What is Proportional Representation?
Since 1998, the geographical constituencies in the Legislative Council Elections in Hong Kong have been elected using ‘proportional representation’ together with the ‘largest remainder method’. How does this system work? Let’s explore it before the coming election.
Proportional representation, proposed in 1846 by the Swiss scholar Victor Considerant, is used in elections with multiple seats. This voting system is presently employed in all elections in Ireland as well as the Senate election in Australia. Parties form lists of candidates; and voters vote for a list rather than for individual candidates. As its name implies, the essence of this method is to allocate seats according to the proportion of votes obtained by the lists^{1}. For instance, if there are 10 seats in a constituency and a total of 100 valid votes, then the quota is the quotient of the number of votes by number of seats, which is 10 in this case^{2}. When the number of votes received by a party list reaches the quota, a seat would be allocated to that party list. Suppose three lists A, B and C join the election and get 50, 30 and 20 votes respectively, then 5 seats would be allocated to A, 3 seats would be allocated to B while 2 seats would be allocated to C.
However, we cannot expect the votes to be so nicely divided all the time. How if, for instance, A, B and C receive 57, 25 and 18 votes respectively?
We could proceed in the following way. Since a list can secure a seat with 10 votes, list A with 57 votes should receive at least 5 seats, and similarly lists B and C should receive at least 2 seats and 1 seat respectively.
The remaining 2 seats would then be allocated using the ‘largest remainder method’ — list A ‘used up’ 50 votes in getting 5 seats, so it has 7 votes left. Similarly B has 5 votes and C has 8 votes remaining. Arranging the number of remaining votes in descending order, we have C (8 votes), A (7 votes) and B (5 votes). The two remaining seats were therefore allocated C and A, which each receives an additional seat.
Therefore, party list A would finally get 6 seats while B and C would each get 2 seats. The first 6 candidates on list A as well as the first 2 candidates on list B and list C would be elected.
This method of allocating seats is known as proportional representation.
Mathematically, the algorithm of proportional representation can be expressed as follows:

Calculate the Quota = .

Divide the number of votes received by a list by the quota. The integral part indicates number of seats allocated to the list in the first round.

Seats unallocated in the first round will be given to party lists with the greatest number of votes remaining (highest remainder). The total number of seats obtained by a list is the sum of the number of seats obtained in the first and the second rounds.
The spirit of proportional representation is to allocate seats in a representative assembly to different parties according to the proportion of votes they obtained in the election. Comparing with the singleseatsinglevote system (this is the system used in the District Council Elections in Hong Kong, in which the whole country or city is divided into constituencies with one seat each, and the candidate with the highest number of votes in each constituency wins), proportional representation can prevent large parties from monopolizing all the seats and hence to ensure that the voice of smaller parties can be heard. On the other hand, this system can also ensure that the candidates who rank top in the lists (e.g. the party leaders) have a pretty good chance to get elected.
When the number of seats is small, proportional representation with the highest remainder method would sometimes cause abnormal seatallocations, as reflected in the following examples.
In the 1998 Legislative Council (LegCo) Elections, there were only 3 seats in the Kowloon East constituency, and yet the system used was proportional representation with the highest remainder method. 3 lists joined the election, namely, lists representing the Democratic Party (DP), the Democratic Alliance for the Betterment of Hong Kong (DAB) and one by the independent candidate Puiyee Fok. The two party lists received similar supports while Fok’s support was much far behind. It appears that Fok was in a very disadvantageous position, but this is not true. Suppose that there are a total of 300000 valid votes, with the DP and the DAB each getting 133000 and Fok getting 34000, the allocation of seats will be as follows.
Quota = 300000 � 3 votes = 100000 votes 

List  # Votes  # seats (first round) 
Votes Remaining  # seats (second round) 
Total Seats 
DP  133000  1 seat  33000  0 seat  1 seat 
DAB  133000  1 seat  33000  0 seat  1 seat 
Puiyee Fok  34000  0 seat  34000  1 seat  1 seat 
Total  300000  2 seats  1 seat  3 seats 
Although each of the two major parties gets almost 4 times as many votes as Fok, each list could still win one seat. This is clearly a big disadvantage of the highest remainder method. In reality, however, Puiyee Fok only received 6349 votes out of a total of 260581, and this prevented the problem from getting wider attention by the public.
Since the implementation of proportional representative in the 1998 LegCo Elections, the number of seats in geographical constituencies have been increasing. As there has been no repartition of constituencies, the seats in each constituency have increased accordingly. For instance, the number of seats in the Hong Kong Island constituency increased from 4 in 1998 to 5 in 2000. By common sense, if the voters’ support for different parties remains the same, an extra seat would mean that an originally unelected candidate could get elected, while those who are originally elected will remain elected.
Nevertheless, dealing with mathematical problems using common sense is a risky act. Let’s take a look at the following example. Suppose there are three party lists A, B and C running for the election and received 100000, 60000 and 20000 votes respectively. Then, in the 1998 election, in which there were only 4 seats, A would get 2 seats while B and C would each get 1 seat.
Quota = 180000 � 4 votes = 45000 votes 

List  # Votes  # seats (first round) 
Votes Remaining  # seats (second round) 
Total Seats 
A  100000  2 seats  10000  0 seat  2 seats 
B  60000  1 seat  15000  0 seat  1 seat 
C  20000  0 seat  20000  1 seat  1 seat 
Total  180000  3 seats  1 seat  4 seats 
In the 2000 election, the number of seats increased to 5. Assuming that the number of votes received by each list remained unchanged, the 5 seats would be allocated as follows:
Quota = 180000 � 5 votes = 36000 votes 

List  # Votes  # seats (first round) 
Votes Remaining  # seats (second round) 
Total Seats 
A  100000  2 seats  28000  1 seat  3 seats 
B  60000  1 seat  24000  1 seat  2 seats 
C  20000  0 seat  20000  0 seat  0 seat 
Total number of votes  180000  3 seats  2 seat  5 seats 
Although the number of votes obtained by each list remained the same, the increase in the number of seats caused list C to lose its only seat! This is clearly unreasonable. Of course, this never happens in reality — it is just impossible for the distribution of votes to remain the same between two elections. Yet this again illustrates a disadvantage of the highest remainder method. Such phenomenon is know as the Alabama Paradox, which is inevitable under such a system.
In the 1998 Elections, Yiuchung Leung, then member of the Frontier, was arranged to run in a list with Emily Lau Waihing in New Territories East, while Cheukyan Lee would run in New Territories West. Leung was dissatisfied with such arrangement, for he had been involved in the local affairs of New Territories West for years. He suggested running with Lee in a list in New Territories West instead. The negotiation ended in failure, and Leung and Lee eventually ran in separate lists in New Territories West. The results were as follows:
Quota = 385220 � 5 votes = 77044 votes 

List  # Votes  # seats (first round) 
Votes Remaining  # seats (second round) 
Total Seats 
DP  147098  1 seat  70054  1 seat  2 seats 
DAB  72587  0 seat  72587  1 seat  1 seat 
Yiuchung Leung  48627  0 seat  48627  1 seat  1 seat 
Cheukyan Lee  46696  0 seat  46696  1 seat  1 seat 
Heung Yee Kuk  25905  0 seat  25905  0 seat  0 seat 
Tinsang Yim  19500  0 seat  19500  0 seat  0 seat 
Others^{3}  24807  0 seat  —  —  — 
Total  385220  1 seat  4 seats  5 seats 
The ‘ruptured partners’ Leung and Lee were both elected. What would happen if they ran in one single list? Assuming that the number of votes obtained by the LeungLee list to be the sum of their individual votes, the result would be as follows:.
Quota = 385220 � 5 votes = 77044 votes 

List  # Votes  # seats (first round) 
Votes Remaining  # seats (second round) 
Total Seats 
DP  147098  1 seat  70054  1 seat  2 seats 
DAB  72587  0 seat  72587  1 seat  1 seat 
LeungLee  95323  1 seat  18279  0 seat  1 seat 
Heung Yee Kuk  25905  0 seat  25905  1 seat  1 seat 
Tinsang Yim  19500  0 seat  19500  0 seat  0 seat 
Others  24807  0 seat  —  —  — 
Total  385220  2 seats  3 seats  5 seats 
The merged list would only get one seat and therefore the second candidate on the list would lose. By contrast, the first candidate on the Heung Yee Kuk list, Waikeung Lam, would have been elected. The ‘rupture’ of alliance has indeed led to a happy ending for both Leung and Lee!
III. To split or not to split?
The LeungLee example does not only illustrate another problem of the largest remainder method, it also gives rise to a new election strategy — splitting lists.
In the New Territories West constituency in the 1998 Elections, 77422 votes were required to secure a seat. Nevertheless, after the first round allocations, candidates might be able to get a seat with fewer remaining votes. If Yiuchung Leung and Cheukyan Lee merged into one list, Waikeung Lam from Heung Yee Kuk could even win a seat with only 25905 votes. Therefore, large parties could use the listsplitting method to make more party members elected with a smaller number of remaining votes. In fact, this strategy was adopted by both the DP and the DAB in some constituencies in the 2000 and 2004 Elections.
On the other hand, there are usually nonmathematical reasons behind splitting lists. One of the major advantages is that the secondtier members in a party could increase public exposure and this helps pave the way for the party’s next generation. For instance, there are 8 seats in the New Territories West constituency in the 2004 Elections. If a party runs in one list, at most 8 people could appear as candidates. If a party runs in two lists, then this number would become 16. (Incidentally, even if there are 8 candidates in a list, those ranked in the bottom are almost sure to lose. Why do they still run in the election? In addition to increasing public exposure, another probable reason might be to make use of the ignorance of the voters. Some may vote for the list merely because they support the candidate ranked last on the list, without realising that by doing so they are actually supporting those ranking top in the list.)
Another advantage of splitting lists is that various resources available for candidates would increase accordingly, for such resources are usually allocated on a list basis. For instance, each list could spend a maximum of $5.5 million, each list could enjoy free postal services twice, each list has 3 minutes to speak in an election forum, etc.
It should be noted, however, that splitting lists is not always good. Being too greedy may do more harm than good. Let’s look at the New Territories West constituency in the 1998 Elections again. As we have seen, if Leung and Lee ran in one single list, then the Heung Yee Kuk list could win a seat. However, if the candidates from Heung Yee Kuk split into two lists, and assuming that each list got half of the original votes, then they would no longer be able to win:
Quota = 385220 � 5 votes = 77044 votes 

List  # Votes  # seats (first round) 
Votes Remaining  # seats (second round) 
Total Seats 
DP  147098  1 seat  70054  1 seat  2 seats 
DAB  72587  0 seat  72587  1 seat  1 seat 
LeungLee  95323  1 seat  18279  0 seat  1 seat 
Heung Yee Kuk I  12953  0 seat  12953  0 seat  0 seat 
Heung Yee Kuk II  12952  0 seat  12952  0 seat  0 seat 
Tinsang Yim  19500  0 seat  19500  1 seat  1 seat 
Others  24807  0 seat  —  —  — 
Total  385220  2 seats  3 seats  5 seats 
Moreover, if a party runs in multiple lists, it is difficult to control which list its supports will vote for. If the distribution of votes deviates from what it desires, the party may even lose seats, not to mention winning more. (Readers could try to construct such examples.) Consequently, some parties may give ‘votesharing instructions’ to its supporters in order to let them know which list to vote for. Some common ways of votesharing include:

By district: A party may divide a constituency into several parts and instruct supporters in individual parts to vote for a particular list.

By gender: The male supporters may be instructed to vote for a list and female for another list.

By birthday: A party may instruct supporters born in different months to vote for different lists.
To conclude, under the proportional representation system, political parties need not only obtain wide support, but must also adopt appropriate strategies. As we have seen, even if the support for different parties remains the same, different combinations and splitting of lists would lead to different outcomes. Therefore, this is not only a political game; it is also a mathematical game.
[1] Largest Remainder Method, http://encyclopedia.thefreedictionary.com/largest%20remainder%20method
[2] Seats and Quotas, http://accuratedemocracy.com/e_shares.htm
[3] Research Paper from HKSAR Provisional Legislative Council Secretariat, http://www.legco.gov.hk/yr9798/english/sec/library/in1_plc.pdf