How List Proportional Representation by Largest Remainder Fails

First published: 11th September 2016

I'm a strong believer in democracy - that people should have a say in how they are ruled, but the devil is in the details. How do you get from millions of individual voices to a functioning legislature, and what unfairness results from the choices made along the way? All methods are imperfect. The simplest method, First Past the Post (FPP) favours a small number of larger parties and a party can take control of the Government with less than half the popular vote, as in the UK 2005 election, where the winning party had only 35% of the vote. Here, I want to discuss what happened with the List Proportional Representation system in Hong Kong. One message is that "the rules matter", and this is a follow-up to Elections in Hong Kong: Making a Simple Decision Complicated where I discussed the Election Committee elections, but, unlike that, this discusion has wider implications. The Election Committee elections are a peculiar Hong Kong system, but the Legislative Council elections use the List Proportional Representation system that is used elsewhere.

The idea of a List Proportional Representation system is that political parties each propose a list of candidates, and voters cast their votes for the list of their choice. When the votes are counted, parties get seats in proportion to their votes. More specifically, how this is implemented in Hong Kong is defined in the Legislative Council Ordinance, Chapter 542 Section 49. This defines a Quota, Q = V / N, rounded down, where V is the total number of votes cast and N is the number of available seats. The number of votes for a list is divided by the Quota, and rounded down to give the number of seats for that list. Any remaining seats are allocated according to the largest remainder. This sounds good and fair.

Looking at the results of the last three Legislative Council Elections, 2008, 2012 and 2016 for Hong Kong Island, in practice the system makes the lists meaningless as parties are pressured into trying to game the system. In those elections, only four lists got over the Quota of votes. Parties that propose multiple lists and efficiently divide their support between those lists are rewarded by more seats. Even in the most recent election, where no list achieved the Quota, some lists had the maximum number of candidates, even though there was zero chance of anyone except the first on the list getting a seat.

Now, I'd like to introduce two measures of how to evaluate the "fairness" of an election. The first is the percentage of voters who are unrepresented, defined as the number of votes cast for lists or candidates that got no seats. In the UK 2005 election, under the FPP system, this was incredibly high at 65%. The second is how much power one vote has in the resulting parliament, measured as a fraction of a Legislator. If a Legislator is elected with, say, 100 votes then each of those voters "owns" one-hundreth of the Legislator. So the Legislator Power of a vote, symbol L, is defined as the number of seats won by a list divided by the total number of votes cast for that list. More usefully, I will use the micro-Legislator, symbol µL, which is one-millionth of L.

Here's the details. The results have been ordered by descending votes, and the parties and candidates are not named because they are irrelevant to this discussion.

2008 Legislative Council Election, Hong Kong Island

From 2008 Legislative Council Election Results: Hong Kong Island


There were six seats available for Hong Kong Island, so the quota was 52 thousand. Only 12% of the voters ended up having no representative. Three lists exceeded the quota, and no list got over twice the quota. The top-ranked list, list 8, did get a second seat because their remainder of 30362 was higher than the top looser's total votes, list 7 with 20523. There was unfairness, a vote for list 9 was only worth 16μL, but a vote for list 2 was worth 32μL, twice as much. A responsible party leader of list 9 could say, "we should split into two lists, if we split evenly we can have 30,500 votes per list and get one seat per list, doubling our seats". If that had happened, then list 8 would have lost it's second seat, but the leader of that party can follow the same logic, and advise splitting into two lists, each with about 41,000 votes, to keep the second seat. The games begin.

But would this happen in practise? Let's look at the next election.

2012 Legislative Council Election, Hong Kong Island

From 2012 Legislative Council Election Results: Hong Kong Island

2 40558104055825
5 33901103390129
7 31523103152332
8 30289103028933
9 27336102733637
4 1866700186670
3 1690000169000
1 29800029800
6 4220 04220

In 2012, there were 7 available seats and the quota was 47 thousand. There were 14 lists, and only one list achieved the quota. There is more unfairness: the unrepresented has grown to 18%, and the value of successful votes varied more: a vote for list 10 was worth 14μL and a vote for list 9 was worth 37μL, over 2.5 times as much.

Was this a temporary aberration or a trend? Let's look at the most recent election.

2016 Legislative Council Election, Hong Kong Island

From 2016 Legislative Council Election Results: Hong Kong Island

1167000670 0

In 2016, we are back to 6 seats and the quota was almost 63 thousand. The number of lists has increased to 15, and one measure of unfairness has risen again: the unrepresented has grown to 27%. This actually means that the number of "wasted" votes was almost double the votes for the top-ranking list. A vote for list 3 was worth 16μL and a vote for list 14 was worth 28μL, the difference there was less.

There are some other important differences in the 2016 election. There was a new force in Hong Kong politics, the Localists, who are mostly youth leaders from the 2014 Umbrella Movement. They can be seen as largely aligned with the pan-democrats, but less likely to appease Beijing. Also, a few days before the election, the candidate for list 8 announced that he was stopping campaigning and his supporters should vote for other democratically-inclined candidates. This illustrates how these rules encourage gaming and give power to (perhaps inaccurate) opinion polls.

Why is List Proportional Representation by Largest Remainder Broken?

These three elections illustrate how the system ceases to be a measure of the people's preferences for representatives, and becomes a measure of the skill of the campaign teams in manipulating the voters to game the system.

  1. The system devolves from "list" to "only the first name matters". As the number of lists proliferate, only the first on each list has a chance of a seat. The rest are there either to "show their support" for the real candidate, or to fool voters into thinking more will be elected.
  2. More lists means that more votes are wasted on unelected candidates. Sure, it isn't the extreme of FPP, but it does tend to grow.
  3. Even successful votes have wildly differing powers.

Good rules encourage playing by the spirit of the game. Here, the spirit of the game is to elect Legislators that match the opinion of the voters. The rules reward the teams with better predictive models and better control of their voters instead.

Can the Rules be Repaired?

They can certainly be improved. How depends on how you want to group people's opinions. One extreme is "everyone's opinion is unique", so, just get every voter in to one meeting and call that the Legislative Council, which is generally considered infeasible. The other extreme is "there's only two sides to the question", FPP systems tend towards this, with just two main parties opposing each other.

Here I make two suggestions for improvement that differ in the grouping:

Single Transferable Vote (STV)

This is well-known, and it emphasises the uniqueness of the candidates. Every candidate can campaign on their own merits, and voters can express their preferences for different candidates secure in the knowledge that, even if their first pick is not elected, their vote is not wasted because their second and subsequent preferences are still considered. With multiple seats, the fractional excess votes from a top candidate will be transferred to other preferences, so there is also no danger of diluting the power of your vote by voting for a popular candidate.

The major problem with STV is the complexity of the counting. You could speed things up by using information technology, but, as an information security professional, I strongly advise against this. The beauty of a manual count is the transparency and repeatability. Ordinary people can watch it happening, and see that it is being done correctly. If there is a problem, a recount can be called. I won't say that computerised systems could not be designed to do this, but it would be difficult and expensive, and current systems have obvious, and hidden, flaws.

Sliding Quota

This method improves on the List Proportional Representation by Largest Remainder to give a result that is closer to the spirit of List PR. The "largest remainder" idea is dropped, instead, the Quota is initially calculated the same way but then, if the number of elected candidates is less than the number of seats, the Quota is reduced by 1 vote. This is repeated until the number of candidates making the Quota is equal to the number of seats.

The method rewards candidates who follow the spirit of List PR, because they can group into like-minded lists without the danger of opponents splitting their lists getting undeserved numbers of seats.


Why did the number of seats for Hong Kong Island change? What are the major factions in Hong Kong politics? What happened in the other Hong Kong Geographical Constituencies? What about the Functional Constituencies? I've left out so much, not because it is unimportant, but because it is not relevant to the discussion.

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