ON RATIONAL NUMBERS, DIVISION ALGORITHMS, AND GREATEST COMMON DIVISORS

An interesting property about the set of integers is the “Division Algorithm” which states that “for any two integers a and b where b is greater than 0, there exist unique integers q and r such that 

 a = bq + r 

where r is greater than or equal to 0 but less than b.” 

The integer q is called quotient, and the integer r is called a remainder while b is the divisor and a is the dividend.

For example, with integers a = 11 and b = 4, we have two unique integers q = 2 and r = 3 also such that 

11 = 4 x 2  + 3.

Note that there is no other integers s and t such that 

11 =  4s + t 

where t is greater than or equal to 0 and t < 4.

Suppose a = -15 and b = 6. 

Although, -15 = 6(-2) + (-3), the remainder cannot be -3 since it is a negative integer. Thus, the quotient cannot be -2. 

Also, although we have -15 = 6(-4) + 9, the remainder cannot be 9 since it is larger than the divisor 6. Thus, the quotient cannot be -4.

The correct combination must be -15 = 6(-3) + 3, and tnhe remainder is equal to 3 it is not a negative integer and it is less than the divisor 6. Thus, the quotient is -3.

The Python programming has built-in commands and operations to find the quotient and the remainder when an integer is divided by a positive integer.To find the remainder between two integers a and b, we can use a % b.

For example, 11 % 4 produces the number 3 as the remainder, The quotient that is equal to 2 can be obtained by (11 - 3) / 4 or simply 11 / 4, if the Python 2.7 version or earlier is used.

A built-in command in Python that produces the quotient and the remainder as a pair is divmod( ... , ... ). 

This divmod(11,4)  will output the pair (2, 3), where q = 2 and r = 3.

A rational number is defined to be a ratio of two integers s and t where t > 0. Thus, a rational number is of the form s / t. 

An integer a is also a rational number since a / 1 is a rational number and it is equal to a.

We can always find a rational number between any two integers 

For example, (3 + 4) / 2 = 7/2  is a rational number between 3 and 4.

Since 7 = 2(3) + 1, we have 7 / 2 = 3 + 1 / 2. The first term in the right-hand side of the equation is an integer and the second-term is a positive rational number less than 1. The second-term is a proper fraction.

Since 34 = 6(5) + 4, we have 34 / 6 = 5 + 4 / 6. The first term is an integer and the second term is a proper fraction which can be reduced to its simplest form 2 / 3. 

Hence, 34 / 6 = 5 + 2 / 3. 

Also,  -2 = 5(-1) + 3. Thus, we have -2 / 5 = -1 + 3 / 5.The first term is also an integer and the second term is a positive rational number less than 1.

Furthermore, 20 = 10(2) + 0 and 20 / 10 = 2 + 0 / 10 = 2 + 0 = 2. The rational number 20 / 10 reduces to an integer 2.

If we have a rational number s / t, then by the Division Algorithm, we have unique integers a and r such that 

s = t x a + r 

where r is greater than or equal to 0 or less than t. 

Thus, the rational number s / t is equal to s / t = a + r / t.

If r = 0, then s / t = a, which is an integer.

The positive integer d is the GCD of  two integers s and t  if d is a common divisor of s  and t and if c is a common divisor of s and t then c is a divisor of d. Thus, d is the largest of all the common divisors of s and t.

If GCD( s,  t ) = 1, then we say that s and t  are relatively prime.

Suppose s = ta +r and r > 0. 

Let d = GCD(r,  t) and let p = r / d and q = t / d. It follows that the GCD(p, q) = 1.

Thus,  s / t = a  +  r / d. Since we can simplify the fraction r / d into a proper fraction p / q where GCD(p, q) = 1.

Hence,   s / t = a + p / q 

where a, p and q are integers with 0 < p < q,  p and q are relatively prime.

Therefore, any rational number can be expressed as an integer or as the sum of an integer and a proper fraction.

by Felix P. Muga II
Associate Professor, Mathematics Department, Ateneo de Manila University
Senior Fellow, Center for People Empowerment in Governance
Chair, Mathematical Sciences Division, National Research Council of the Philippines

The Mathematics of Kits

Next topic:
Immediate - "The Rational Number as the Sum of an Integer and a Proper Fraction";
Main Objective: "Digitizing a Rational Number"

THREE METRICS FOR THE PCOS MACHINES

A PCOS MACHINE BY SMARTMATIC

In measuring the accuracy of the PCOS COUNT we propose that the COMELEC shall establish a metric for three areas where the PCOS is making a decision when it scanned a ballot. These are: 

1.  Decision on whether a ballot is valid or not,
2. Decision on whether an over-voting occurs in a certain position, and
3. Decision on whether an oval has a valid mark or not.

A Metric for the PCOS Decision on the Validity of a Ballot

A ballot contains around 20 to 25 votes, about 10 different elective positions, and around 400 to 500 ovals.

 If a PCOS machine decides that a ballot is invalid when it is not, then about 20 to 25 votes are lost and the PCOS COUNT will be incorrect. A PCOS machine may decide that a ballot is valid even if it is not and proceed to count the votes in the ballot. Thus, a metric shall be established to measure the accuracy of the PCOS Machines in determining whether a ballot is valid or not.

In finding the accuracy of the PCOS machines on the validity of a ballot we have to determine the maximum number of ballots that a PCOS machine will scan since any accuracy rating that can be established in this number of ballots, then it will be true for a smaller number of ballots.

I think it is safe to say that a PCOS machine will scan less than 1,000 ballots in the coming elections. Hence, COMELEC shall require that the error rating of the PCOS MACHINES in determining whether a ballot is valid or not shall be at most 1 error out 1,000 ballots.

Therefore, the COMELEC shall require that the accuracy rating of the PCOS MACHINES in determining whether a ballot is valid or not is 99.9%.

A Metric for the PCOS Decision on Over-Voting

A ballot may contain about 10 elective positions. An elective position may have one vote as in the case of the Party List Position or of the Member of the House of Representative in a Legislative District. It may contain an elective position that allows at most 12 votes as in the case of the Senatorial Position.

If a voter voted for more than the required number of votes in an elective position, then an over-voting occurs. If it happens then the vote or votes in that elective position will not be counted. However, a PCOS MACHINE may decide that no over-voting occurs even if there is one, and may proceed to count the vote or votes in that position. The PCOS MACHINE may also decide that there is an over-voting in an elective position even if the voter voted for less than the allowed number of votes.

Thus, we have to devise a metric to measure the accuracy of the PCOS MACHINE in determining whether there is an over-voting or not in an elective position.

A PCOS MACHINE may scan on the maximum about 1,000 ballots. Since a ballot may contain at most 10 elective positions, a PCOS MACHINE is giving a decision whether an over-voting occurs to about 10,000 elective positions. Thus, the COMELEC shall require an error rating of at most 1 error out of 10,000.

Therefore, the COMELEC shall require an accuracy rating of 99.99% for deciding over-voting in an elective position.

A Metric for the PCOS Decision on the Validity of a Mark

If a ballot is valid and no over-voting occurs, then we shall proceed to determine the accuracy of the PCOS COUNT on the votes of all the candidates appearing in the ballot.

A PCOS MACHINE is designed to scan a ballot and determine whether an oval has a valid mark or not. An oval has no valid mark if it is empty or if the amount of shading does not pass a required threshold which can be calibrated in the PCOS MACHINE.

If a PCOS MACHINE detects a valid mark on an oval, it will add one vote to the candidate that corresponds to the validly marked oval. If the PCOS MACHINE decides that the oval has no valid mark, then no vote will be added to the candidate that corresponds to the oval.

An error occurs when a PCOS MACHINE add one vote to a candidate, when the oval corresponding to the candidate is not validly marked. This PCOS COUNT error is called a FALSE POSITIVE ERROR.

Another error occurs when a PCOS MACHINE does not add a vote to candidate, if the oval corresponding to the candidate is validly marked. This PCOS COUNT error is called a FALSE NEGATIVE ERROR.

The RMA commissioned by the COMELEC in 2010 and the mock election supervised by CSER last month compared the PCOS COUNT and the MANUAL COUNT on the votes of each candidate. This COMPARISON TECHNIQUE cannot detect really detect the difference of the PCOS COUNT from the MANUAL COUNT.  If we assume that the MANUAL COUNT is the TRUE COUNT then the COMPARISON TECHNIQUE cannot detect the errors of the PCOS COUNT.

For example, in a precinct with 100 ballots, the PCOS COUNT counted 90 votes for candidate Y and the MANUAL COUNT for candidate Y showed 90 votes also, then the COMPARISON TECHNIQUE will conclude that the PCOS COUNT and the MANUAL COUNT are the same and the PCOS COUNT is a 100% match to the MANUAL COUNT on the votes of candidate Y.

If the MANUAL COUNT is the TRUE COUNT, then the COMPARISON TECHNIQUE will conclude that the PCOS COUNT is 100% accurate on the votes of candidate Y.

However, it is possible that the PCOS COUNT counted correctly the first 90 ballots and computed 85 votes for candidate Y. It is also possible that in the next five ballots, the PCOS COUNT committed 5 FALSE POSITIVE ERRORS and added 5 votes to candidate Y even if the ovals corresponding to candidate Y were not validly marked. Thus, candidate Y has 90 votes after counting the 95 ballots. It is also possible that in the last five ballots, the PCOS COUNT committed 5 FALSE NEGATIVE ERRORS and did not add 5 more votes to candidate Y even if the ovals were validly marked. Therefore, it is possible that PCOS COUNT and MANUAL COUNT matched even if the PCOS COUNT committed 10 ERRORS. This is possible, if the number of FALSE NEGATIVE ERRORS and the number of FALSE POSITIVE ERRORS are equal.

Since the PCOS MACHINES scanned 100 ballots, it made decision on whether 100 ovals corresponding to candidate Y were validly marked or not. Since the PCOS COUNT committed 10 errors, it follows that the ERROR RATING OF THE PCOS COUNT on the votes of candidate Y is 10 out of 100 or 10%.

Therefore, the accuracy rating of the PCOS COUNT on the votes of candidate Y is 90% and not 100% as shown by the COMPARISON TECHNIQUE. If candidate Y is a Party List Candidate where 2% is assured of at least one party list seat, then the PCOS COUNT clearly disenfranchised the voters of candidate Y and denied the candidate of at least one seat.

The COMELEC must therefore do away with the COMPARISON TECHNIQUE and devise a mechanism to detect all types of ERRORS that a PCOS MACHINES will commit so that a reliable accuracy rating will be devised.

To determine the accuracy rating of the PCOS COUNT on the votes of all the candidates, then we have to determine the maximum number of ovals that a PCOS MACHINE will scan in an election and we have to determine the total number of FALSE POSITIVE ERRORS and FALSE NEGATIVE ERRORS committed by the PCOS COUNT on all of the ovals scanned by the PCOS MACHINE. This can be done by using the ballots, their corresponding ballot images generated by the PCOS MACHINES, and the AUDIT LOG of the PCOS MACHINES that shows how each ballot is interpreted by the PCOS MACHINE.

We assume a maximum of 500 ovals per ballot. Since we also assume a maximum of 1,000 ballots scanned per PCOS MACHINE, the maximum number of ovals shall be 500,000.

Thus, the COMELEC shall require an error rating of at most 1 error per 500,000 ovals. This is equivalent to an accuracy rating of 99.9998% for the PCOS COUNT on deciding the validity of a mark of an oval.

Felix P. Muga II

Associate Professor, Mathematics Department, Ateneo de Manila University

Senior Fellow, Center for People Empowerment in Governance

Mathematics of Politics

DESIGNING CLUSTERS FOR SUPERCOMPUTING

Cluster computing is now an accepted form of supercomputing. A cluster is a collection of computing nodes interconnected by a high-speed network using a network switch or a number of network switches. Each cluster node is either a workstation, a PC, or a symmetric multiprocessor.

A few years back we proposed a new cluster design that is symmetric and every pair of computing nodes is connected by a single switch.

We based the construction of a cluster using a graph with diameter 1 or 2. The graph in the figure below is called a circulant graph of order 8. It consists of 8 nodes denoted by 0, 1, 2, 3, 4, 5, 6 ,7 and 12 edges connecting each of the pairs {0, 1}, {0, 4}, {0, 7}, {1, 2}, {1, 5}, {2, 3}, {2, 6}, {3, 4}, {3, 7}, {4, 5}, {5, 6}, {6, 7}.

We construct a cluster with 8 network switches and 48 computers. Each network switch has 24 ports and each computer has 4 Network Interface Cards (NICs)  to connect to 4 network switches. The cluster is shown below with 8 switches denoted by S0, S1, S2, S3, S4, S5, S6 and S7 and the 48 computers are denoted by 0, 1, 2, ..., 47.

The constructed cluster has the following properties:

1. There are 48 computers with 8 network switches.
2. With 48 computers, we have a total of 1,128 pairs of computers. All of these pairs are connected by at least one switch. In fact, there are 288 pairs joined by exactly one switch, 720 pairs joined by exactly two switches, and 120 pairs joined by exactly 4 switches.
3. If a 100 Mbps ethernet is used, the bidirectional bandwitch per pair of computers is 391.48Mbps.
4. The bisection bandwidth of the cluster reaches 9.6Gbps.
5. The bisection bandwith per computer reaches 200 Mbps.

by Felix P. Muga II

Associate Professor, Mathematics Department, Ateneo de Manila University

Senior Fellow, Center for People Empowerment in Governance

The Mathematics of Networks

THE NATURAL NUMBERS, THE WELL-ORDERING PRINCIPLE, AND THE INTEGERS

Natural numbers were developed for counting objects. The ancient Egyptians invented a number system that started with a symbol for number 1. This symbol represented one object that is counted. Two similar symbols were used to represent two objects counted. There will be nine of this symbol if there are nine objects counted.

Another symbol distinct from the previous symbol was used to represent ten objects counted.  

Ancient Egyptian Numerals (  http://en.wikipedia.org/wiki/Egyptian_numerals)

Later on the Babylonians considered zero (0) as a number.

Much later the set of natural numbers is considered to be the set consisting of the counting numbers 1,2,3, …, and so on including the number 0.

An important property of the set natural numbers is the “Well-Ordering Principle” which states that every non-empty subset of the natural numbers has a least element.

Suppose we have a subset S = {x, y, z} of three natural numbers.

Then by the well-ordering principle,  S has a smallest element, say x. Then x < y and x < z.

Let us consider the subset T = {y, z}. Again, by the well-ordering principle, T has a least element. Let us assume that it is y. Then y < z.  

Therefore, we have x < y < z. This means that the elements x, y and z of S are well-ordered.

When the number 3 is subtracted from 5, the difference which is equal to 2 is a natural number. However, if the number 5 is subtracted from 3, the difference is not a natural number. Hence, the negative of a non-zero natural number is invented and we have the set of integers.

 The set of integers were developed to include the natural numbers and their negatives. Note that the negative of 0 is still 0.

An interesting property about the set of integers is the “Division Algorithm”which states that “for any two integers a and b with b > 0, there exist unique integers q and r such that  a = bq + r where r is greater than or equal to 0 but less than b.”

 Repeated applications of the Division Algorithm will give us the greatest common divisor of two integers and the k-ary expansion of a natural number. 

by Felix P. Muga II
Associate Professor, Mathematics Department, Ateneo de Manila University
Senior Fellow, Center for People Empowerment in Governance

The Mathematics of Kits

Next topic:
Immediate - "A Rational Number Expressed as an Integer or as Sum of an Integer and a Positive Rational Number Less Than 1"
Main Objective: "Converting a Rational Number into a Sequence of Kits"

ACCURACY OF THE COUNTING MACHINES

If a machine is designed and constructed for counting, then it is important to measure its counting accuracy. If  we do not allow a machine that is built for counting money to commit an error, then it is equally that we do not allow a machine that counts our votes to miscount these votes.

A vote counting machine may commit a false positive error when it decides that a ballot has one vote for a candidate even if the ballot does not have a valid mark (or a valid vote) for the candidate. Thus, the vote counting machine erred when one vote is added to this candidate when the candidate should not have.

A vote counting machine may commit a false negative error when it decides that a ballot has no vote for a candidate even if the ballot has a valid mark (or a valid vote) for the candidate. Thus, no vote is added to this candidate. Thus, the vote counting machine erred when no vote is added to this candidate when the candidate should have a vote.

A technique by comparing the total votes of a candidate using the MANUAL COUNT (which is the TRUE COUNT) and the total votes of a candidate using the MACHINE COUNT can not detect the MISCOUNTED VOTES or errors of the MACHINE COUNT if the counting machine committed both the false positive errors and the false negative errors on the votes of this candidate.

Suppose that a ballot box contains 18 ballots and one COUNTING MACHINE is use to read the votes of these 20 ballots. Further suppose that the COUNTING MACHINE reads ten ballots and assigns 10 votes to candidate Y and the COUNTING MACHINE correctly counted these votes for candidate Y. 

Then the COUNTING MACHINE reads 5 additional ballots and it added five votes to candidate Y. Now, the MACHINE COUNT has 15 votes for candidate Y. However, each of the five ballots has no valid vote for candidate Y. The MACHINE COUNT has miscounted five votes for candidate Y. 

Hence, The are 5 FALSE POSITIVE ERRORS of the MACHINE COUNT on the votes of candidate Y. 

Then the COUNTING MACHINE reads 5 more ballots and each of these 5 ballots has a valid vote for candidate Y. But, the machine did not add 5 more votes to candidate Y. Thus, MACHINE COUNT has still 15 votes for candidate Y.  The MACHINE COUNT has committed 5 FALSE NEGATIVE ERRORS.

After reading all the 20 ballots, the MACHINE COUNT has 15 votes for candidate Y. The TRUE COUNT for candidate Y is also 15 votes. 

The absolute difference between the MACHINE COUNT and the TRUE COUNT for candidate Y is | 15 -15 | = 0. Thus by the COMPARISON TECHNIQUE, the MACHINE COUNT has no error and the ACCURACY RATING of the MACHINE COUNT on the votes of candidate Y is therefore 100%.

But, the MACHINE COUNT committed 5 FALSE POSITIVE ERRORS and 5 FALSE NEGATIVE ERRORS on the votes of candidate Y or a total of 10 COUNTING ERRORS.

The COUNTING MACHINE appreciated 20 ballots for candidate Y and it committed a total of 10 COUNTING ERRORS.

Therefore, the Error Rating of the MACHINE COUNT on the VOTES of candidate Y must be 10/20 or 50% and not 0%. 

the COMPARISON TECHNIQUE on the MACHINE COUNT and the TRUE COUNT on the votes is useless since it could not determine accurately the ACCURACY RATING and the ERROR RATING of the MACHINE COUNT.

Therefore, the COMELEC must STOP using the COMPARISON TECHNIQUE in determining the ACCURACY RATING OF the PCOS MACHINES.

Any procedure that determines the accuracy rating of the PCOS MACHINES must take into account the FALSE POSITIVE ERRORS and the FALSE NEGATIVE ERRORS of the MACHINE COUNT. It can be determined using the AUDIT LOG of the PCOS MACHINE, the BALLOT IMAGES and the  REAL BALLOTS.

Felix P. Muga II
Associate Professor, Mathematics Department, Ateneo de Manila University
Senior Fellow, Center for People Empowerment in Governance

The Mathematics of Politics

ON BITS AND RATIONAL NUMBERS

We, human beings, have ten fingers. This might be the reason why we are used to count by ten digits or decimal digits. 

Computers are designed to count in twos only; 0 and 1. These are called as binary digits or bits. 

A sequence of bits is a binary expansion of a number. For example, the sequence 110 is the binary expansion of the number 6. This is determined by computing  (1 x 4) + (1 x 2) + 0.

Hence, a binary sequence of the form 

is equivalent to the number computed in 

This binary expansion is unique to every number and every number has a unique binary expansion.

A rational number of the form s/t where s and t integers, t greater than 0 can be represented by a binary expansion. For example, the rational number 1/4 is represented uniquely by 0.01 and the rational number 11/4 is represented uniquely by 10.11 since 19/8 = 1 x 2 + 0 + (0 x 1/2) +  (1 x 1/4) + (1 x 1/8). 

The sequence 10 in 10.011 is called the whole part and the sequence 011 after the binary point is called the fractional part of the binary expansion.

Computers are designed so that it can store a binary sequence of a finite length in its memory. 

For example, if the computer is designed to store 12 bits of the fractional part only, then the rational 1/5 or 0.2  will be stored as 0.001100110011.This is only equivalent to 0.19970703125 and not 0.2. Therefore, the 12-bit computer can not store the binary expansion of 1/5 completely. 

Note that the fractional part of the binary expansion is periodic with period 0011 of length 4. Hence, an exact binary expansion for 1/5 is 0.(0011) where the sequence inside the parenthesis is a periodic sequence.

Note that the number 5 and 2 are relatively prime, that is, their greatest common divisor is equal to 1. 

Thus, a  fractional part of the binary expansion has a period if the denominator of the rational number has a factor which is relatively prime to 2.

If two numbers x and y are relatively prime, then we can find the smallest positive integer m such that the number 

is divisible by y.

This smallest positive integer m is called the multiplicative order of x under modulo y.

The multiplicative order of 2 under modulo 5 is 4, since 4 is the smallest positive integer such that 2^4 -1 = 15 is divisible by 5. 

We can obtain 15 from 5 by multiplying 5 by 3.

Thus, we have

The binary expansion of 3 is equal to 11. Since 4 is the multiplicative order of 2 under modulo 5, we can write the binary expansion of 3 as 0011 instead of 11. 

Therefore, 

by Felix P. Muga II
Associate Professor, Mathematics Department, Ateneo de Manila University
Senior Fellow, Center for People Empowerment in Governance

The Mathematics of Kits (specifically Bits)

A FAILURE OF PARTY LIST ELECTION IN 2013?

The House Committee on Suffrage and Electoral Reforms conducted a mock election on July 24-25, 2012  at the Nograles Hall, South Wing Building of the House of Representatives to demonstrate the counting capability of the SMARTMATIC-supplied PCOS machines for the 2013 elections.

South Wing Building , House of Representatives, Philiippines (www.congress.gov.ph)

The MANUAL COUNT had 768 votes while the PCOS COUNT had 770 votes for the Party List Position with 55 Party List Candidates. See Table 1. 

TABLE 1.    PARTY LIST POSITION
No.	Candidate	Manual Count	PCOS Count
1	BROWNBEAT ALLSTARS	80	79
2	DAGTANG LASON	25	25
3	GREYHOUNDZ	12	13
4	HALE	29	29
5	HOTDOG	68	68
6	IAXE	6	6
7	IMAGO	10	10
8	K AND THE BOXERS	3	3
9	KABISAYAAN	6	6
10	LA LIGA FILIPINA	8	8
11	LIYAH	1	1
12	MAHARLIKA	0	0
13	MANOMANO	2	2
14	MASTAPLANN	2	2
15	MEN OPPOSE	36	36
16	MOCHA GIRLS	61	60
17	NEW MINSTRELS	26	27
18	NINO and THE FORCE	36	36
19	P.O.T.	33	32
20	PACK OF WOLVZ	17	17
21	PAREDES	13	13
22	PASSAGE	9	9
23	PEDICAB	8	8
24	PIGS WITH PEARLS	5	5
25	PINAY	11	10
26	POWER FOUR	5	6
27	PROTEIN SHAKE	4	4
28	RAGE	2	2
29	RETROSPECT	30	30
30	ROCKSTEDDY	21	23
31	SALINGKET	22	19
32	SANDWICH	26	27
33	SENSITIVO	14	15
34	SESSION ROAD	13	13
35	SIDE A	13	13
36	SIN	4	4
37	SINOSIKAT?	9	10
38	SKY CHURCH	3	3
39	SOPIZ	0	0
40	SUNFLOWER DAY CAMP	5	5
41	TAGALOG REPUBLIC	5	5
42	THE GO GIRLS	19	18
43	THE HARBOR BABIES	12	12
44	THE MEMBERS	5	5
45	THE WUDS	4	5
46	TRUE FAITH	6	6
47	UP DHARMA DOWN	11	10
48	VIRUS ARTISTS	1	2
49	VOYZ AVENUE	1	2
50	WADAB	2	2
51	WATAWAT	3	3
52	WINK	5	5
53	WOLFGANG	4	4
54	X AXIS	0	0
55	ZELLE	12	12
		768	770

The MANUAL COUNT and the PCOS COUNT differed in 17 of the Party List Candidates and the total number of absolute differences is 20. See Table 2.

TABLE 2.   PARTY LIST POSITION
No.	Candidate	Manual Count	PCOS Count	Absolute Difference
1	SALINGKET	22	19	3
2	ROCKSTEDDY	21	23	2
3	BROWNBEAT ALLSTARS	80	79	1
4	GREYHOUNDZ	12	13	1
5	MOCHA GIRLS	61	60	1
6	NEW MINSTRELS	26	27	1
7	P.O.T.	33	32	1
8	PINAY	11	10	1
9	POWER FOUR	5	6	1
10	SANDWICH	26	27	1
11	SENSITIVO	14	15	1
12	SINOSIKAT?	9	10	1
13	THE GO GIRLS	19	18	1
14	THE WUDS	4	5	1
15	UP DHARMA DOWN	11	10	1
16	VIRUS ARTISTS	1	2	1
17	VOYZ AVENUE	1	2	1
				20

If we assume that the MANUAL COUNT is the TRUE COUNT of the Party List Votes in the mock election, then the PCOS COUNT miscounted 20 out of 768 party list votes. This gives an error rating of 20/768 or 2.60417%.

If these PCOS machines with 2.60417% ERROR RATING will be used in the 2013 Party Election, then the COMELEC cannot determine with certainty the WINNERS for three seats, two seats and one seat.

Therefore, we might have a FAILURE of  PARTY LIST ELECTION in 2013.

How do we avoid this disaster?

We recommend the following:

1. the COMELEC shall enforce the 99.995% accuracy rating specified in the Terms of Reference for the bidders of the 2010 Automated Election System. A 99.995% accuracy rating implies an error rating of at most 0.005%.
2. An accuracy rating of the PCOS machines shall be determined. COMELEC can ask the National Research Council of the Philippines (NRCP), an advisory arm of the Department of Science and Technology to do the certification.
3.  COMELEC shall promulgate a mechanism for the RECOUNT OF THE BALLOTS if the percentage share of votes of a Party List Candidate PLUS or MINUS the ERROR RATING determined and certified by NRCP shall cause to win an additional seat or to lose one seat for the said Party List Candidate.

by Felix P. Muga II
Associate Professor, Mathematics Department, Ateneo de Manila University
Senior Fellow, Center for People Empowerment in Governance

Mathematics of Politics