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The other day, Felix P. Muga II sent me his reply to Artemio V. Panganiban’s article, Law, mathematics and the party-list system.

I’ve tended to agree with Professor Muga’s solution to the party list problem. Here is his reply in verbatim, or you can download a copy, Muga Reply To Panganiban

July 16, 2007
On the Party-list Seat Allocation Issue
A Rejoinder to CJ Panganiban’s article on “Law, Mathematics and the Party-List System” (PDI, July 15, 2007)

By Felix P. Muga II

In his article, “Law, Mathematics and the Party-List System” (PDI, July 15, 2007, page 15), former Supreme Court (SC) Chief Justice Artemio Panganiban enumerates four inviolable parameters to determine the winners in a “Philippine-style” party-list election
and the number of seats each winner shall receive. These parameters are:

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;

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; (and)

Fourth, proportional representation—the additional seats which a party is entitled to shall be computed ‘in proportion to their total number of votes.’

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

Two parameters, say, n ≤ 3 and n ≥ 1 where n is an integer, are consistent, since a number that is between 1 and 3 inclusive, satisfies the two parameters.

However, two parameters, say, n ≤ 3 where n is an integer, and n = k where k is a rational number, are inconsistent since there is no number that satisfies the two parameters when k ≥ 4.

The .2TOTAL parameter rephrased

The .2TOTAL parameter states that “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”.

However, 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 phrase “shall not exceed twenty-percent” is the final substance of the .02TOTAL parameter so that the 3-SEATCAP parameter will be consistent with the .02TOTAL parameter.

The PR parameter violated

But the High Court fails to make the 3-SEATCAP consistent with the PR parameter. The PR parameter is 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

Thus, by the principle of proportional representation, the (ideal) number of seats of a qualified party is given by:

(ideal) no. of seats = its % share of the total votes × 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. Suppose that a qualified party in the 2007 election obtains 8.0 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 will be awarded three seats only. However, by the principle of proportional representation, its (ideal) number of seats is 8.0% x 55 = 4.4 or at least four seats. There
is a difference of 1 seat which is equivalent to at least (1/55) x 8,070,680 = 146,739 disenfranchised voters.

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

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

If the Panganiban Formula is applied to the latest Party-List Tally (Canvass Report No. 27), a “solution” is obtained with 21 seats where Buhay has 3 seats, Bayan Muna, Cibac and Gabriela have 2 seats each, and Apec, A Teacher, Akbayan, Alagad, Butil,
Anakpawis, Coop-Natcco, Abono, Agap, Arc, and An Waray have one seat each. The solution is consistent with the .2TOTAL, .02THRESH and the 3-SEATCAP parameters. However, the formula violates the PR parameter by at least 28 seats. This 28-seat
violation is equivalent to at least (28/55) × 8,070,680 = 4,108,701 disenfranchised voters. Hence, the Panganiban Formula is not a correct solution to the seat allocation problem of the Philippine party-list system.

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 since it is the cause of the inconsistency of the system. Also, it is the reason, why 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 will 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 who may dominate the partylist 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 26.187407 percent obtained by Bayan Muna in 2001 did not qualify it to be a major political party. This figure was a special case since a number of parties were disqualified in the said SC decision and the total number of party-list votes was reduced from
15,118,815 to 6,523,185 votes. In fact, in the 2004 election, Bayan Muna garnered 9.458493 percent and in the recent election, it is projected that Bayan Muna will have 6.72 percent of the votes.

The .02TOTAL parameter should be rephrased back to the original intention of the Constitution. In can now be stated, thus: “The combined number of all party-list congressmen shall be twenty percent of the total membership of the House of Representatives, including those elected under the party-list”.

The .02THRESH may be retained but it should be clarified that the 2 percent is a formal threshold and not an informal one as misinterpreted by the 2-4-6 Comelec Formula. A formal threshold is the minimum share needed to qualify for a seat while the informal threshold is the minimum share needed to win one seat. The correct value of the informal threshold is 1/ (total number of party-list seats).

Since the 3-SEATCAP parameter is discarded, we have the PR parameter as the third parameter.

Our Solution: The Largest Remainder Method

In Germany’s Bundestag parliament, the number of party-list seats of a qualified party in a given Land is determined by the Largest Remainder Method (the LR Method) and subtracting from the number determined by the LR Method, the number of legislative
seats it won in single-member districts. This is the Niemeyer Formula. Although, we cannot apply the said Formula to our system, we can adopt the LR Method.

A number of party-list systems in the world are using the LR Method. These are 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.

1. 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.

2. 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.

3. 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.

******

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.