How Proportional Representation Elections Work

(Douglas J. Amy’s article direct from Mount Holyoke College’s PR Library in April 2005.)

We in the United States are very used to our single-member district, winner-take-all style of elections. We've all grown up with a system where we elect members of our legislatures one at a time in small districts, with the winner being the candidate with the most votes.

This system seems so "natural" that proportional representation (PR) elections may at first appear a bit strange to us. Adding to the potential confusion is the fact that there are several different kinds of PR systems in use around the world.

But in reality, the principles underlying proportional representation systems are very straightforward and all of the systems are easy to use.

The Basic Principles of PR

The basic principles underlying proportional representation elections are that all voters deserve representation and that all political groups in society deserve to be represented in our legislatures in proportion to their strength in the electorate. In other words, everyone should have the right to fair representation.

In order to achieve this fair representation, all PR systems have certain basic characteristics -- characteristics that set them apart from our current election system.

First, they all use multi-member districts. Instead of electing one person in each district, as we do here in the U.S., several people are elected. These multi-member districts may be relatively small, with only three or four members, or they may be larger, with ten or more members.

(The figures below illustrate districting maps for a hypothetical 50-person state senate.  Figure 1 shows 50 single-seat districts, as is common with plurality-majority systems. Figure 2 depicts 10 five-seat PR districts, and Figure 3 shows 5 ten-seat PR districts.)

            Fig-1.                                   Fig-2.                                         Fig-3.             

The second characteristic of all PR systems is that they divide up the seats in these multi-member districts according to the proportion of votes received by the various parties or groups running candidates.

Thus if the candidates of a party win 40% of the vote in a 10 member district, they receive four of the ten seats -- or 40% of the seats. If another party wins 20% of the vote, they get two seats, and so on.

That, in a nutshell, is how proportional representation works. But while all PR systems have the same goals of ensuring that all voters receive some representation and that all groups are represented fairly, various systems do have different ways of achieving these goals. So it is helpful to see how different kinds of PR systems work in practice.

Types of PR Systems

1) Party List Voting.

2) Mixed-Member Proportional Voting.

3) Single Transferable Vote or Choice Voting.

1) Party List Voting

Party list voting systems are by far the most common form of proportional representation. Over 80% of the PR systems used worldwide are some form of party list voting. It remains the system used in most European democracies and in many newly democratized countries, including South Africa.

How It Works. Legislators are elected in large, multi-member districts. Each party puts up a list or slate of candidates equal to the number of seats in the district. Independent candidates may also run, and they are listed separately on the ballot as if they were their own party (see below).

On the ballot, voters indicate their preference for a particular party and the parties then receive seats in proportion to their share of the vote. So in a five-member district, if the Democrats win 40% of the vote, they would win two of the five seats. The two winning Democratic candidates would be chosen according to their position on the list.

There are two broad types of list systems: closed list and open list.

In a closed list system--the original form of party list voting--the party fixes the order in which the candidates are listed and elected, and the voter simply casts a vote for the party as a whole.

This is shown in the first ballot below, which illustrates an election for the House of Representatives in a five-seat district. Voters are not able to indicate their preference for any candidates on the list, but must accept the list in the order presented by the party.

Winning candidates are selected in the exact order they appear on the original list. So in the example here, if the Democrats won two seats, the first two candidates on the pre-ordered list--Foster and Rosen-Amy--would be elected.

Closed Party List Ballot.

Most European democracies now use the open list form of party list voting. This approach allows voters to express a preference for particular candidates, not just parties. It is designed to give voters some say over the order of the list and thus which candidates get elected.

One version of this is illustrated in the ballot below. Voters are presented with unordered or random lists of candidates chosen in party primaries. Voters cannot vote for a party directly, but must cast a vote for an individual candidate. This vote counts for the specific candidate as well as for the party. So the order of the final list completely depends on the number of votes won by each candidate on the list.

The most popular candidates rise to the top of the list and have a better chance of being elected. In our example, if the Democrats won 2 seats, and Volz and Gentzler received the highest and next highest number of individual votes, they would rise to the top of the list and be elected. This example is similar to the system used in Finland and widely considered to be the most open version of list voting.

Open Party List Ballot.

A variety of different formulas exist for accomplishing the actual allocation of seats to the parties. One of the simplest seat allocation formulas is the called the "largest remainder formula."

In this approach, the first step is to calculate a quota, which is determined by taking the total number of valid votes in the district and dividing this by the number of seats. In the example in the table below, 100,000 votes were cast and ten seats are to be filled. 100,000/10 = 10,000 which is the quota.

The quota is then divided into the vote that each party receives and the party wins one seat for each whole number produced. So the Republican party received 38,000 votes, which is divided by 10,000 to produce three seats with a remainder of 8,000.

After this first allocation of seats is complete than the remainder numbers for the parties are compared and the parties with the largest remainders are allocated the remaining seats. In our example, two seats remain to be allocated and the Republicans and Moll, the independent candidate, have the largest remainders, so they get the seats. Ultimately all the parties end up with the number of seats that as closely as possible approximates their percentage of the vote. 

Largest Remainder Approach to Seat Allocation.

2) Mixed-Member Proportional Voting

Mixed-member proportional representation goes by a variety of other names, including "the additional member system," "compensatory PR," the "two vote system," and "the German system."

It is an attempt to combine a single-member district system with a proportional voting system. Half of the members of the legislature are elected in single-member district plurality contests. The other half are elected by a party list vote and added on to the district members so that each party has its appropriate share of seats in the legislature.

Proponents claim that mixed-member proportional voting (MMP) is the best of both worlds: providing the geographical representation and close constituency ties of single-member plurality voting along with the fairness and diversity of representation that comes with PR voting.

This system was originally invented in West Germany right after World War Two, though since then it has also been adopted in several other countries, including Bolivia and Venezuela.

It is still one of the least used PR systems, but in recent years it has begun to garner a great deal of attention. In fact, it is now one of the "hottest" systems being considered by those involved in electoral design. In part this growing attention is a result of MMP’s unique claim to be a "compromise" between the two main rival systems.

In the 1990s New Zealand abandoned its traditional single-member plurality system for MMP. Hungary also adopted this approach. Most recently, the newly formed parliaments of Scotland and Wales used this system for their first elections.

How It Works. People cast votes on a double ballot--see the ballot below. First, on the left part of the ballot, they vote for a district representative. This part of the ballot is a single-member district plurality contest to see which person will represent the district in the legislature. The person with the most votes wins. Typically half of the seats in the legislature are filled in this way. So in a hypothetical 100-member state legislature, the winners of these district contests would occupy 50 of the seats.

On the right part of the ballot--the party list portion--voters indicate their choice among the parties, and the other half of the seats in the legislature are filled from regional lists of candidates chosen by these parties. The party lists are closed in the German version.

These party list votes are counted on a national basis to determine the total portion of the 100-seat legislature that each party deserves. Candidates from each party's lists are then added to its district winners until that party achieves its appropriate share of seats.

The following table illustrates how this process works for our hypothetical election. The Democrats won 40% of the party list votes in the 100-member state legislature, so they would be entitled to a total of 40 of the 100 seats. Since they already elected 28 of their candidates in district elections, they would then add 12 more from their regional party lists to come up to their quota of 40 seats.

Allocation of Seats in MMP.

In the German version two electoral thresholds are used, either of which a party must overcome to be allotted seats in the legislature. A party must either get 5% of the nationwide party list vote or win at least three district races in order for it to gain any seats in the legislature.

In our hypothetical case, the New Party did not win any district seats, but they did win over 5% of the nationwide vote, so they deserve their share of legislative seats--which in this case would be six seats, all of which would be filled from the regional party lists.

3) Single Transferable Vote or Choice Voting

This system of proportional representation is known by several names. Political scientists call it "the single transferable vote." It is called the "Hare-Clark system" in Australia. In the United States, electoral reform activists have taken to calling it "choice voting." Currently this system is used to elect parliaments in Ireland and Malta.

In Australia it is used to elect the federal Senate, as well as the legislatures in several states there. It is also the PR system that was used in a number of cities in the United States during the twentieth century, including New York, Cincinnati, Cleveland, Toledo, and Boulder. It continues to be used today in Cambridge, Massachusetts for elections to their city council and school board.

Senator Penny Wong and her partner Sophie.
(Wong was a minister in Gillard's Government.)

(Blogger’s notes: One of the most senior Socialist politicians from Australian Labor Party Ms. Penny Wong -- A Malaysia-born Chinese-Australian openly-lesbian woman and a Senator -- was said to have stated that she would never possibly be elected into Australia's Lower House the House of Representative as it uses traditional Winner-Takes-All system in all single-district electorates which normally have very conservative white-Australians as majority. Instead she always contested only for the Senate which uses state-wide Proportional Representation system and always got elected again and again.)

How It Works. The voting process is illustrated by ballot below. All candidates are listed in the same place on the ballot. Instead of voting for one person, voters rank each candidate in their order of choice. So if you like Campbell best, you would mark the "1" after his name. If you liked Gomez second best, you would mark "2" by his name, and so on. You can rank as few or as many as you want.

This ballot illustrates the use of the AccuVote system used in Cambridge, Massachusetts to elect its city council and school board. Voters fill in the ranking numbers as they would for standardized tests taken in school, which allows for computerized vote counting and ballot transfers.

Choice Voting Ballot.

As the name "single transferable vote" implies, this systems involves a process of transferring votes. To understand how the transfer process works, it may be best to start out with a simple analogy.

Imagine a school where a class is trying to elect a committee.

Any student who wishes to run stands at the front of the class and the other students vote for their favorite candidates by standing beside them. Students standing almost alone next to their candidate will soon discover that this person has no chance of being elected and move to another candidate of their choice to help him or her get elected.

Some of the students standing next to a very popular candidate may realize that this person has more than enough support to win, and decide to go stand next to another student that they would also like to see on the committee.

In the end, after all of this shuffling around, most students would be standing next to candidates that will be elected, which is the ultimate point of this process.

In the single transferable vote, votes are transferred around just as the students moved from candidate to candidate in the analogy. The exact order of the transfer process is illustrated in figure below. An example of how the votes are actually transferred is shown in the table that follows.

For the sake of simplicity, assume that there is a three-seat district in which six people are running for office.

The first step in the process is to establish the threshold: the minimum number of votes necessary to win a seat. The threshold usually consists of the total number of valid votes divided by one plus the number of seats to be filled, plus one vote.

The formula looks like this: Threshold = (valid votes/1+seats) +1 vote. So in our three-seat districts with 10,000 voters, a candidate would need 10,000/1+3 (which is 2,500) plus one more vote, for 2,501.

Diagram of Ballot Transfer Process.

The second step is to count all the number one choices to see if any candidates have reached the threshold of 2,501. As shown on the table below, the Democrat Gomez has 2,900 voters and he is declared elected.

But Gomez actually has 399 more votes than he needs to win. These votes are considered wasted if they stay with Gomez, so they are transferred to the second choices on the ballot. (There are several ways to do this, but we needn’t get into those details here.)

In the second count, we see the effect of this transfer. The other Democratic candidate, Campbell, gets 300 of those second choice votes, and the independent candidate, Daniels, gets the other 99. The vote totals are now recalculated to see if anyone is now over the threshold.

No one is, so the next transfer takes place. The candidate with the least chance to win is eliminated and his or her votes are transferred to their second choices. This candidate is Higgins, the Republican, and 500 of his votes are transferred to the other Republican candidate, Dains; and the other 100 votes are given to Daniels.

Again the votes are recounted to see if anyone has reached the threshold. Dains has reached it with 2,800 votes and so she is declared elected. Once again her excess votes are redistributed to their second choices--200 to Graybeal, and 99 to Daniels.

But still no one has reached the threshold, so again the lowest candidate is eliminated and those votes transferred. That candidate is Campbell, the Democrat, and 100 of his votes go to Graybeal, and 600 go to Daniels. This puts Daniels, the independent candidate, over the threshold with 2,698 votes, and she is the last one elected.

Ballot Count and Transfer process.

This transfer process is a bit complicated, so why does it exist? The transfer process was invented primarily to reduce the problem of wasted votes -- votes that are cast but do not actually elect anyone.

Plurality-majority systems routinely waste large numbers of votes and this is why they are prone to such problems as party misrepresentation, and the underrepresentation of political minorities, racial minorities, and women.

The transfer process in STV is designed to ensure that the fewest votes are wasted and that the maximum number of people gets to elect a representative to office. It acknowledges that there are two kinds of wasted votes: votes for candidates that stand little chance of winning, and votes in excess of what a winning candidate needs.

Transferring these votes to their next ranked choice makes it more likely that they will actually contribute to the election of a candidate.

Simpler Than They Look

Again, to American eyes, these various PR systems often appear at first to be overly-complex and confusing. And while the mechanics of seat allocation can sometimes be complicated, the actual voting process is not intimidating at all and can be easily utilized by the average citizen.  

Voters need not understand all the mathematics of these systems to use them effectively.  To use an analogy:  you don't have to understand how all the electronic components in your car radio work in order to use it to find the kind of music you like. 

The party list system, the mixed-member system, and the choice vote system have all been used for decades in other Western democracies. Voters in these countries have had no trouble using these systems, as indicated by the very high voters turnout rates that these PR countries enjoy. Certainly we could expect that American voters would easily master the use of these systems as well.

For more detailed descriptions of the workings of various proportional representation systems, see Douglas J. Amy,  Behind the Ballot Box:  A Citizen's Guide to Voting Systems.

Extra-long Senate Ballot for the state of New South Wales (NSW) in Australia.

(Blogger’s Notes: I’ve been living in Australia since 1988 and voted in all Australian Federal and State and Local elections as here in our Dictatorially-Democratic Australia if one doesn’t vote one is punished with a hefty fine and if one doesn't pay that not-voting fine in time they take away one’s Driver License.

Really scary as one can easily loss one’s job or one’s small business by losing one’s own driver license. And that is the sole reason the voters turnout rates in Australian elections are almost 100% while Americans are calling a very-low 30% voters turnout in their elections a great turnout.

So I was forced to vote and also forced to learn both the Winner-Takes-All voting system used in Australian House of Representatives the Lower house, and the Proportional Representation voting system used in Australian Senate the Upper House.

I’m posting this excellent article by Dougla J. Amy of Mount Holyoke College USA explaining the Proportional Representation voting system, since Burma’s Upper House the Nationalities’ Parliament has just voted on June-11 to switch to Proportional Representation voting system from current Winner-Takes-All voting system.

Once a Burmese knows the Proportional Representation voting system very well he or she can also understand why our Nobel-Laureate Aung San Suu Kyi and her so-called democracy-allies like Min Ko Naing's 88-Generationers are so much bitterly against the PR voting system since her NLD is expecting to win majority in most electoral districts in coming 2015 General Election in Burma under current Winners-Take-All voting system.

He or she can also understand why all race-based ethnic insurgent groups are bitterly rejecting the PR voting system as in all those race-based states and regions the main ethnic-race is just a very thin majority as there are other races including Burmese after more than 60 years of Burmanization efforts by Burmese Army. For example the Kachins in Kachin State are only about 40%  while Burmese are 20% and the Shans nearly 30%.)