Problems of the Panganiban Formula: Underallocation, Nonallocation, Overallocation

The Panganiban Formula was introduced by the Supreme Court when it ruled on the case Veterans Federal Party vs. Comelec (2001). In the said formula, a party-list group is entitled to one seat if it reaches the two-percent threshold of all the party-list votes. The additional number of seats that the party-list group shall receive is determined as follows:

1. if the party-list group has the most number of votes (called as the first party) and if it obtains
a. at least 6% of the total number of party-list votes, then it shall receive two more seats; or
b. at least 4% but less than 6% of the total number of party-list votes, then it shall receive one more seat, or
2. if the party-list group obtains the same number of votes as the first party then it receives the same number of additional seats as the first party, or
3. if the party-list group obtains at least one-half but less than the number of votes of the first party, then it shall receive one more seat, or
4. if the party-list group obtains less than one-half of that of the first party, then it does not receive any additional seat.
Party-List Canvass Report No. 30, the latest Party-list Tally of the Comelec shows that there are 17 party-list groups that obtain at least 2% of the total party-list votes where Buhay is the first party with 7.418753% of the total party-list votes. If we apply the Panganiban Formula we have the following results.

The 1987 Constitution mandates that the party-list shall constitute 20% of the total number of representatives in the House of Representatives including the party-list. There are 220 legislative districts now, the total number of party-list seats available is 220/4 = 55 seats. Note that 55/(220+55) = 0.2 or 20%.

Hence, the Panganiban Formula allocates only 23/55 = 0.41818182 or 41.82% of the total number of seats available. This is a case of underallocation.

In the 2007 party-list election, ninety-three party-list groups were allowed to participate by the Comelec. Thus, there is a possibility that no party will be able to reach the 2%-threshold. If this happens, no party-list group will receive a seat and the party-list will not be represented in the House of Representatives. This is a case of noallocation.

If the first party obtains 6% of the total party-list votes and 27 party-list groups obtains 3% of the the total party-list votes, then there will be 28 party-list groups that are entitled to a seat and their combined share of the party-list votes is 27(3%) + 6% = 87%. Using the Panganiban Formula, the first party shall receive 3 seats and the 27 party-list groups shall receive 2 seats each. The Panganiban Formula will allocate 3+54 = 57 seats. This allocation is two seats over the available number of seats.

Actually, the largest possible allocation of the Panganiban Formula is 65 seats where the first party obtains 6% and there are 31 party-list groups with 3% of the total party-list votes. This allocation is 10 seats over the available number of seats.

These are cases of overallocation.

On The Two-Percent Formal Threshold: A Restrictive Mechanism for the Marginalized and Under-Represented

courtesy of www.congress.gov.ph


A formal vote threshold is the share of the total votes that a party has to obtain in order to qualify for a seat. In some countries, the formal threshold is higher to lessen the number of parties in the parliament and making it easier to form a government. In Germany, it has 5-percent, Poland has 7-percent and Turkey has 10 percent.

According to a blogger, Global Economy Matters, “the ten percent threshold was introduced by the 1980-83 military government to prevent a recurrence of the excessive parliamentary fragmentation and resulting governmental instability that characterized Turkish politics for much of the 1970s - has had a mixed record over the years.” (http://globaleconomydoesmatter.blogspot.com/2007/07/turkeys-early-parliamentary-election-of.html).


courtesy of www.congress.gov.ph

The formal vote threshold for the Philippine party-list system can be deduced from Section 11 of the Party-List System Act (Republic Act 7941). The said provision specifies that a party-list group that obtained at least 2% of the total votes cast for the party is entitled to a seat.

The party-list system in the Philippines is instituted to give the marginalized and under-represented sectors of its society representation in the House of Representatives and it comprises 20% of the entire number of representatives as specified by the Constitution. Although, it has not been reached since party-list elections have been conducted since 1998.

With the two-percent threshold only a small number of parties are able to obtain a party-list seat. In the last election, there are 17 out 93 participating party-list groups are represented.

If the formal vote threshold is removed and the Largest Remainder Method will be used, about 40 parties will gain seats for the 2007-2010 House of Representatives. If the formal threshold is fixed at one-half of the informal threshold which is 1/(total number of party-list seats) about 36 parties can make it. See our computations below.


 Formal Threshold at Four-Percent


Formal Threshold at Two-Percent



Formal Threshold at 0.9091-Percent



No Formal Threshold

The 2007 Party-List Election in Japan

courtesy of http://www.sangiin.go.jp/eng/index.htm

Last Sunday, July 29, 2007, Japan held an election for the House of Councillors (Senators) which is the upper chamber of the Japanese Parliament or the National Diet.

A total of 121 seats were contested where 73 seats will be filled in forty-seven prefectural districts and 48 seats will be allocated by proportional representation on a national basis.

Every voter has to cast two votes for the National Diet. The first vote is for a single candidate in a prefectural district. The candidates with the largest number of votes in each district, up to the number of seats to be filled, are elected to office. The second vote is for the party-list system. The 48 seats will be allocated by the Highest Average Method devised by the Belgian Mathematician Victor D’ Hondt.

All the elected Senators will serve for 6 years. The total number of members in the upper chamber is 242 and half of them are elected every three years.

Seven parties are qualified to receive a party-list seat. These are:

1. Democratic Party of Japan (DJP) with 23,256,242 votes
2. Liberal Democratic Party (LDP) with 16,544,696 votes
3. New Komeito Party (NKP) with 7,762,324 votes
4. Japanese Communist Party (JCP) with 4,407,937 votes
5. Social Democratic Party (SDP) with 2,637,716 votes
6. People’s New Party (PNP) with 1,269,220 votes
7. New Party Nippon (NPN) with 1,770,697 votes

The total number of votes of all qualified parties is 57,648,832 representing 97.853050% of all the votes cast for the party-list. The qualified parties received all 48 party-list seats available.

The Highest Average Method

In the Highest Average Method, successive quotients or averages are calculated. It is determined by dividing the number of votes of each party by a sequence of integer divisors 1, 2, 3, 4, and so on.

The list of successive quotients for each qualified party using the divisors from 1 up to 25 is given in the table below.




The top 48 quotients shall determine the 48 seats to be allocated. Hence,

1. Democratic Party of Japan (DJP) shall be awarded with 20 seats,
2. Liberal Democratic Party (LDP) with 14 seats,
3. New Komeito Party (NKP) with 7 seats,
4. Japanese Communist Party (JCP) with 3 seats
5. Social Democratic Party (SDP) with 2 seats
6. People’s New Party (PNP) with 1 seat, and
7. New Party Nippon (NPN) with 1 seat

The table below shows the rank of each quotient corresponding to the seat number that each party obtained.


Analysis

The Highest Average Method allocates the total number of available party-list seats. The seat allocation error  is zero as well as the degree of negation. This means that the Highest Average Method affirms the principle of proportional representation on the parties participating the 2007 Japan party-list election. See table below.

where
TQPV is the total number of votes of all qualified parties,
Actual Number of Seats is the actual allocation of the
                 Highest Average Method,
Ideal Number of  Seats is the obtained by the product
                  of the  % Based on TQPV and 48, 
Seat Allocation Error is the Ideal Number of Seats 
                    minus the Actual Number of  Seats, and the
Degree of Negation is the absolute value of the integer 
                    part of  the Seat Allocation Error.

The index of proportionality of a given method is determined by dividing the sum of the absolute value of the seat allocation error by 96 and the result is subtracted from 1. A value of 1 or 100% means full proportionality and a value of 0 means a disproportionate method. The index of proportionality of the Highest Average Method on the 2007 Japan party-list election is equal to 97.088495%.

The Niemeyer Formula Revisited

courtesy http://www.bundestag.de

We examine a formula that was mentioned earlier when the party-list system was tackled in the drafting of the 1987 Constitution and the Party-List Law.

In his dissenting opinion in the case Veterans Federation vs. Comelec (G.R. 136781, October 6, 2001), Justice Mendoza states that

“Rep. Tito R. Espinosa, co-sponsor of the bill which became R.A. No. 7941, explained that the system embodied in the law was largely patterned after the mixed party-list system in Germany. Indeed, the decision to use the German model is clear from the exchanges in the Constitutional Commission between Commissioners Blas F. Ople and Christian S. Monsod. The difference between our system and that of Germany is that whereas in Germany half (328) of the seats in the Bundestag are filled by direct vote and the other half (328) are filled through the party-list system, in our case the membership of the House of Representatives is composed of 80 percent district and 20 percent party-list representatives.

The party-list system of proportional representation is based on the Niemeyer formula, embodied in Art. 6(2) of the German Federal Electoral Law, which provides that, in determining the number of seats a party is entitled to have in the Bundestag, (the total number of) seats should be multiplied by the number of votes obtained by each party and then the product should be divided by the sum total of the second votes obtained by all the parties that have polled at least 5 percent of the votes. First, each party receives one seat for each whole number resulting from the calculation. The remaining seats are then allocated in the descending sequence of the decimal fractions.

The Niemeyer formula was adopted in R.A. No. 7941, §11. As Representative Espinosa said:

MR. ESPINOSA: This mathematical computation or formula was patterned after that of Niemeyer formula which is being practiced in Germany as formerly stated. As this is the formula or mathematical computation which they have seen most fit to be applied in a party-list system. This is not just a formula arrived at because of suggestions of individual Members of the Committee but rather a pattern which was already used, as I have said, in the assembly of Germany.”

Allocation of Seats in the 2005 Bundestag Election Using the Niemeyer Formula


courtesy of http://www.bundestag.de

In the 2005 Bundestag election, five parties, namely the Social Democratic Party (SPD), the Christian Democratic Union (CDU) and its Bavarian counterpart, the Christian Socialist Union (CSU), which forms a joint parliamentary group with the CDU but runs as a separate party, the Free Democratic Party (F.D.P), the Left Party and the Alliance 90/The Greens received at least five percent of all valid second votes cast, and were thus entitled to participate in the proportional allocation of seats at the federal level. The accumulated number of second votes of the 5 parties is 45,271,207. None of the other parties like the NPD and the Republicans that participated in the election reached the five percent threshold or received any constituency mandates; therefore, these were excluded from the apportionment process.



In the 2005 Bundestag Election, the number of seats mandated by the German Law is 598.

To determine the number of seats that qualified parties are entitled to, we compute the following:

1. The total number of available seats (598) should be multiplied by the number of votes obtained by each party and then the product should be divided by the sum total of the second votes obtained by all the parties that have polled at least 5 percent of the votes.

This is the ideal number of seats. See table below.



2. Each party receives one seat for each whole number resulting from the calculation.

This is the first round of seat allocation.


3. The remaining seats are then allocated in the descending sequence of the decimal fractions.

The total number of available seats is not filled up hence we have the second round of seat allocation to distribute the remaining number of seats which is 598 – 596 = 2.


Hence, the qualified parties are entitled to the number of seats given below:

This is the first stage of the Niemeyer Formula. The second stage of the Niemeyer Formula is to the allocation of the number of list seats per Land to each qualified party given the number of direct seats it obtained in the Land and the number of seats it is entitled to receive in that Land.

The Federal Republic of Germany is divided into 16 Lands. There are two votes in the Bundestag election, the first vote refers to a vote for a candidate of a party directly elected in the legislative district. The second vote refers to the party-list.

1. The number of direct seats from the legislative districts that a qualified party obtained in a given Land is subtracted from the number of seats that a qualified party is entitled to obtain in that Land.

     a. If the difference is positive, then it is the number of list seats that will be awarded to the     qualified to fill up the number of seats that the qualified is entitled to receive.

     b. If the difference is negative, no list seats will be awarded.

2. The sum of the number of direct seats and the number of list seats of a qualified party in a given Land is the actual number of seats that the qualified party receives in the Land.

If the number of direct seats is larger than the number of seats that a qualified party is entitled to receive, then there are overhang seats and the qualified party may actually receive more seats than the number of seats it is entitled to receive. The German Law allows this anomaly.

The Land level allocation of seats of each party is given below:

a) Christian Socialist Union (Table 6)

b) Social Democratic Party of Germany (Table 7)


 c) Free Democratic Party (Table 8)

 d) The Left Party (Table 9)


e) The Alliance 90/Greens Party (Table 10)



Hence, in the 2005 Bundestag election, 614 seats were allocated by the Niemeyer Method where the number of list seats is 315 and the number of direct seats is 299. See the Table 11.



Niemeyer Formula Clarified

  The Niemeyer Formula consists of two stages. In the first stage, the number of seats that each qualified party is entitled to receive is determine using the Largest Remainder Method. The second stage in the Formula is to determine the actual number of list seats that a qualified party shall receive per Land using the number of seats obtained in the first stage and the number of seats the qualified party won per Land.

 So the Formula that was referred to in the discussion of the Constitution, the deliberation of RA 7941 and the ruling of the Supreme Court is not really the Niemeyer Formula but the Largest Remainder Method.

Analyzing the Proposed Amendments to RA 7941

 April 25, 2005
Dr. Felix P. Muga II
Mathematics Department
Ateneo de Manila University

Introduction

In the present Congress, 6 House Bills are proposed to amend RA 7941, otherwise known as the Party-List System Act. The amendments aim to correct its imperfections so that it will be true to its purpose of broadening the representation of the marginalized and under-represented sectors of our society.

RA 7941 declares that the party-list system is a mechanism of proportional representation in the election of members to the House of Representatives and reserves 20% of the total seats of the House to the party-list system.

This means that in the 2004 party-list election, 53 seats in the House of Representatives are reserved for the party-list. However, only 24 are proclaimed by the COMELEC. The entire party-list seats were not filled up also in the 1998 and the 2001 party-list elections.

The problem lies in the seat allocation method of RA 7941.

The Seat Allocation Method of RA 7941

The seat allocation method of RA 7941 is a variant of the Largest Remainder Method (LR Method) used by some countries with a party-list system like Germany, Republic of Korea, Russia, Taiwan, Ukraine, Germany, Mexico, Czech Republic, Iceland, and Slovenia.

In the LR Method, the allocation of seats consists of two rounds. The first round computes the automatic number of seats that a party will win. It is based on the integral part of the quotient where the number of votes received by the party is divided by a minimum quota. The second round distributes the remaining number of seats not allocated by the first round using the remainders that are arranged from the highest to the lowest. The parties with the largest remainders win one seat each until the remaining seats are allocated.

The minimum quota can be expressed as an informal threshold in percent or as a winning minimum percentage. The natural value of the winning minimum percentage is given by

Natural Winning Minimum Percentage = 1/Total No. of Party-List Seats Available

The quotient is multiplied by 100%.

In most of the countries that have a party-list proportional system, a formal vote threshold is instituted. This formal threshold is the minimum share of the vote required by law to qualify for a seat. Note that the two thresholds are distinct. It is a possible that a party may pass the formal vote threshold but may not win a seat.
In the seat allocation method of RA 7941 the informal vote threshold is 2% but there is no formal vote threshold. Based on the 2% winning threshold, a party that garners 5% of the total votes wins 2 seats and a party that obtains 6% of the total votes is awarded 3 seats. But because of the 3-seat cap, a party that obtains 11% of the total votes only receives 3 seats.

The number of seats that can be allocated by the seat allocation method of RA 7941 is up to 50 since 1/2% = 50.

Hence, the 2% winning minimum percentage and the 3-seat cap distort the principle of proportional representation of the party-list system and cause the failure to broaden the representation of the marginalized and the under-represented sectors.

Hence, there is a need to amend the existing party-list law and find the best seat allocation method for the party-list system.

Our search for the best method must be guided by three parameters called the index of representation, index or proportionality, and the index of elasticity.

The Three Parameters for Analysis

We shall analyze the 6 proposed amendments using 3 parameters that we call as the index of representation, index of proportionality, and the index of elasticity.

1) The Index of Representation
The index of representation is the amount in percent that is relative to the total number of seats allocated for the party-list system. Its value ranges from 0% to 100%. A 100% index of representation implies that the seat allocation method fills up the total number of party-list seats.
The index of representation of an allocation produced by a seat allocation method applied on a given party-list election is equal to the quotient when the total number of seats allocated by the method is divided by the total number of available seats for the party-list. The quotient is multiplied by 100%.

The seat allocation method of RA 7941 has one round of seat allocation. The number of seats obtains by a party is the integer portion of the quotient when the share of votes of the party is divided by 2%. The share of votes of the party is obtained by dividing its number of votes by the total votes of the party-list.

In the 2004 party-list election as shown in COMELEC Report no. 20, the seat allocation of RA 7941 produced 24 seats. This means that the index of representation of the seat distribution based on RA 7941 is 24/53 x 100% = 45.283%.

2) The Index of Proportionality

The party-list system is based on the principle of proportional representation. This is emphasized in section two (Declaration of Policy) of RA 7941.
The principle of proportional representation means that a party with 1% of the total votes receives 1% of the total seats; a party with 2% of the total seats gets 2% of the total seats; a party with 3% of the total seats obtains 3% of the total seats; and so on.
The degree in which proportionality is achieved is measured by the index of proportionality. Its value ranges from 0% to 100%. If the index of proportionality is 100% this means that full proportionality is reached.
Since the total number of party-list seats is less than 100, full proportionality may not be achievable. Hence, an index of proportionality that is at least 90% for an allocation produced by a seat allocation method is desirable.
To compute for the index of proportionality, we shall adopt the formula proposed by Richard Rose, Neil Munro and Tom Mackie in 1998. This formula is known as the Rose Index of Proportionality and is a standardized version of the Loosemore-Hanby index, see “Electoral Engineering Voting Rules and Political Behavior” by Professor Pippa Norris.
The (Rose) index of proportionality is computed by finding the absolute or positive difference between the shares of votes and the corresponding shares of seats, added up together multiplied by 0.5 and the product is subtracted from 1. The result is multiplied by 100%.
In the paper “On the Seat Allocation Method of the Philippine Party-List System”, this author shows that the index of representation and the index of proportionality are related by the formula

Index of Proportionality = (1 + Index of Representation)/2 × 100%

Hence, index of proportionality based on RA 7941 is (1 + 24/53)/2 × 100% = 72.642%.

3) The Index of Elasticity

A second-chance party is a small party-list organization or group whose share of votes is below that of the actual winning minimum percentage but passes the formal threshold and wins a seat in the second round. A seat allocation method is elastic if it is flexible enough to give smaller parties a chance to win a seat. This is consistent with the purpose of the 1987 Constitution to give a chance to the marginalized and the underprivileged sectors of the society a voice in congress.

The index of elasticity is equal to the number of second-chance parties divided by the number of first round winning parties. The quotient is multiplied by 100%.

A value that is equal to 0% means that the system is inelastic, but this will imply a very high index of proportionality when the index of representation is almost perfect.

A value that is equal to 100% or more means that the number of second-chance parties is at least equal to the number of winning parties in the first round of seat allocation. However, the index of proportionality may suffer even though the index of representation is almost perfect.

In the last party-election, the seat allocation method of RA 7941 produces no second-chance parties. Hence, the index of elasticity of the allocation produced by RA 7941 is 0%.

Analysis of the 6 House Bills

1) House Bill 341

House Bill 341 authored by Rep. Aimee Marcos proposes to entitle each voter 13 votes for the party-list election, one vote for a candidate in the national or regional political party and one vote for a candidate for each of the 12 sectors.

However, House Bill 341 does not provide a mechanism to allocate the total seats available for the party-list system. It cannot use the seat allocation method of RA 7941 because of the differences in the ballot structure where RA 7941 has one second vote while the Marcos Proposal has 13 second votes for the House of Representatives.

Because of the absence of a clear mechanism for the allocation of seats, we cannot analyze the Marcos Proposal based on the three parameters that we presented.

2) House Bill 409

The seat allocation method of House Bill 409 introduced by Rep. Roseller L. Barinaga provides a 2% formal vote threshold and a 10-seat cap for a winning party-list party.

The seats are allocated mainly in two rounds. The number of seats that are distributed to a given party in the first round is equal to the integer portion of the quantity calculated in the following formula:
Total No. of Available Seats × Party’s Share of the Total Votes of all Qualified Voters
where the party’s percent share of the total votes is equal to the number of the party’s votes divided by the total number of votes to be tallied.

If there are remaining seats after the first round, these remaining seats are allocated in the next round similar to the second round of the Largest Remainder Method using the decimal fractions of the quotient obtained in the formula above.

If in case of equal decimal fractions, the assignment of last seat shall be decided by the COMELEC using drawing lots.

If there are winning parties affected by the 10-seat cap, then there are unfilled seats left and a third round will take place to distribute the remaining seats to the highest ranking parties based on the number of votes each party garners.

The Barinaga Proposal uses the natural value of the winning minimum percentage in the allocation of seats, since the total number of party-list seats available is the reciprocal of the natural value of the winning minimum percentage.

Our analysis shows that a seat allocation based on the House Bill 409 or the Barinaga Proposal as applied to the 2004 party-list election results in 16 winning parties that occupy 53 seats. This gives a perfect (100%) index of representation, a very high (96.327%) index of proportionality, but a 0% index of elasticity.

3) House Bill 2451

The seat allocation method of House Bill 2451 introduced by Reps. Edgar L. Valdez, Ernesto C. Pablo, and Sunny R.A. Madamba, provides a formal vote threshold that is equal to the natural value of the winning minimum percentage, a 6-seat cap and two rounds of seat allocation.

In the first round, all parties that qualify, that is, those parties whose shares of votes pass the formal vote threshold, are guaranteed one seat each. The remaining number of seats is distributed in the second round.

The number of seats that a party may win in the second round is equal to the integer portion of the product calculated as follows:
Remaining No. of Seats × Party’s Share of the Total Votes of all Qualified Parties
This round may not distribute all the seats remaining from the first round. Hence, it is possible that the Valdez et al Proposal may not fill up the entire number of seats available for the party-list.

The seat allocation based on the Valdez et al Proposal on the 2004 party-list election produces 44 seats. This means that the allocation has an 83.019% index of representation, an 87.997% index of proportionality and 0% index of elasticity.

Two seat allocation methods are equivalent if the number of seats of a party in the first method is equal to the number of seats in the second method.

If there are no winning parties affected by the cap or limit on the number of seats that a party may win, the Valdez et al Proposal and the Barinaga Proposal are equivalent with the following adjustments:

1. The quantity below shall be added to the product in the second round of the Valdez et al Proposal

No. of Guaranteed Seats × Party’s Share of the Total Votes of all Qualified Parties - 1

2. The two proposals shall have the same cap on the number of seats, and

3. The two proposals shall have the same formal vote threshold

4. A new round of seat allocation similar to the second round of the Barinaga Proposal will be added to the Valdez et al Proposal to allocate the remaining number of seats.

4) House Bill 2734
House Bill No. 2734 proposed by Reps. Satur C. Ocampo, Teodoro A. Casiño, Joel G. Virador, Crispin B. Beltran, Rafael V. Mariano, and Liza L. Maza has two rounds in the allocation of party-list seats, a 6-seat cap, a 2% actual winning minimum percentage and no formal vote threshold.

In the first round of seat allocation, the number of seats that a party may win is equal to the integer portion of the quotient when the party’s share of votes is divided by 2%.

The remaining number of seats after the first round is allocated in the second round to those parties that received less than 6 seats including those parties that do not win a seat in the first round. The allocation in this round is similar to the second round of the Barinaga Proposal.

In case of equal fraction, the allocation of the last seat shall be decided by the COMELEC by drawing lots.

The allocation based on the Ocampo et al Proposal results in 37 winning parties occupying 53 seats. 16 of the parties are first round winners while 21 are second-chance parties. This results in a perfect (100%) index of representation, 86.203% index of proportionality and a very high 131.250% index of elasticity.

If there are no winning parties affected by the cap or limit on the number of seats that a party may win, the Ocampo et al Proposal and the Barinaga Proposal are equivalent with the following adjustments:

1. the Ocampo et al Proposal shall adopt the natural value of the winning minimum percentage,

2. the two proposals shall have equal formal vote threshold, and

3. the two proposals shall have the same cap on the number of seats.

5) House Bill 3302

House Bill Number 3302 introduced by Reps. Loretta Ann P. Rosales, Mario Joyo Aguja and Anna Theresia Hontiveros-Baraquel has three rounds of seat allocation, adopts 1.8% both as a formal vote threshold and the actual winning minimum percentage and a 6-seat cap for the number of seats that a party may win. The parties who are qualified to win are called the winning minimum percenters. These parties receive one guaranteed seat each in the first round.

The remaining number of seats from the first round is allocated to the qualified parties in the second round. Each party is awarded a number of seats equal to the integer portion of the quotient when the party’s share of all the votes of the winning minimum percenters is divided by 1.8% less 1.

If there are remaining seats after the second round, these are allocated in the third round similar to the second round of the Barinaga Proposal.

In case of equal fractions, the assignment of the last seat shall be decided by the COMELEC based on the comparison of actual votes.

If the Rosales et al Proposal is applied to the 2004 party-list election, the index of representation is 98.113%, since a total of 52 seats out of 53. It has is a high index of proportionality of 93.987%. There are 19 winning parties and there are no second-chance parties. Hence, the index of elasticity is 0%.
If there are no winning parties affected by the cap or limit on the number of seats that a party may win, the Rosales et al Proposal and the Barinaga Proposal are equivalent with the following adjustments:

1. The two proposals shall have equal formal vote thresholds, and

2. The Rosales et al Proposal shall use the natural value for the actual winning minimum percentage.

6) House Bill 3474

House Bill No. 3474 introduced by Rep. Guillermo P. Cua has three rounds of seat allocation, adopts a 3-seat cap and uses 1.8% as a formal vote threshold.

In the first round, all parties that qualify to win a seat are awarded one guaranteed each.

The remaining number of seats is distributed in the second round where each party receives a number of seats equal to the integer portion of the product when the remaining number of seats is multiplied to the party’s share of the total votes of all the qualified parties.

If there are remaining seats after the second round, a third round similar to the second round of the Barinaga Proposal occurs.

In case of equal fractions, the assignment of the last seat shall be decided by COMELEC by comparison of actual votes where the last remaining seat is awarded to the higher votes.

The allocation of Cua Proposal produces an 86.679% index of representation, an 86.232% index of proportionality of 93.987%. Since there are no second-chance parties, the index of elasticity is 0%.

If there are no winning parties affected by the cap or limit on the number of seats that a party may win, the Cua Proposal and the Barinaga Proposal are equivalent if the following adjustments are made:

1. The quantity below shall be added to the product in the second round of the Valdez et al

No. of Guaranteed Seats × Party’s Share of the Total Votes of all Qualified Parties -1

2. The two proposals shall have the same cap on the number of seats, and

3. The two proposals shall have equal formal vote thresholds.

Recommendation

We find that a method which uses 1) an extra round to allocate the remaining party-list seats, 2) a natural value for the winning minimum percentage, and 3) a higher cap in the number of seats results in a very high if not a perfect index of representation and a very high index of proportionality. This is shown in the Barinaga and the Rosales et al Proposals.

We also find that a method which uses a zero value of the formal vote threshold and an extra round result in a very high index of elasticity, but reduces the index of proportionality. See the Ocampo et al Proposal.

As the value of the formal vote threshold is made smaller, the shares of the total seats of the highest ranked parties decrease. This means that a small value of the formal vote threshold serves as a natural cap for the number of seats that a party may win.

For example, when the formal vote threshold of the Barinaga Proposal is at 2%, the number of seats received by Bayan Muna is 8 out of 53 seats. But when it is lowered to 1%, Bayan Muna receives 6 seats only and there are 11 second-chance parties.

Hence, a smaller value of the formal vote threshold means an improve value in the index of elasticity and a substitute for the cap on the number of party-list seats.

A value that is greater than ½ of the natural value of the winning minimum percentage or ½ of the reciprocal the total number of available seats for the party-list produces a number of second-chance parties and a high index of proportionality. This claim is proven in our paper entitled “On the Seat Allocation Method of the Party-List System in the Philippines” which can be accessed at http://www.math.admu.edu.ph~fpmuga.

Therefore, we recommend the following:

1. the adoption of a formal vote threshold that is equal to
½ × (1/Total Number of Seats Available for the Party-List ) × 100%
The percentage value is round up to two decimal places, that is, the digits to the right of the decimal point starting from the third are dropped and 1 is added to the second decimal place.

2. the use of two rounds of seat allocation similar to the Barinaga Proposal.

If we apply this recommendation to the 2004 party-list election using COMELEC Report No. 20, the formal vote threshold is equal to 0.95%, since 0.5 × 1/53 =0.94340%. There are 32 parties that qualify to win a seat where 21 parties are first round winners and 11 are second-chance parties. All the 53 seats are filled up. The seat allocation based on this recommendation has a perfect (100%) index of representation, a 92.667% index of proportionality and a 52.381% index of elasticity. See Table 1.

The Largest Remainder Method at 2% Formal Vote Threshold and 3-Seat Cap on the Party-List Canvass Report No. 29 Dated July 11, 2007

The Mendoza Dissenting Opinion Based on the Party-List Report No. 29 Dated July 11, 2007 (3-Seat Cap)

13th Congress Proposed Amendment (2nd Reading) to RA 7941 Based on the Party-List Canvass Report No. 29 Dated July 11, 2007 (with 6-Seat Cap)

The Largest Remainder Method at 2% Formal Vote Threshold on the Party-List Canvass Report No. 29 Dated July 11, 2007 (No Seat Cap)

The Panganiban Formula on Party-List Canvass Report No. 29 Dated July 11, 2007 (3-Seat Cap)

The Negligence and the Violation of the State on the Proportional Party-List System

by Felix P. Muga II

Our party-list system is envisioned to be proportional. Section 2 (Declaration of Policy) of the Party-List Law (R.A. 7941) declares that “the State shall promote proportional representation in the election of representatives to the House of Representatives through a party-list system of registered national, regional and sectoral parties or organizations or coalitions thereof….”

The number of party-list seats available in every party-list election is determined by Section 5(2) of Article VI of the 1987 Constitution which asserts that “The party-list representatives shall constitute twenty per centum of the total number of representatives including those under the party list. For three consecutive terms after the ratification of this Constitution, one-half of the seats allocated to party-list representatives shall be filled, as provided by law, by selection or election from the labor, peasant, urban poor, indigenous cultural communities, women, youth, and such other sectors as may be provided by law, except the religious sector.”

This means that out of five congressmen, one comes from the party-list and four come from the single-member legislative districts. Thus, the total number of party-list seats available is equal to one-fourth of the total number of legislative districts all over the country.

Evidence of Neglect

The State has conducted 4 party-list elections already since 1998. In the 1998, 2001, 2004 and 2007 elections, the total number of party-list seats available was 52, 52, 53 and 55, respectively. However, the Commission on Elections proclaimed only 14 (about 5.4%), 20 (about 7.7%), and 24 (about 9.1%) party-list congressmen in 1998, 2001 and 2004 elections. In the recent 2007 elections, the Comelec decided to implement the Panganiban Formula and based on the latest Comelec Tally (Report No. 29 dated July 11, 2007), there will be a total of 23 party-list congressmen (about 8.4%) that will be proclaimed. The State neglected to fill up even one-half of the seats allocated to party-list representatives.

Evidence of Violation

The principle of proportional representation asserts that the share of the total seats of a party that is qualified to receive a seat is equal to its share of the total votes of all the parties that are qualified to receive a seat.

Hence, by this principle, the (ideal) number of seats that a qualified party is entitled to receive is equal to its share of the total votes of all the qualified parties multiplied by the total number of seats available.
Thus, if a party receives 10% of the total votes of all qualified parties then by the principle of proportional representation, it is entitled to 10% x 55 = 5.5 seats where 55 is the total number of party-list seats available.

If a seat allocation formula actually allocates 5 or 6 seats, the seat allocation error of the formula with respect to the principle of proportional representation is either 0.5 or -0.5 seats, respectively. The absolute value of this error is less than one seat. Thus, the error is negligible, and we say that the Formula affirms the principle. Otherwise, if the absolute error is at least one seat then the Formula violates the principle.

If a Formula violates the principle of proportional representation, then there are votes that are disenfranchised. The number of disenfranchised votes of a qualified party is computed in this way: the whole part of the absolute error is divided by the total number of party-list seats and the result is multiplied by the total number of votes of all qualified parties.

Thus, in our previous example, if the total number of votes of all the qualified parties is 8,500,000 votes and the party is actually given 3 seats, then the absolute value of the seat allocation error is 5.5 – 3 = 2.5. Thus, the number of disenfranchised votes of the said party is at least (2/55) x 8,500,000 = 309,090 votes.

In the 1998 up to 2004 party-list elections, the simplified Comelec Formula was used wherein a party with at least 2% but less than 4% of the total votes is given one seat, if the party obtains at least 4% but less than 6% of the total votes is given two seats, and if the party obtains at least 6% is given 3 seats.

The simplified Comelec Formula violated the principle of proportional representation in the 1998, the 2001, and the 2004 party-list elections by at least 29 seats, 26 seats, and 21 seats, respectively. The violations of the Comelec Formula resulted to at least 1,912,508 disenfranchised votes in the 1998 elections, 2,529,735 disenfranchised votes in the 2001 elections, and 3,301,625 disenfranchised votes in the 2004 elections.

The Panganiban Formula also violated the principle of proportional representation in the 2007 elections. If we based our computation on the latest Comelec Tally (Report No. 29 dated July 11, 2007), the violation of the Panganiban Formula amounts to at least 23 seats and this is equivalent to at least 3,662,334 disenfranchised votes.

The Root Cause of the Violation

The violation of the Comelec Formula and Panganiban Formula on the principle of proportional representation is caused by the following factors:
1. 3-seat cap
2. first-party-rule of the Panganiban Formula
3. 2-4-6 rule of the Comelec Formula

The major cause of the violation is the 3-seat cap. This is shown in our previous example.

The Panganiban Formula is also bound to violate the principle of proportional representation because it uses the 3-seat cap and it based the computation of proportionality on the number of votes and the number of additional number of seats of the first party but not on the number of votes of all qualified parties and the total number of seats available.

The application of the simplified Comelec Formula will also result in the violation of the principle because it uses the 3-seat cap and 2% as the minimum share needed to win 1 seat. This is not correct mathematically and the correct one is 1/(total number of seats). Thus, in the 2007 election, the correct threshold is 1/55 or about 1.818182%.

The State Shall Rectify This Error
It is the primordial duty of the State to respect and uphold the sanctity of our votes. In the proportional party-list system every votes count. If a system of parameters that governs the seat allocation procedure of our party-list system is inconsistent, no formula can be designed to give a correct solution that upholds the principle of proportional representation. A formula that claims to follow this system of parameters will produce a disproportional solution. Hence, the votes for the party-list system will be disenfranchised.

The State shall rectify the error committed by its branches of government on our proportional party-list system. We take note of the error committed by the legislature when it introduced the 3-seat cap, the executive when it neglected to veto this provision, the judiciary when it justified the existence of the 3-seat cap and even introduced a system of 4 inviolable parameters that is inconsistent, and the Comelec when it implemented an erroneous 2-4-6 rule and the absurd first-party rule.

The State shall reject the 3-seat, the first party rule and the 2-4-6 rule and come up with a seat allocation formula that upholds the policy of the Party-List Law and satisfies the 20% mandate of the 1987 constitution.

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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.

Party-List Seat Allocation Issue

By Felix P. Muga II

On June 24, the Philippine Daily Inquirer published an article, “Party-List System: Mathematical Absurdity” (page 14) where the author, lawyer Oscar Franklin B. Tan, proposed a seat allocation formula to correct the mathematical absurdity of the party-list system. The proposed formula yields 55 seats which is the total number of party-list seats available in the 2007 party-list election as mandated by the 1987 Constitution.

Is Tan’s formula a solution to the seat allocation problem that has plagued our party-list system since 1998?

I believe that the proposed formula is an unacceptable solution to the seat allocation problem of the party-list system in the Philippines. The proposed formula violates the principle of proportional representation as provided for by the Party-List Act (R.A. 7941).

Instead, I propose another seat allocation formula that, I believe, affirms the policy of the Party-List Act and complies with the mandate of the 1987 Constitution that the party-list shall constitute 20% of the total members in the House of Representatives.

Vote threshold clarified

Almost all of the party-list systems in the world impose a formal vote threshold to determine which parties are entitled to a legislative seat. For example, the party-list systems in Germany, Republic of Korea, Russia, Taiwan, Hungary, New Zealand, Thailand and Ukraine have a formal vote threshold of 5%. In Mexico and Denmark, the formal vote threshold is 2%. If a party does not reach the formal vote threshold, then it has no right to receive a seat.
In her book on Electoral Engineering: Voting Rules and Political Behavior (ISBN: 0521536715, Cambridge University Press, 2004), Pippa Norris says
“The formal vote threshold is the minimum share of the vote (in the district or nation) required by law to qualify for a seat, and this is distinct from the informal threshold or the actual minimum share of the vote required to win a seat.”

In Germany, the law mandates the Bundestag parliament to have at least 598 members. Its informal vote threshold is 1/598 or 0.167224%. Thus, a qualified party in Germany that obtains at least 5% of the total party-list votes is assured of at least 29 party-list seats, as measured by 5% x 598 = 29.9, based on the principle of proportional representation. In fact, the Alliance90/Greens Party with 8.119883% of the total party-list votes was awarded 50 party-list seats in 2005.

In the Philippines, is the 2-percent threshold in the Party-List Act a formal or informal threshold? Or both? Section 11 of the Law states:

“xxx
The parties, organizations, and coalitions receiving at least two percent (2%) of the total votes cast for the party-list system shall be entitled to one seat each:

provided, that those garnering more than two percent (2%) of the votes shall be entitled to additional seats in proportion to their total number of votes:
xxx”

In his proposed formula, Attorney Tan believes that the 2% is an informal vote threshold. Hence, he is right in rejecting its implementation because the real informal threshold in the 2007 election is 1/55 or 1.818182%.

What if the Supreme Court interprets the 2% to be a formal threshold? Then the proposed formula shall be applied to the top 15 party-list groups that obtained at least 2% based on the Party-List Canvass Report No. 25 of the Comelec. Even if all the 15 parties shall be awarded 3 seats each, the total number of party-list seats (55 in 2007) cannot be filled since 15x 3 = 45. This will be contrary to the claim of the proponent that the proposed formula will be able to allocate the 20% mandate of the 1987 Constitution. It is in this vein that the proponent calls for the rejection of the 3-seat cap.

Features of the proposed formula

Tan’s proposed formula will generally work if there is no formal vote threshold. This is the first feature of the proposed formula. There are party-list systems in the world that have zero formal vote thresholds such as in Belgium, Peru and Switzerland.

The proponent may argue that the 2-percent formal threshold should be junked since it is restrictive and will allow a few party-list groups to win a seat. Thus, the 2-percent formal vote threshold defeats the purpose of allowing the marginalized and the underrepresented sectors to be represented in Congress.

Another argument is that the 2-percent formal threshold will disenfranchise the voters of An Waray (1.97%), FPJPM (1.89%) and AMIN (1.84%) since one seat in the 2007 party-list election is equivalent to 1/55 or 1.818182%.

The second feature of the proposed formula is the nth-party rule with 3-seat cap. In the 2007 election, this is called as the 32nd-party rule since it uses the 1.03% of the 32nd ranked party-list AGHAM as the basis of the proportionality. This is in contrast to the Panganiban Formula which uses the first-party rule where the basis of the proportionality is the party with the highest number of votes.

Problem with 2-digit decimal expansion

The proponent uses two-digit expansion when a party’s number of votes is divided by the chosen party’s number of votes. Since two seats are not yet filled when the formula was applied on the latest Comelec data (Canvass Report No. 25), the two unfilled seats were assigned to AKBAYAN (0.97) and An Waray (0.91).

But if eight-decimal digit expansion is used, the top 2 parties with highest ranking decimal components are Gabriela with 0.99632511 since (4.129778%)/(1.033394%) = 3.99632511 and Akbayan with 0.96890099. Since Gabriela has 3 seats already, then it cannot be awarded one additional seat.

The proponent is silent on how to award the remaining seat if this type of problem arises. Will it be given to An Waray with 0.91003223 or to the third ranking party based on decimal component Yacap with 0.92291198? In fact An Waray is 6th in the rank. The 4th and 5th ranking parties are TUCP with 0.92284144 and ANAK with 0.91557631.

Let us assume that this problem is settled and eventually 55 seats are awarded to the parties.

Proportional representation

The proponent agrees that there are two conditions to be met for a seat allocation method to be acceptable:

1. 20% of the total members of the House of Representatives must come from the party-list; and
2. The seat allocation must adhere to the principle of proportional representation.

If there is no formal vote threshold, generally the first parameter will be satisfied.

The problem lies in the second parameter. The principle of proportional representation asserts that



“The number of seats awarded to a qualified party shall be proportional to the number of votes obtained by the qualified party…”

Euclid (c. 325 BC – 265 BC) in Book V of Elements defines the following:
“xxx
3. A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
xxx
5. Let magnitudes which have the same ratio be called proportional.
xxxx”
The two magnitudes that have the same ratio are:
• Number of seats awarded to a qualified party with respect to the total number of seats available for the party-list system; and
• Number of votes obtained by the qualified party with respect to the total number of votes obtained by all the parties that are qualified to receive a seat.

This means that the percentage of seats awarded to a qualified party based on the total number of party-list seats available (TPLS) is equal to the percentage of votes it garnered based on the total number of votes of all parties who are qualified to receive a seat (TQPV), or
This equation becomes



This is the ideal number of seats that a qualified party is entitled to receive based on the principle of proportional representation. The actual number of seats that a qualified party receives is the number of seats allocated by the existing seat allocation formula. Hence,




If there is no formal vote threshold then TQPV is equal to TPLV which denotes the total number of votes of the party-list.

In the latest Comelec data, TPLV = 13,719,165 and in the 2007 election, TPLS = 55.
The number of votes of Buhay in the report is 1,111,035. Hence, the ideal number of seats of Buhay based on the principle of proportional representation is 4.45412859.

The difference between the ideal number and the actual number of seats of a qualified party is called the seat allocation error of the existing allocation formula on the qualified party. It is determined by

seat allocation error = ideal no. of seats - actual no. of seats

If actual number differs from the ideal number by less than one, then the allocation formula affirms the principle of proportional representation. Otherwise, if the difference is one or more then it violates the said principle.

The integer part of the absolute value of the seat allocation error is called the degree of negation of the formula.

For example, the ideal number of seats of Buhay is 4.45412859 but it is awarded 3 seats only by the 32nd-party rule. Then the seat allocation error is 1.45412859. Thus, the degree of negation of 32nd-party rule on Buhay is 1. This is equivalent to (1/55) x 13,719,165 or about 249,439 voters of Buhay are denied of representation in Congress because of the violation of the 32nd-party rule on the principle of proportional representation.

The seat allocation error of the 32nd-party rule differs by more than one seat on Apec, A Teacher and Akbayan. See Table below.



The actual number of seats allocated by the 32nd –party rule on Apec, A Teacher and Akbayan is more than the ideal number of seats based on the principle of proportional representation. The difference is at least one in each party. This means that the voters of other parties are denied of representation in Congress since the seats that are due to the parties they voted for are transferred to Apec, A Teacher and Akbayan by the 32nd-party rule. The number of voters that are disenfranchised in this way is about (3/55) x 13,719,165 or about 748,318.

Therefore the 32nd-party rule violates the principle of proportional representation by at least 4 seats.

The 32nd-party rule contradicts this principle because of the following factors:

1. Imposition of the 3-seat cap; and
2. Using the nth-party as the basis of proportionality.

The imposition of a cap clearly violates the principle of proportional representation when the ideal number of seats is one more than the cap.

The votes of the nth-party (or the first party of the Panganiban Formula) must not be used as the basis of proportionality. Proportional representation must be based on the total number of votes of all the parties that are qualified to receive a seat and on the total number of seats available for the party-list because the principle dictates that the

Our proposed formula

Our proposed formula has two rounds of seat allocation to the qualified parties:

1. In the first round, the number of seats that is allocated to the qualified parties is equal to the whole part of the ideal number of seats based on the principle of proportional representation;
2. If the total number of seats does not reach the total number of seats available for the party-list, then a second round of allocation is conducted, thus
a. The qualified parties are ranked from the highest to the lowest based on the decimal fractions of the ideal number of seats.
b. One seat of the remaining number of seats is given each to the highest ranking qualified parties based on the decimal fractions until all the seats are filled up.

Assuming that there is no formal vote threshold as in the case of the 32nd-party rule, the allocation of seats should be:

   
 Our proposed formula satisfies the first condition of an acceptable seat allocation method since it earmarks the total number of seats available for the party-list.

Note that the difference between the ideal number of seats based on the principle of proportional representation and the actual number of seats allocated by the proposed formula is less than one on each qualified party. This means that the proposed formula affirms the principle of proportional representation.

If the 3-seat cap of the Party-List Act will be imposed on our proposed formula after we have allocated the seats, then Buhay and Bayan Muna will receive 3 seats each instead of 5 and 4, respectively. The degree of negation of the 3-seat cap will be 3 seats. This is equivalent to (3/55) x 13,719,165 or about 748,318 voters that are disenfranchised because of the violation of the 3-seat cap on the principle of proportional representation.
Since the total number of seats is reduced from 55 to 52 after the 3-seat cap is imposed, the 20-percent mandate of the Constitution is violated. In this case, the 3-seat cap does not only violate the policy of the Party-List Act but is also unconstitutional. Therefore, the 3-seat cap must be rejected.

Our proposed formula at 2% formal vote threshold

Suppose that the Supreme Court interprets the 2-percent threshold as formal. Then our proposed formula with 2% formal vote threshold will still satisfy the 20%-mandate of the 1987 Constitution and the principle of proportional representation. The allocation reaches 55 seats and the difference between the ideal number and actual number of seats is less than one seat. See Table 4.



Our proposed method is not the only formula with a formal vote threshold of 2% that satisfy the 20%-mandate of the 1987 Constitution and the principle of proportional representation. We claim, however, that our formula gives the “best index of proportionality” since the 7 qualified parties that were chosen in the second round to be given one additional seat have the highest decimal fractions. In this way, the seat allocation error on each qualified party is minimized, i.e. the error approaches zero, while filling the entire number of party-list seats.

The index can be computed using the formula:


where the computed value is expressed in percentage and it ranges from 0% to 100%.
The index is 100% if full proportionality is achieved. The index is 0% if a party with no votes is awarded all the available seats. The index of proportionality of our proposed formula with 2% formal vote threshold is 96.132874%. Further discussion on this topic can be obtained in my papers “On the Seat Allocation Method of the Party-List System in the Philippines, Loyola Schools Review, Vol. 4, 2005” and “Amending Republic Act 7941, Otherwise Known as the Pary-List System Act”, Matimyas Matematika, Vol 28, Nos. 1-3, 2005.


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Felix P. Muga II is an associate professor at the Mathematics Department of Ateneo de Manila University and is a Fellow of the Center for People Empowerment in Governance (CenPEG). He obtained his B.S. Mathematics (Magna Cum Laude) at Silliman University and his Ph.D. in mathematics at the University of the Philippines in 1995 and was awarded one of the Ten Outstanding Young Scientists by the National Academy of Science and Technology (NAST) in 1998. He has a written a number of articles on the proportional party-list system since 2005. These articles can be downloaded at http://www..math.admu.edu.ph/~fpmuga.

On the Philippine Party-List System and the 4 Inviolable Parameters

by Felix P. Muga II 
July 17, 2007

In his article, “Law, Mathematics and the Party-List System” (PDI, July 15, 2007, page 15), former Supreme Court (SC) Chief Justice Artemio Panganiban enumerated a system of 4 inviolable parameters that determines the seat allocation method of the Philippine party-list system.

1. the twenty percent allocation (0.2TOTAL) —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;

2. the two percent threshold (0.02THRESH) —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 (sec. 11 of the Party-List Law, RA 7941);

3. the three-seat limit (3SEATCAP) —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 (sec. 11 of RA 7941); and

4. proportional representation (PR) —the additional seats which a party is entitled to shall be computed ‘in proportion to their total number of votes.’ (secs. 2 and 11 of RA 7941).

Chief Justice Panganiban failed to mention, however, whether the four inviolable parameters are consistent with each other.

A system is consistent if and only if there is at least one solution. On the other hand, a system is inconsistent if and only if it has no solution.

I believe that the system of four inviolable parameters mentioned by Chief Justice Panganiban is inconsistent. Hence, no formula can be designed to give a correct solution.

0.2TOTAL 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 0.2TOTAL parameter does not faithfully follow this formulation. I believe the said parameter was rephrased so that the 0.2TOTAL and 3SEATCAP parameters will be consistent.


Inconsistent PR and 3SEATCAP Parameters

The PR parameter is based on the principle of proportional representation which is the public policy of the Party-List Law or RA 7941. This principle asserts that the qualified party’s share of the total seats is equal to its share of the total votes of all qualified parties.

The qualified party’s share of the total seats is computed by dividing the number of seats it received by the total number of seats.

The qualified party’s share of the total votes is determined by dividing the number of votes it garnered by the total number of votes of all qualified parties.

Hence, by the PR parameter, the (ideal) number of seats that a qualified party shall be entitled to receive is equal to its share of the total number of votes of all qualified parties multiplied by the total number of party-list seats.

For example, in the 2007 party-election, the total number of party-list seats is 55. Thus, if a qualified parties has 10 percent of the votes of all qualified parties, then by the PR parameter, its (ideal) number of seats is 10% x 55 = 5.5. This means that the said party is entitled to at least 5 seats but not more than 6 seats.

If the 3SEATCAP parameter is imposed, then the said party cannot have more than 3 seats.

Since the two parameters are inviolable and both parameters produce inconsistent solutions where the 3SEATCAP parameter demands a number of seats that cannot exceed 3 while the PR parameter demands a number of seats that is either 5 or 6, it follows that the 3-SEATCAP and the PR parameters are inconsistent.

As a result, a formula that adheres with the 3SEATCAP is bound to violate the PR parameter.
The Panganiban Formula Violates the PR Parameter

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. The solution is consistent under the 0.2TOTAL, .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 Panganiban Formula which assigns 3 seats to Buhay violates the PR parameter on Buhay by at least 4 seats.

The formula also violates the PR parameter on Bayan Muna by at least 4 seats also, on Cibac by at least 3 seats, on Gabriela, APEC, A Teacher and Akbayan by at least 2 seats each, on Alagad, Butil, Batas, Anakpawis, COOP-NATCCO, Abono, Agap, ARC and An Waray by at least 1 seat each.

Thus, the total number of seats violated by the Panganiban Formula over all the qualified parties is at least 28 seats. This is equivalent to at least (28/55) × 8,070,680 = 4,108,701 disenfranchised voters.

Since the system of parameters is inconsistent, no seat allocation formula can be designed to produce a correct solution. Thus, a formula that adheres to the 3SEATCAP parameter will disenfranchise a number of party-list voters if the PR parameter produces more than 3 seats for a qualified party.

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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.