One Vote, Wrong System
24 September 2004

In the aftermath of the 2004 Legislative Council elections, it is high time that Hong Kong reviewed its unusual and unfair method of proportional representation, put in place by the unelected Provisional Legislative Council in 1997. As we will show, the devil is in the details, which mathematically favour short lists, 1-person lists and lunatics. They also exclude voters from candidate choice within parties. We make several proposals for electoral reform.

Wrong Quota

Hong Kong operates, for the half of its 60-seat Legislative Council that is now directly elected, a closed party-list proportional representation system using what is known as a "largest remainder" seat allocation method. The 30 seats are divided into 5 geographic constituencies with seats allocated broadly proportional to the population of those areas, resulting in constituency sizes of 4,5,6,7 and 8 seats. Electors cast a single vote for a list consisting of 1 or more candidates.

The total number of votes in each constituency is divided by the number of seats, to determine a "quota". Any list which gets 1 or more quota of votes is allocated one seat per quota, then for those lists which have unelected candidates, the remaining votes are ranked by size, and the remaining seats are allocated to the lists with the largest remainders. The type of quota used in Hong Kong is known as the Hare quota after English solicitor Thomas Hare (1806-1891).

As we will show, this is the wrong type of quota for Hong Kong's system, because the votes discarded are non-transferable, so there is no allocation of unused remainders or from an exhausted list, where all the candidates on that list have been elected and there are votes left over, nor are votes transferred from the minority lists which are not elected.

The combination of the Hare quota and the non-transferable vote has some mathematical consequences:

1. The first candidate on each list does not need to get a full Hare quota to be sure of election. For example, in a S=4 seat constituency with V=100,000 votes, the Hare Quota is 25,000, but any list with 20,001 votes can be certain of their first candidate being elected, because it is impossible for 4 other lists to each have more votes than that. This lower target is known as the Droop quota, after English lawyer and mathematician Henry Droop (1831-1884) and the general formula is V/(S+1) votes, rounded up to the nearest vote. This is always less than the Hare quota, V/S, and in constituencies with the smallest number of seats the percentage difference between Hare and Droop is greatest.

2. When there are not more lists (L) than seats, that is, if L<=S, then the number of votes needed for election of the first candidate reduces even further. For example, if there are 4 lists of candidates for 4 seats and 100,000 votes, then we know that at least one list will get a full Hare quota of 25,000, removing those votes from the remainders. So that will leave 3 seats and not more than 75,000 votes. Any list which gets at least 18,751 votes can be certain of their first candidate being elected because it would be impossible for 3 other lists to each have more than that number. The general formula for this quota is V *(L-1)/SL (rounded up), as long as L<=S.

3. Note that these lower targets are mathematical absolutes, regardless of voting patterns. In practice, the more quotas that are filled by the most popular party lists, the easier it gets for the first candidate on an unpopular list. This is simply because each successive candidate elected by quota on a list absorbs a full Hare quota, whereas the first candidate on a competing list only needs a Droop quota at worst.

4. The use of the "largest remainder" is also favourable to small lists, because in the worst case, if all except 1 quota is absorbed by the lists, then that only leaves 1 seat to fight over with the remainders. If the remainders are evenly distributed amongst the lists, then a candidate could be elected with only 1/L quotas or V/LS votes. For example, in a 8-seat constituency with 12 lists, the minimum vote for a winning candidate could be as low as 1/96 of the vote, or about 1.05%, giving him 12.5% of the seats. This encourages people to split their lists, because the more lists you have, the greater the number of tickets you have in the "remainder lottery". It also encourages lunatics who have a theoretical chance of being elected with a tiny vote but are almost certain to result in wasted votes that might otherwise have gone to the leading parties.

5. So you see that our system favours, in the following order: lunatics, 1-person lists and short lists over long lists. Not only do 1-person lists have an easier target to ensure election, but their surplus votes tend to draw support away from the popular parties, and for the complete outsiders, they have just a chance of being elected (probably for 1 term only, unless they turn out to be sane after all).

Highest average

Within the constraints of non-transferable voting, there is a better way to handle the remainder vote than the current "largest remainder" used in HK. The alternative is the "highest average" and in essence it allocates seats to lists so as to achieve the highest average votes per seat.

There are two ways to do this, and the fairest and most popular method is the Sainte-Lague method, published in 1910 by the French mathematician Andre Sainte-Lague (1882-1950). It is also known as Webster's method. This method results in rounding to the nearest whole seat for each list. The alternative is the d'Hondt method described in 1878 by Belgian mathematician Victor d'Hondt (1841-1901). This rounds down and so generally favours large parties over small ones.

The Sainte-Lague method in effect finds the highest quota which, when divided into the actual votes for each list and rounded to the nearest whole number, allocates all the seats. The d'Hondt method finds the highest quota which, when divided into the actual votes for each list and rounded down to a whole number, allocates all the seats.

Example 1: HK Island

Now for some real-life examples. Let us first consider the 2004 race for HK Island, where anti-democrat Rita Fan was running alone. She was one of 6 lists for 6 seats. As a consequence, regardless of how people voted, she only needed 5/36=13.89% of the vote to be certain of election. When you further consider that, due to their popularity, 3 of the other lists were nearly certain to each fill a Hare quota (absorbing exactly half the votes), then you realise that there were only 3 seats and 50% of the vote left to fight over, and she only needed 12.5% to be sure of election.

In the end, she got 18.54% of the vote, which was 21,399 votes more than she needed. She almost certainly drew her surplus wasted votes from people who would otherwise have voted for the DAB list. Here are the actual votes (pro-democracy lists in blue, anti-democracy in red, unknowns in green):

Look also at the pan-democrat strategy. In HK Island they split their 4 main candidates onto two separate lists, headed by barrister Audrey Eu Yuet Mee and the Democratic Party's Yeung Sum and with a third member on the Democratic Party list. The split lists was a risky gamble which did not pay off, but you can't blame them for trying, because the main strategic goal of the pan-democrats in this election was to give themselves a slight chance of getting 30 or 31 of the 60 votes in LegCo, even if it meant increasing the downside risk. Politically, there isn't much difference between 24 and 26 votes (they won 25) but 30 or 31 would have made a big difference. Other factors, such as the pecking order for 3rd and 4th place on a combined list, also may have kept them apart.

The leading 2 candidates on the 2 lists were near-certain of election, absorbing a Hare quota, or 16.67% each. They also knew that the anti-democrat DAB would fill 1 Hare quota. So the pan-democrats next 2 candidates each needed only 12.5% of the vote to be sure of a win. So if the votes were evenly split on the two lists, they could be certain of victory with 58.33% of the vote, and they also had 2 tickets in the remainder lottery.

By comparison, if they had run on a single list, then to get 4 candidates elected, they would have filled 3 Hare quotas with 50% of the vote, one quota would go to the DAB, and then the pan-democrats would need 11.11% to be certain of the 4th seat, for a total target of 61.11%.

So you can see that because of HK's biased Hare quota, the two-list strategy required a lower percentage of pan-democrat support than a 4-in-1 list, but it did require that the votes be split evenly to have the highest chance of success.

In the end, the twin lists polled a combined 58.07%, but their distribution was so skewed that the remainder system allowed the second DAB candidate to win the 6th seat by 815 votes, or 0.23%. This skew has been attributed to a last-minute switch by voters, egged on by the DP, who saw that opinion polls put one list ahead of the other and chose to switch their votes to the trailing list. So many people did this that it skewed the result in the opposite direction to the opinion poll. In fact, if another 1,887 people had switched to the DP, they would have got a 3rd seat on the remainder, bringing the pan-democrats to 4.

While the media were quick to jump on the result as faulty strategy, it was really the faulty system that was to blame, as it incentivises the splitting of lists and penalises the all-in-1 approach. It was always a long shot to expect 200,000 voters to split their votes evenly on two lists.

But let's see what happens if you use the Droop quota of 50,586, removing the bias in the Hare quota of 59,016 which favours short lists:

As you can see, the DP would have won a 3rd seat on its remainder, bringing the pan-democrat total to 4. Both the Sainte-Lague and d'Hondt methods of highest average would also have given the DP 3 seats, taking the pan-democrats to 4 in total. Similarly, if they had run a combined list and attracted the same votes, they would also have won 4, using either Hare or Droop.

Example 2: NT East

In another example, in New Territories East, there were only 6 lists for 7 seats, which meant that tycoon James Tien and veteran protester Leung Kwok Hung, each running on their own lists, only needed 5/42=11.90% of the vote, or 51,311 votes, to be certain of election. In the end, they got 15.91% and 14.14% of the vote respectively, which was 17,249 and 9,614 votes more than they needed.

Example 3: NT West

Here, there were 8 seats, 12 lists and 30 candidates! Albert Chan Wai Yip, running alone, won 36,278 votes or just 7.83% of the vote, less than the 11.11% Droop quota but enough to win a seat, which of course is 12.5% of the seats.

Closed Lists v Open Lists

Another defect in the list system is that unless the candidate runs on a solo list, you cannot vote for a particular candidate, only for a party. Hong Kong has what is known as a "closed list" voting system. Many jurisdictions which have the party-list system have "open lists" so that with a single tick you can vote for a particular candidate, and the vote also counts for the party list. The votes within the list then determines the order in which candidates on a party list are awarded seats, while still giving the list the same number of votes.

Such a system, if employed in Hong Kong, would remove any arguments about the pecking order when pro-democracy candidates team up and run on the same list, because the voters would determine the order. If you combine that with a change from the Hare quota to the Droop quota, then the system would be fairer to all parties, large and small.

Singe Transferable Vote - the way forward

Although HK's party-list system could be corrected with the use of the Droop quota and an Open List, it would still be structurally defective in the sense that voters cannot express any order of preference between lists or between candidates running on different lists. This means that any vote for an outside list with no real hope of election is a wasted vote (unless they get very lucky in a largest-remainder system). Votes are also wasted when a list of candidates is exhausted by quota.

While maintaining proportional representation, a better way to give representation to electors would be to abandon the party list system and move to a system which allows electors to express an order of preference for candidates. In multi-seat constituencies, this is known as the "Single Transferable Vote" or STV system. An elector places a "1" next to his favourite candidate, a "2" next to his second choice, "3" against the third choice, and so on. The elector does not have to prioritise the entire list - she can stop at any point after indicating her first choice. The elector does not have to stick with a particular party - the candidates can be ranked in any order, regardless of party. It also allows those who wish to lodge a protest vote for a minority candidate to still put a leading party's candidate as their second choice.

When the votes are counted, the first choices are ranked, and any candidate with a Droop quota is elected. His surplus votes, and the votes of the lowest ranked candidate, are then transferred to the second-choice candidates (where the elector has expressed a preference). Anyone with a quota is elected, and the process repeats itself until all the seats are allocated. In single seat constituencies, this type of election is known as the "Alternative Vote" (AV) system. The counting mechanics are fairly complicated and need to be centralised, but they are perfectly manageable and well documented. The use of machine-readable voting papers can accelerate the count while still providing an auditable paper trail.

From the elector's point of view, the voting process is simple. Anyone who ranks all the candidates (or all but one) can then be certain that their vote has counted towards determining the outcome of the election, because their vote will end up either electing their first choice or being transferred to another candidate. It encourages voter participation and tends to lead to higher turn-outs. The STV system is used by, amongst others, Ireland (since independence in 1922),  Malta and the Australian Senate, and is gaining increasing popularity due to the fairness and increased participation and choice that it offers electors. The Scottish Parliament recently adopted STV for local council elections.

It may also surprise you to learn that the AV system (which is the STV system for a 1-seat constituency) is actually used in Hong Kong for 4 of the 30 Functional Constituencies: Heung Yee Kuk, Agriculture and Fisheries, Insurance and Transport. At least, it would be used if these constituencies ever have contested elections, which they don't most of the time as the electorates are so small (ranging from 149 to 182, and mostly corporates many of which have common control) that they just agree amongst themselves rather than put it to a vote. But if anyone tells you that STV wouldn't work in HK, just remind them that we've already got it.

Conclusions and recommendations

We have shown that the Hare quota and largest remainder system of Hong Kong's proportional representation are structurally biased in favour of short lists, 1-person lists and lunatics, while failing to allow electors to vote for an individual candidate. The quota problem could be easily addressed by a legislative change to the Droop quota, and we could easily move to an open-list system, but it would still leave the defects of the largest-remainder, which can result in a lottery for the remaining seats in which a minority candidate with minimal support can be elected if the distribution of remainders is even. If the party-list system is to be retained, then it would be better to forget fixed quotas and move to the Sainte-Lague method of seat allocation, which allocates seats based on largest average votes.

A more forward-looking approach, in line with Hong Kong's ambitions to be a world-class city, would be to abandon party lists and move to a Single Transferable Vote system in which every vote can count, and electors express an order of preference for the candidates. So, in order of preference, our recommendations are:

1. Scrap party lists in favour of a Single Transferable Vote system

2. Failing that, open the party lists and scrap the largest remainder in favour of the Sainte-Lague method of seat allocation

3. Failing that, as a minimum change, open the party lists and change from the Hare quota to Droop quota

Of course, none of this discussion changes the fact that a legislative assembly can hardly be called "proportional" so long as the general public only gets to elect half its members. That is a separate problem, but the correction and enhancement of our proportional representation system can take place in parallel with the push for universal suffrage.

To read more on election systems:

• International Institute for Democracy and Electoral Assistance
• Administration and Cost of Elections Project
• Electoral Reform Society (UK)

© Webb-site.com, 2004

Topics in this story

• Electoral reform

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