified to receive a seat.

The inconsistency of the 3-SEATCAP with the PR parameter can be shown by citing a counter-example. In the 2007 election the total number of party-list seats is 55 and suppose that a party obtains 10 percent of the total number of votes of all qualified parties which is 8,070,680 based on the Party-List Canvass Report No. 27 dated June 29, 2007. By the 3-SEATCAP parameter, the party cannot have more than 3 seats. However, by the PR parameter, its number of seats is 10% x 55 = 5.5 or at least five seats (but not more than 6 seats).

Hence, the 3-SEATCAP is inconsistent with the PR parameter.

Since the system is inconsistent, no formula can be formulated that complies with all the four parameters.

The Panganiban Formula cannot produce the correct solution. If it is applied to the latest Party-List Tally (Canvass Report No. 27), a “solution” is obtained with 21 seats where Buhay has 3 seats. The solution follows the .2TOTAL, .02THRESH and the 3-SEATCAP parameters.

By the PR parameter, Buhay with 14.2643% of the total votes of all qualified parties will get about 14.2643% × 55 = 7.84536495 seats. The formula violates the PR parameter by more than 4 seats and the total number of seats violated is 28 over all the qualified parties. This is equivalent to at least (28/55) × 8,070,680 = 4,108,701 disenfranchised voters.

If there is no correct solution, the party-lists cannot be correctly represented in the House of Representatives.

Formulating a consistent system

I believe that the 3-SEATCAP parameter should be rejected by the Supreme Court since it is the cause of the inconsistency of the system. Also, it is the reason that the .2TOTAL parameter was rephrased.

Moreover, the basis for the 3-seat limit does not exist anymore. The imposition of a limit on the party-list seats was dealt with in the Constitutional Commission in the context of a two-party system. In the proposal, the seats for the major political parties would be limited to enable sectoral or regional groups to have the majority of the seats for the party-list.

In this regard, a commissioner said, “This way, we will open it up and enable sectoral groups, or maybe regional groups, to earn their seats among the fifty. When we talk about limiting it, if there are two parties, then we are opening it up to the extent of 30 seats” (Records of the ConCom, pp. 85-86).

The final draft of the 1987 Constitution mentioned no cap because its framers eventually resolved not to adopt the two-party system and leave the development of a free and open multi-party system to “the choice of the people.”

It may be argued that there are still major political parties which may dominate the party-list system even if the two-party system is not adopted. However, the SC in Bagong Bayani vs Comelec (G.R. No. 147589, June 25, 2003) disqualified the major political parties from participating in the party-list election.

The .02TOTAL parameter should be rephrased back to the original intention of the Constitution.

The .02THRESH and the PR parameters shall be retained.

**Proposed solution: LR Method**

In Germany, the Niemeyer Formula computes the number of party-list seats in a given Land (or State) of a qualified party by first determining the number of seats the party is entitled to in the Land using the Largest Remainder Method (LR Method) and subtracting from this number the number of single-member legislative seats it won in the Land. Although, we cannot apply the Niemeyer Formula to our system, we propose that the Supreme Court or the 14th Congress adopt the LR Method that the Niemeyer Formula utilized.

The LR Method is also used by the party-list systems of South Korea, Russia, Taiwan, Ukraine, Germany, Mexico, Iceland, and Slovenia.

The Largest Remainder Method is applied as follows:

Suppose that in a party-list election, there are 10 available seats and 3 qualified parties after applying the .02THRESH parameter where Party A has 10,500 votes, B has 8,800, and C has 3,700. Thus, the total number of votes of all the qualified parties is 23,000.

- The total number of available seats shall be multiplied by the number of votes obtained by each qualified party. The product shall be divided by the total votes obtained by all the qualified parties. Thus, party A has (55 ×10,500) / 23,000 = 4.57, B has 3.83, and C has 1.61. This is the ideal number of seats of each qualified party based on the principle of proportional representation.
- Each qualified party receives one seat for each whole number resulting from the calculation in (1). Thus, the number of seats of parties A, B and C is 4, 3, and 1, respectively. Since the total number of seats allocated is 8, there are 2 remaining seats.
- The remaining seats are then allocated in the descending sequence of the decimal fractions of the qualified parties.

The decimal fraction of parties B, C, and A is 0.83, 0.61, and 0.57, respectively. Thus, parties B and C shall be given one additional each.

Hence, parties A, B and C are given 4, 4, and 2 seats, respectively and all the available seats are distributed. Hence the LR Method affirms the .2TOTAL parameter. Since the ideal and the actual number of seats of A is 4.57 and 4, respectively, the magnitude of the seat allocation error is less than one seat. This is also true for parties B and C. Therefore, the Largest Remainder Method affirms the PR parameter. The proofs of these claims can be found in my homepage.

If the Largest Remainder Method is applied to the Party-List Canvass Report No. 27, we have the following allocation: Buhay, with 8 seats; Bayan Muna, 7; Cibac, 5; Gabriela and Apec, 4 seats each; A Teacher, Akbayan, Alagad, Butil and Batas, 3 seats each; and Anakpawis, Coop-Natcco and Abono, 2 seats each. The total number of seats allocated is 55 seats.