MathDB – Mathematical Articles

More on Proportional Representation —
Implications of the Results of the Legislative Council Election 2004

1. Introduction

In the article “What is Proportional Representation?”, we mentioned that proportional representation with the largest remainder method was used in the election of geographical constituencies in the Legislative Council (LegCo) in Hong Kong since 1998. The shortcomings of the method were discussed and supported by examples. Further analysis of the proportional representation will be made in this article based on the results of the geographical constituencies in the LegCo Election 2004.

2. Results of the LegCo Election 2004

1. Hong Kong Island Constituency

Number of seats: 6; number of valid votes: 354095

Securing a seat requires 354095 ÷ 6 = 59016 votes.

Candidate list # votes # seats
(first round)
# seats
(second round)
Total # seats
DAB 74659 1 seat 15643 1 seat 2 seats
Rita Fan Hsu Lai-tai
65661 1 seat 6645 0 seat 1 seat
Tsang Kin-shing et al.
5313 0 seat 5313 0 seat 0 seat
Democratic Party 131788 2 seats 13756 0 seat 2 seats
Wong Kam-fai
2830 0 seat 2830 0 seat 0 seat
Audrey Eu Yuet-mee  and Cyd Ho Sau Lan



73844 1 seat 14828 0 seat 1 seat
TOTAL 354095 5 seats 1 seat 6 seat

*DAB = Democratic Alliance for the Betterment of Hong Kong (民建聯)

In the article “What is Proportional Representation?”, we showed that splitting the candidate list could increase the number of seats a party could get with the risk of “vote sharing”. What happened in the Hong Kong Island Constituency was a typical failure of vote-sharing. The democratic camp aimed at 4 seats, and split into two lists: the list of Democratic Party and the list led by Audrey Eu Yuet-Mee (余若薇). With the slogan “1 + 1 = 4”, they urged the voters to “vote in pairs”, that is to form groups of two  with one vote for Democratic Party and the other for Eu.

Contrary to what Democratic Party wished, pre-election polls showed that Eu’s list took a great lead ahead the Democratic Party list, on which Martin Lee Chu-ming (李柱銘) was the second candidate and therefore faced the prospect of losing in the election. Consequently Democratic Party urged voters to vote solely for the Democratic Party list so as to ensure Lee’s victory. Some newspaper even went to the extent of having the headline “Lee in emergency”.

Unfortunately, this striking headline became the bane of Democratic Party. Many Democratic Party supporters cast their votes to the Democratic Party list instead of Eu’s as a result of the exaggerating expression of Lee’s possible loss, and the outcome took the other side of the equilibrium. The Democratic Party list gained 13756 votes more than what it needed for securing two seats. While these votes were not able to gain a third seat for the Democratic Party list, Eu’s list had only 14828 votes remaining after the first seat was taken, less than that of DAB which had 15643 remainder votes. As a result, this difference of 815 votes led to the loss of Cyd Ho Sau-lan (何秀蘭), the second candidate on Eu’s list, while Choy So-yuk, the second candidate on DAB’s list, secured her seat in the Council. Obviously, if the “emergency” news went to an even further extent, Democratic Party could have got a third seat; and if the tactic had been used more moderately, each of DP’s and Eu’s list could have got 2 seats as desired. In either case, Choy would have lost in the election. It was indeed quite a misfortune for the situation to end up between the two scenarios.

In fact, the “emergency effect” had its precedent in the LegCo election 2000. In the election Democratic Party split into three lists in the New Territories East Constituency, led by Lee Wing-tat (李永達), Ho Chun-yan (何俊仁) and Albert Chan Wai-yip (陳偉業) respectively. Pre-election polls indicated that Lee took the lead, followed by Ho and Chan was last. So Chan used the same “emergency” tactic, causing many supporters of Lee to vote for Chan instead, and eventually Lee lost his seat rather unexpectedly. From these incidences we can see that there is a certain risk in splitting list — if the vote-sharing fails, the results can be startling.

2. Kowloon East Constituency

Number of seats: 5; number of valid votes: 293702

Securing a seat requires 293702 ÷ 5 = 58741 votes.

Candidate list

# votes # seats
(first round)
# seats
(second round)
Total # seats

Democratic Party

56409 0 seat 56049 1 seat 1 seat


55188 0 seat 55188 1 seat 1 seat

Albert Cheng Jinghan et al.

73424 1 seat 14683 0 seat 1 seat

Alan Leong Kah-kit (梁家傑)

56161 0 seat 56161 1 seat 1 seat


52520 0 seat 52520 1 seat 1 seat


293702 1 seat 4 seats 5 seats

*FTU = The Hong Kong Federation of Trade Unions (工聯會)

In Kowloon East, it was commonly predicted that democratic camp could get 3 seats, while the pro-government camp could get the remaining two. Nonetheless, the situation was not as clear as it should be due to the competition within the camps.

In the pro-government camp, Chan Yuen-han (陳婉嫻) and Chan Kam-lam (陳鑑林) from DAB were on two separate lists in the election. High support for Chan Yuen-han according to pre-election polls favoured the entrance of the second candidate on her list, Lam Man-fai , into the Council and therefore reduced the chance for Chan Kam-lam.

Meanwhile, the democratic camp was eyeing for the fourth seat. Yet no agreement could be made on whether the fourth seat should be targeted by Democratic Party list or that of Albert Cheng Jinghan. In fact, neither side wanted to give in, and thus could not unite to strive for the extra seat.

The election results showed certain “emergency effect” within the pro-government camp. ChanYuen-han, though secured a seat, was strikingly beaten by Chan Kam-lam. Contrary to the failure in Hong Kong constituency, this “emergency effect” led to an optimal result for the pro-government camp. Competition within the democratic camp, on the other hand, did not bring any surpise.

3. Kowloon West Constituency

Number of seats: 4; number of valid votes: 227694
Securing a seat requires 227694 ÷ 4 = 56924 votes.

Candidate list

# votes # seats
(first round)
Votes remained # seats
(second round)
Total # seats

Lau Yuk-shing et al.

1824 0 seat 1824 0 seat 0 seat

Frederick Fung Kin-kee (馮檢基)

46649 0 seat 46649 1 seat 1 seat

Lau Chin-shek

43460 0 seat 43460 1 seat 1 seat


61770 1 seat 4846 0 seat 1 seat

Democratic Party

60539 1 seat 3615 0 seat 1 seat

Liu Sing-lee (廖成利)

13452 0 seat 13452 0 seat 0 seat


227694 2 seats 2 seats 4 seats

It was widely expected that the the results of LegCo election in 2000 would repeat — 3 seats for the democratic camp and 1 seat for DAB. In the 1998 and 2000 elections, DAB got 21.7% and 23.5% of the total votes respectively. Anticipating the prospect of DAB’s lapse below 20% in vote share, the democratic camp aimed at taking all four seats in the Kowloon West constituency.

Mathematically, if DAB got lower than 20% of votes and the democrats got more than 80%, then by splitting into 4 lists the democrats could get all four seats provided that each list got no less than 20% of total votes. But how could the democrats ensure the votes be distributed uniformly among the four lists, even if they managed to get over 80% of the votes?

In the Kowloon West election in 2000, the elected candidates from the democratic camp included Frederick Fung Kin-kee, James To Kun-sun (涂謹申) and Lau Chin-shek. Originally,  the democratic camp planned to have Cyd Ho Sau-lan to ‘land on’ Kowloon West, but the proposal was strongly opposed by Frederick Fung, allegedly due to the possibility that he would lose in the election.

Cyd Ho eventually did not ‘land’, but there was then no consensus among the democratic camp on the actual strategy towards taking all four seats. During the nomination period, Frederick Fung, who originally strongly opposed the four-list-strategy, suddenly declared that the Hong Kong Association for Democracy and People’s Livelihood (ADPL, 民協) to which he belonged would join the election in two separate lists, led by himself and Liu Sing-lee respectively, in the aim of a democratic grand slam. This act of reversal was immediately criticised by James To from the Democratic Party, who said that no conclusion had yet been reached among the democrats and that he had been looking for a partner of higher popularity such that he could gain two seats with his list.

Mathematically, To’s suggestion was extremely unwise, as its success required at least 81.3% of votes for the democrats (against the 80% in case the democrats had 4 separate lists) to be distributed in a ratio very close to 7:3:3 in order to gain all 4 seats (the reader may try to show this). Nonetheless, taking into consideration other political factors, this method had its own strength. In fact, given the very low support for Liu, it was essentially impossible for the 80% of votes to be shared equally amongst the 4 lists. On the other hand, it was true that To’s high support from the public together with a suitable partner could guarantee two seats. Unfortunately, rivalry among the pan-democrats themselves (to the extent that they could not even manage to give the electorates a guideline on how they should vote to give the desired vote distribution as what was done in Hong Kong constituency) destined the four-list-strategy to fail.

At the initial stage of the election campaign, To had been taking great lead among the other pan-democrat candidates, and ADPL could possibly end up with nothing. On day, the ADPL called a press conference, allegedly to declare full support solely on Fung’s list and give up Li’s. Ironically, several hours before the conference, the news that To was involved in a scandal spread through the city. Along with the immediate lapse of To’s popularity, ADPL’s press conference was canceled and the ‘giving up Liu for Fung’ tactic was never mentioned again. As for the supporters for the democrats in Kowloon West, they eventually received no guideline on how the votes should be shared.

To’s scandal seriously impaired the democrats’ popularity in the election, as could be seen from the surprisingly high vote share of 27.1% for the DAB.  Nonetheless, the election result was not at all surprising, with the four incumbents successfully defending their seats.

4. New Territories East Constituency

Number of seats: 7; number of valid votes: 431007
Securing a seat requires 431007 ÷ 7 = 61573 votes

Candidate list

# votes # seats
(first round)
# seats
(second round)
Total # seats

Leung Kwok-hung

60925 0 seat 60925 1 seat 1 seat

Tso Wung-wai

14174 0 seat 14174 0 seat 0 seat

7.1 United Force

168833 2 seats 45687 1 seat 3 seats

Liberal Party

68560 1 seat 6987 0 seat 1 seat

Andrew Wong Wang-fat (黃宏發)

23081 0 seat 23081 0 seat 0 seat


95434 1 seat 33861 1 seat 2 seats


431007 4 seats 3 seats 7 seats

New Territories East was perhaps the most exciting — and queer — constituency of all in the 2004 elections. To start with, the Liberal Party sent only one candidate, James Tien Pei-chun, who managed to get 15.9% (over one-seventh) of the total votes. It certainly was not impossible for them to gain a second seat should there be one more candidate was on the list. So why was it not done? Probably it was because LP could not find another  popular candidate, and with an unsuitable candidate Tien might end up with fewer votes at the end. It was said that Tien actually invited Allen Lee Peng-fei (李鵬飛), the former Chairman of the party, to join the list, but was denied as Lee “did not want to risk winning”. Without Lee, the only remaining choice for the Liberal Party would be Lau Hing-kei (劉慶基) who is nicknamed ‘Cha Siu Ping (叉燒炳)’ (i.e. the TV drama actor Lau Tan(劉丹)). Perhaps Tien thought that Lau was not popular enough and could become a ‘negative asset’ for him, so he decided to go on his own. The same one-candidate scenario might also have happened to Andrew Wong Wang-fat who got 14.6% of total votes in 2000 — but with the surprising fact that he lost the election this time.

But the most peculiar of all should be the single “diamond list” of the democrats. Under the largest remainder method, the effectiveness of such strategy is really doubtful since every single candidate in the in the initial segment had to ‘exhaust’ one-seventh of the total votes to become elected, and could not get a seat using a smaller ‘remainder’.

The results actually demonstrated the complete failure of the “diamond list” strategy. The 7.1 United Force eventually got only 3 seats, far from the initial aim of “securing 4 and striving for 5”. Apart from the above mathematical reason, the most primary reason of its failure can easily be seen from the votes gained by Leung Kwok-hung Leung. With only 5.4% votes in 2000, his support jumped to 14.1% this time. Why? The rationale behind this is simple: in 2000 the pan-democrats split into 3 lists (2 for Democratic Party and 1 for the Frontier (前線)), and when supporters ‘shared votes’ on their own, most gave one vote to Democratic Party and one to the Frontier. But this time, with the democrats in one single list, many voters who did not know the underlying mathematics well made the vital vote of casting one vote for the 7.1 United Force and one for Leung Kwok-hung. As a matter of fact, if some 10000 votes of Leung Kwok-hung could be ‘transferred’ to Andrew Wong Wang-fat, Wong would displace Lee Kwok-ying (李國英), the second candidate on DAB’s list, to get elected. It was possible that many votes turned from Wong to Leung in anticipation that Wong would surely get elected to prevent ‘wastage’ as there was no second candidate on Wong’s list. Furthermore, Tso Wung-wai, also a university professor, probably attracted some of Wong’s votes, which contributed to Lee’s victory.

It was also generally believed another non-mathematical reason for the failure of the 7.1 United Force was that the seven candidates, from varying sectors of the democratic camp, differed in political views and platforms. Some voters might dislike a particular candidate on the list and decided to abandon the list altogether. This provides justification for Tien and Wong to go on their own.

5. New Territories West Constituency

Number of  seats: 8; number of valid votes: 463413

Securing a seat requires 463413 ÷ 8 = 57927 votes

Candidate list

# votes # seats
(first round)
Votes remained # seats
(second round)
Total # seats

Albert Chan Wai-yip

36278 0 seat 36278 1 seat 1 seat

Democratic Party (1)

62500 1 seat 4573 0 seat 1 seat

Democratic Party (2)

62342 1 seat 4415 0 seat 1 seat

Leung Yiu-chung et al.

59033 1 seat 1106 0 seat 1 seat

Chow Ping-tim

1725 0 seat 1725 0 seat 0 seat

Stephen Char Shik-ngor (查錫我)

9116 0 seat 9116 0 seat 0 seat


115256 1 seat 57329 1 seat 2 seat

Ng Tak-leung

1920 0 seat 1920 0 seat 0 seat


50437 0 seat 50437 1 seat 1 seat

New Century Forum

4511 0 seat 4511 0 seat 0 seat

Lee Cheuk-yan et al.

45725 0 seat 45725 1 seat 1 seat


14570 0 seat 14570 0 seat 0 seat


463413 4 seats 4 seats 8 seats

The contest at New Territories West was relatively less exciting. With the highest number of seats, the constituency attracted a total of 12 lists, some probably in the hope of winning one seat with a small ‘remainder’. With the lists were so scattered, people were concerned whether an incident similar to Lee Wing-tat’s loss in the 2000 election would repeat.

The election results turned out with no surprise. Despite the lack of a vote-sharing strategy among the pan-democrats, they managed to get 5 seats , one from each of the lists led by Lee Cheuk-yan, Leung Yiu-chung, Ho Chun-yan (何俊仁), Albert Chan Wai-yip and Lee Wing-tat . The remaining three seats were shared by DAB (2 seats) and the Liberal Party (1 seat).

What’s worth noting is that the Democratic Party had 2 lists in this constituency and divided the district into 2 parts, letting each list run their campaign in their respective “area”. At the end the votes obtained by Lee Wing-tat’s and Ho Chun-yan’s lists differed by only 158, which showed that the Democratic Party had done an excellent job in the vote-sharing work.

3. Conclusion

Under a perfect electoral system, election results should not be altered by strategic factors such as splitting lists or vote-sharing. In precise mathematical terms, this is to say that once each voter has a certain ordered preference, a perfect electoral system should produce a unique result. Obviously proportional representation with the largest remainder method could not achieve this aim. Moreover, while the essence of proportional representation is that “number of votes gained is proportional to number of seats taken”, this is practically impossible when the total number of seats is small. In general, such a system could be efficient only if there are more than 10 seats. However, in the 2004 election, the even largest constituency New Territories West had only 8 seats, and therefore the election result was far from the ideal case “number of votes gained is proportional to number of seats taken”.

Finally, we should realise that better electoral systems do exist — one example is ‘single transferable vote’, which is basically an improved version of proportional representation. Nonetheless, the voting method is very complicated — voters have to provide their ordered preference for all candidates (they could also choose not to rank candidates they dislike), after which a rather complicated mechanism would decide which candidates win. Put simply, if there are n seats in a constituency, then any candidate who gets more than 1/n of the total votes wins, and the excess votes would be ‘transferred’ to those candidates with lower preference. At the same time, the candidate with smallest number of votes would be knocked off, and his votes would be also be ‘transferred’ to those candidates with lower preference. These two processes iterate until all the seats are assigned.

The 2008 LegCo election is soon to come. Let’s see whether insufficiencies of the current electoral system would lead history to repeat.

4. Reference

Official website of Legislative Council Election 2004:

Hong Kong Legislative Election 2004, from Wikipedia:,_2004