A mathematical quirk of our electoral system

Chan Jia Hui

It takes a few mathematical steps to work out Electoral Mathematics.

It takes a few simple workings to demonstrate that is possible for a party that gained 51% of the popular vote to occupy 4 seats in parliament.

One day, I was remarking over MSN messenger to a writing staff of the New Asia Republic (NAR) on the disproportionate nature of seats won by the opposition in relation to proportion of votes won.

The opposition won 39.9% of votes and occupied 6 seats in our parliament. My maths junkie counterpart from NAR remarked that actually it was not considered extreme.

The latter revealed, to my utter disbelief, that it is mathematically possible for a political party to win 51% of the total share of votes but end up occupying 4 seats in parliament. That chap showed me why in the form of a mathematical proof.

Using data from singapore-elections.com, the total number of eligible voters for General Elections 2011 is 2,350,873. We have to assume a very close fight in all wards, with the party losing by a close margin of 49.5% for all the other wards. However, the losing party did extremely well in a total white-wash victory for one 4 seater ward.

We assume that the ward in which the party did well in is Moulmein-Kallang Group Representative Constituency (GRC). We also assume no absentee voters for simplicity’s sake. The other obvious assumption is that winner is based on the first-past-the-post format of contest where majority takes all. Lastly, the nature of contest explored in this proof is between two political parties.

Total number of eligible voters = 2,350,873
51% of the voters supporting the party = 1198945.23
Total number of eligible voters for Moulmein-Kallang GRC = 87,595
Total number of voters in the other wards other than Moulmein-Kallang = 2,350,873 – 87,595 = 2,263, 278
Assuming the party obtained a consistent margin of 49.5% for each of the other wards it contests in, the total number of voters voting for that party in the other wards = 0.495 x 2,263,278 = 1120322.61
Therefore, the total number of voters in Moulmein-Kallang who gave the 4 seats to the party =
1198945.23 – 1120322.61 = 78622.62 of 87,595 total voters.

Hence, this is an extreme case of a party having only 4 out of the 87 seats in parliament despite having 51% of the voters supporting that party. Such can emerge when there are close fights in other wards with wafer thin marginal defeats in addition to a big margin victory in one GRC.

One scenario of contest discussed by observers online is a win-all-lose-all merging of all wards into a mega-GRC. This means all 87 seats are contested on the basis of the outcome of the votes casted by all 2,359,873 voters.

However, if we base our analysis on the mathematical proof above, each nature of contest can have a different outcome. If the format of contesting 15 GRCs and 12 single member constituencies (SMC) is used, the party with 49% of the popular vote will gain 83 seats.

If the contesting format is a winner-takes-all-loser-loses-all mega-GRC, the party with 51% of the popular votes will come up tops.

Hence, going back to the situation where the party wins 51% of popular votes but ends up with 4 seats, a popular albeit important reason observers give is due to GRC. That is quite possible. The party could have won certain sub-divisions of a GRC (s), but nevertheless, when the total votes are tallied, they only scored 49.5%.

However, if we assume that the GRCs are split into sub-divisions and even all of them obtained 49.5% of the votes, it is still possible end up with 4 seats in parliament.

There is attributed to another more important reason, which is often overlooked but important nonetheless – the voters to seats ratio. Now, we go back to the mathematical proof, and modify some variables. Say, Moulmein-Kallang has 200,000 voters to 4 seats. This makes a ratio of 50,000 voters per seat.

Total number of voters in the other wards = 2,350,873 – 200,000 = 2,150,873
If party scores a constant 49.5% on every other ward, total number of voters voting for the party in the other wards = 0.495 x 2,150,873 = 1064682.135
Assuming 90% of voters at Moulmein-Kallang voted for the party = 0.9 x 200,000 = 180,000
Total number of voters voting for the party is = 180000 + 1064682.135 = 1244682.135
Percentage of total voters voting for the party = 1244682.135/2350873 x 100 = 53%

The voters to seat ratio effect is demonstrated by my maths-junkie peer who showed that it is possible to occupy 44 seats out of 87 (a little more than half) in parliament by contesting in areas that have a total of 1,082,783 voters. These 1,082,783 voters spread out over 7 GRCs and 11 SMCs made up a mere 46% of the total eligible voters (2,350,873). The other 54% of voters are spread out over the rest of the GRCs and 1 SMC.

The table shows respective wards, the total number of voters in each, and the voters to seats ratio (rounded to nearest whole number).

GRCs

Aljunied GRC (5 seats), 143,148 voters – 28,630 voters/seat
Ang Mo Kio GRC (6 seats), 179,071 voters – 29,845 voters/seat
Bishan-Toa Payoh GRC (5 seats), 122,492 voters – 24,498 voters/seat
Choa Chu Kang GRC (5 seats), 158,648 voters – 31730 voters/seat
East Coast GRC (5 seats), 120,324 voters – 24,065 voters/seat
Holland-Bukit Timah GRC (4 seats), 91,607 voters – 22,902 voters/seat
Jurong GRC (5 seats), 125,276 voters – 25055 voters/seat
Marine Parade GRC (5 seats), 154,451 voters – 30,890 voters/seat
Moulmein-Kallang GRC (4 seats), 87,595 voters – 21,899 voters/seat
Nee Soon GRC (5 seats), 148,290 voters – 29,658 voters/seat
Pasir Ris-Punggol GRC (6 seats), 168,971 voters – 28,162 voters/seat
Sembawang GRC (5 seats), 142,459 voters – 28,492 voters/seat
Tanjong Pagar GRC (5 seats), 139,771 voters – 27,954 voters/seat
Tampines GRC (5 seats), 137,532 voters – 27,506 voters/seat
West Coast GRC (5 seats), 121,045 voters – 24,209 voters/seat

SMCs (1 seat)

Bukit Panjang SMC – 33,053 voters
Hong Kah North SMC – 27,701 voters
Hougang SMC – 24,560 voters
Joo Chiat SMC – 22,069 voters
Mountbatten SMC – 23,731 voters
Pioneer SMC – 25,745 voters
Potong Pasir SMC – 17,327 voters
Punggol East SMC – 33,281 voters
Radin Mas SMC – 31,014 voters
Seng Kang West SMC – 26,882 voters
Whampoa SMC – 21,631 voters
Yuhua SMC – 23,199 voters

Courtesy of www.singapore.elections.com

What my peer demonstrated was to pick 7 GRCs with relatively lower voters to seat ratios. The 7 GRCs are Holland-Bukit Timah, Bishan-Toa Payoh, East Coast, Tampines, Moulmein-Kallang, Jurong and West Coast. 5 of these GRCs each have 5 seats whilst Moulmein-Kallang and Holland-Bukit Timah have 4 each. This makes a total of 33 seats.

11 SMCs were picked, except for Punggol East GRC which has the highest number of voters at 33,281. This sums up to a grand total of 44 seats.

Assuming the party scored 51% for every ward it contests for the 7 GRCs and 11 SMCs, that translates to 552219.33 out of 1,082,783 possible voters.

Thus, by targeting SMCs and GRCs with lower voters to seat ratios, it is possible for a party to occupy half the seats in parliament with much less than half the popular vote, based on the votes casted on polling day. In this case, we assume that the party does not send candidates to contest the other wards. In reality, voters to seat ratio can be an issue due to the differing residential densities of the wards.

It is interesting to know of the mathematical possibility that a party may win 51% of the popular vote but occupy a mere 4 seats in parliament. The undeniable popular reason that observers come up with is the GRC. They are largely correct.

However, there is another more important reason, which is the voters to seat ratio that can explain such an ‘aberration’. The ratio can be influenced by residential developments or when residents shift from one place to another. More importantly, the drawing of boundaries by the elections department also has a huge bearing on the voters to seat ratio of different wards.

As the table above show for GRCs, the voters to seat ratio can range from as low as 21,899 voters/seat for Moulmein-Kallang GRC to a high 31,730 for Choa Chu Kang GRC. Even the SMCs demonstrated a varying trend where Potong Pasir has the lowest total number of eligible voters at 17,327 to Punggol East’s highest of 33,281 voters.

The writer is thankful to New Asia Republic’s staff for providing mathematical insights. Photo courtesy of lucius_clay, Flickr Commons.