The mathematics of elections

In this segment, we feature blogs that carry insightful and considered posts. This week we highlight The Mathematics of Elections by Alex Au (14 Mar 08) in

EVER SINCE THE stunning gains made by opposition parties in Malaysia’s general election, held 8 March 2008, a few Singaporean blogs (and comments in their talkback train) have been almost ecstatic at this example of an incumbent government humbled.

Some have been vociferous in calling on Singaporeans to follow this example and vote for the opposition in our next election. A handful went further, cussing the “66.6%” who voted for the People’s Action Party (PAP) in 2006.

Frankly, I think we should be a bit more circumspect. The electoral “tsunami” in Malaysia has brought Parti Islam Se-Malaysia (PAS), a religious party, into power either by itself or in coalition in 4 of Malaysia’s 13 states. In 3 of those states, PAS politicians became Chief Ministers. For one who believes strongly in the separation of Church and State, this is not an outcome that thrills me in any way.

Fortunately, Singapore does not have any religiously-based party on offer, but this is not to say that all the available secular opposition parties are necessarily any good. I think it is important for a discerning voter to know what he is voting for as much as what he is voting against.

However, this essay does not intend to discuss the pros and cons of political parties. Instead, I hope to throw some light on the way voter sentiment translates into seats, using the available data from Malaysia’s 2004 and 2008 elections and Singapore’s 2006.

In the process, I will demonstrate how the differences between Singapore’s and Malaysia’s electoral systems affect the way votes are translated into seats in each country, and I will reveal in the process two, probably discouraging and counter-intuitive, insights:


(1) There is not a lot of difference in party preferences in Malaysia 2008 compared to Singapore 2006; and

(2) Even if Singaporeans’ preferences (ruling party:opposition) were to be like Malaysia 2008, we would still end up with a parliament that is 90% PAP, meaning the PAP would still be able to change the constitution at will.


What it shows is that our problem is not so much that people are unwilling to vote for our opposition parties, despite the ranting we see on many blogs, but that our electoral system in Singapore makes such votes nearly meaningless. Consequently, it shows the need for us to put more priority on electoral reform, rather than just exhorting people to vote for the opposition.

You don’t have to support any particular party to support electoral reform, just as you don’t need to support any particular political program to support clean elections. You only have to believe in the principle that elections should produce results that more or less reflect the spread of opinions among citizens.

In my view, Singapore’s system does not do that, and is in fact designed not to.

* * * *

Malaysia’s electoral system resembles the British system far more than ours. The country is divided into 222 electoral constituencies, each electing one member of parliament on a first-past-the-post basis.

The average number of voters in a constituency is about 50,000.

In a general election that involves 2-sided contests in nearly all the seats — and this was the case in Malaysia, due to electoral pacts made among the 3 main opposition parties — the percentage split of ruling party:opposition votes, while varying from constituency to constituency, will tend to be distributed in manner resembling a Gaussian (bell) curve.

The mean of that curve would be the overall vote share obtained by the winning party. The standard deviation, i.e. the spread, of the curve will depend on how similar or dissimilar the constituencies are to each other in their political preferences, plus regional particularities.

In this diagram, you’ll see the difference in hypothetical election outcomes in countries with widely diverse and mostly similar constituencies. The height of the curve at any given point along the X-axis represents the number of constituencies with that particular vote-share.

Without doing a detailed analysis of data from the Malaysian elections, I really don’t know what the standard deviation is in that country’s experience, but going by Singapore’s single-member constituencies in our 2006 general election, I am estimating the spread between -2SD and +2SD (technical language to indicate the overall spread from nearly one end of the bell curve to nearly the other end) to be about 40 percentage points.

To put it in another way, it means the ruling party’s nearly worst performing candidates poll about 40 percentage points less than their nearly best performing candidates. Actually, I suspect it may even be wider than that in Malaysia due to greater heterogenicity and regional particularities. For example, Mukhriz Mahathir, the son of the former Prime Minister, won his seat for BN with 82% of the vote in his constituency of Jerlun, about 30 percentage points higher than BN’s national average. However, for the purposes of this rough-and-ready paper, this estimate of a 40-percentage point spread (i.e. 20 points above and below the mean) will suffice.

* * * * *

Obviously, in a 2-sided contest, if one party fails to get 50% of the vote in a constituency, then it loses it to its opponent. The diagram on the right shows what happened in Malaysia’s 2004 general election where the ruling Barisan Nasional (BN) received an average of 63% of the popular vote and won 91% of the seats in Parliament.

In the diagram, seats won by the BN are represented by the colour orange.

The diagram on the left shows what happened in the recent election. The BN obtained 51 or 52% of the popular vote (figures vary depending on the source) but won 63% of the constituencies.

Actually, going by the model, they should have won fewer (i.e. the opposition “tsunami” should have been even bigger). What cushioned BN’s setback in terms of parliamentary seats was the fact that the Borneo states of Sabah and Sarawak are over-represented in the legislature. These two states have about 21% of Malaysia’s population, but 25% of the seats.

In the 2008 election, BN’s support in Sabah and Sarawak remained solid, and so the ruling coalition benefitted from it.

* * * * *

Turning our attention to Singapore’s 2006 general election, we need to distinguish between the Single-Member Constituencies (SMC) and the Group Representation Constituencies (GRC).

There are only 9 SMCs in Singapore, with an average of 24,000 voters per constituency.

All were contested in the last election, in straight fights. There were no three-cornered contests.

The highest PAP vote-share was about 77% in Bukit Panjang and the lowest was 37% in Hougang, indicating a spread of 40 percentage points. The PAP’s mean vote-share in the 9 SMCs was 61%.

The PAP won 7 of the 9 SMCs (78%). According to the model, the PAP should have won more than that, but as there were only 9 SMCs, variation can result with small numbers.

GRCs in Singapore require a party to put up a slate of 5 or 6 candidates, who all get into Parliament as a team so long as the party polls more votes than any other team in the same GRC. There are 14 GRCs in Singapore’s electoral map, but only 7 were contested in 2006, again all were straight fights.

GRCs have an average of 139,000 voters, much bigger than the average for Malaysian constituencies (even though Malaysia has a population 7 or 8 times bigger than Singapore). The law of large numbers tends to dilute the impact of dissenting or marginal voices, leading to greater homogeneity (see an extreme example in the yellow box at right).

This tendency towards similarity is reinforced by the huge impact that public housing has. With 85% of Singaporeans living in public housing, and all types of housing spread over all GRCs (not to mention the strict racial quotas implemented identically across all public housing estates), there is a strong homogenising effect across the various GRCs.

The result can be seen in the narrow spreads of the 7 GRCs contested. The PAP had an average vote-share (the “mean”) of 67% in the GRCs, with the lowest, at 56% in Aljunied, and the highest, at 77%, in Sembawang. The spread between the lowest and highest is only 21 percentage points, half the spread seen among the SMCs.

What this means is that for the opposition to win 1 or 2 GRCs, the narrow bell curve has to move considerably to the left. The mean may have to move from 67% to about 55-57% before part of the bell curve slips under the 50% mark.
* * * * *

At this point, some readers may say, hey, if the BN can suffer a 11-percentage point fall in vote-share in 2008, the PAP can too. This is where I’d invite you to look at overall vote-share again.

A key difference between Singapore and Malaysia is that here, voting is compulsory, but not in Malaysia. In their recent general election, turnout was said to be 70%, meaning about 30% of voters did not bother to go to the polls, whereas in Singapore, with the law over our heads, turnout is typically in the 95-97% range.

The world over, the ones most likely to stay at home are those who are the most apathetic and least politicised. If they are forced to vote, they are much more likely to vote for the devil they know than the one they don’t. In other words, they tend to vote for the status quo, or the incumbent party.

My guess is that if Malaysia had compelled by law that 30% to vote, a large number of them would have cast their votes for the BN. Putting it another way: If Malaysia had compulsory voting, BN would have scored about 60% of the popular vote, instead of the 51-52% as now reported.

60% is not terribly different from the 66.6% that the PAP scored in 2004, is it? So how come Malaysia’s election can produce a result, in terms of seat allocation, that is so different from Singapore’s?

The answer is apparent by now: The electoral system here makes it much harder for the opposition than in Malaysia. The GRCs and compulsory voting introduce significant features in the mathematics that work against the opposition.

Does this mean that compulsory voting is unfair? No. You can argue for and against it. With it, the depoliticised and apathetic citizens get to influence the result, even though they never gave a moment’s thought to the issues. Without it, the zealous may get disproportionate weight should large numbers not bother to vote.

The “evil” of GRCs is more obvious. It does nothing except to skew the result. (As for minority-race representation, there are many other ways of achieving it, but I shan’t go into the details here.)
* * * *

Now, imagine at the next election, the PAP had an average vote share similar to the BN (but with compulsory voting), i.e. 60%.

How many SMCs would they lose? With a spread of 40 percentage points typical of SMCs, the PAP might lose no more than 2 or 3. (Consider this: In the 2006 Singapore general election, after Hougang and Potong Pasir, which were won by the opposition, the next best opposition performance in an SMC was Choa Chu Kang, where Steve Chia polled 39.6%. Thus, even if the PAP had lost about a further 7 percentage points nation-wide, they would still likely have retained Choa Chu Kang.)

How many GRCs would they lose? With a narrower spread of 21 percentage points typical of GRCs, the PAP may well win all GRCs again. At worst, they may lose one, giving the opposition a yield of 5 or 6 seats.

(Consider this: The best 2006 opposition performance in a GRC was Aljunied, where the Workers’ Party team won 43.9% of the vote. It would be touch and go whether they would have won it if the overall PAP vote-share was 60% instead of 66.6%.)

So in total, the opposition may win about 7-10 electoral seats out of 84 (around 10%) in the event that the PAP’s vote-share resembled the BN’s (as modified by compulsory voting). Contrast this to the BN’s actual result (without compulsory voting) in 2008, losing 37% of parliamentary seats to the opposition.

It’s time we opened a debate about our electoral system, separate from any debate about the merits of this or that political party.

However, some may argue that no change will happen unless we first throw out the PAP and vote in an opposition party, but let me caution you: The system is such that it rewards the winner. So if another party wins an election, it will from that point on, be the beneficiary of our unique electoral system. Why should they change it?

That’s why the issue of electoral reform has to be kept distinct from the question of which party to support, and should be dealt with now.


Read also:
Any lessons from the Malaysian general election for us? (10 Mar 08)
The mathematics of elections 2 (7 Apr 08)
Political activism, cyber or otherwise (14 Apr 08)

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