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Electoral Systems

© Copyright 2000, Charles King
A country’s electoral system is the method used to calculate the number of elected positions in government that individuals and parties are awarded after elections. In other words, it is the way that votes are translated into seats in parliament or in other areas of government (such as the presidency). There are many different types of electoral systems in use around the world, and even within individual countries, different electoral systems may be found in different regions and at different levels of government (e.g., for elections to school boards, city councils, state legislatures, governorships, etc.).

Electoral systems can be divided into three general types:

1. Plurality electoral systems

Also called “first-past-the-post” or “winner-take-all” systems, plurality systems simply award a seat to the individual candidate who receives the most votes in an election. The candidate need not get a majority (50%+) of the vote to win; so long as he has a larger number of votes than all other candidates, he is declared the winner. Plurality systems normally depend on single-member constituencies, and allow voters to indicate only one vote on their ballot (by pulling a single lever, punching a hole in the ballot, making an X, etc.) Plurality electoral systems also tend to encourage the growth of relatively stable political systems dominated by two major parties (a phenomenon known as “Duverger’s Law”). 

Such an electoral system, though, clearly does not represent the interests of all (or even most) voters. In fact, since a candidate need have only a plurality of votes to be elected, most voters may actually have voted against the winner (although their votes are split among several candidates). 

Elections for the House and Senate in the United States and for the House of Commons in the United Kingdom use the plurality system. The US presidential election is also generally considered a plurality system, but the existence of the Electoral College actually makes it a strange hybrid of plurality and majority systems.

2. Majority electoral systems

Also called “second ballot” systems, majority electoral systems attempt to provide for a greater degree of representativeness by requiring that candidates achieve a majority of votes in order to win. “Majority” is normally defined as 50%-plus-one-vote. If no candidate gets a majority of votes, then a second round of voting is held (often a week or so after the initial ballot). In the second round of voting, only a select number of candidates from the first round are allowed to participate. In some countries, such as Russia, the top two vote-getters in the first round move on to the second round. In other countries, such as France, all candidates with a minimum threshold percentage of votes (in the French case, 12.5% of all registered voters) move on to the second round. Like plurality systems, majority systems usually rely on single-member constituencies, and allow voters to indicate only one preference on their ballot. 

Presidential elections in Austria, Finland, Portugal, Russia and other east European states, as well as presidential and National Assembly elections in France, make use of various forms of majority electoral systems. The US Electoral College also has components of a majority system, because a presidential candidate must get 50%-plus-one electoral votes (270 out of 538) in order to win. If no candidate reaches the 270 mark, the election is decided by the House of Representatives. In determining who votes for whom in the Electoral College, though, the US presidential race is a strict plurality system: The candidate who gets a plurality of the popular vote in a state gets all that state’s electoral votes.

3. Proportional representation

Also known as “PR”, proportional representation is the general name for a class of voting systems that attempt to make the percentage of offices awarded to candidates reflect as closely as possible the percentage of votes that they received in the election. It is the most widely used set of electoral systems in the world, and its variants can be found at some level of government in almost every country (including the United States, where some city councils are elected using forms of PR). 

The most straightforward version of PR is simply to award a party the same percentage of seats in parliament as it gets votes at the polls. Thus, if a party won 40% of the vote it would receive 40% of the seats. However, there are clear problems with such a system: Should parties that receive only 0.001% of the vote also be represented? What happens if the voting percentages do not translate evenly into seats? How do you award a party 19.5 seats if it got 19.5% of the vote? More sophisticated PR systems attempt to get around these problems. Two of the most widely used are discussed below.

Party list sytems

Under party list forms of PR, voters normally vote for parties rather than for individual candidates. Under a closed party list system the parties themselves determine who will fill the seats that they have been allocated; voters vote only for a particular party, and then it is up to the party to decide which party members will actually serve as representatives. Legislative elections in Israel and Germany are conducted according to such a system. Under an open party list system, voters are given some degree of choice among individual candidates, in addition to voting for entire parties. Denmark, Finland, Italy, Luxembourg and Switzerland all have versions of open party list systems.

Under all party list systems, though, one still needs some method for allocating seats to individual parties. One commonly used method is named for the nineteenth-century Belgian mathematician Victor d’Hondt, and is normally referred to as a “highest average method using the d’Hondt formula.” 

For example, assume that we have an election with 1,000 total voters in which five parties (A, B, C, D, and E) have gained 100 (10%), 150 (15%), 300 (30%), 400 (40%), and 50 (5%) votes, respectively. Assume also that, in our electoral constituency, there are 3 seats up for election; that all votes cast are valid; and that the electoral system has a 7% vote threshold. (That is, parties must get at least 7% of the total valid votes cast in order to participate in the distribution of seats.) Party E would thus be elimiated from competition at the outset. The d’Hondt method of seat allocation then proceeds in the following steps. 

1. Place the total number of votes garnered by the competing parties (A, B, C, and D. E has been eliminated) in a row.

  100  150  300  400

2. Divide each figure in the row by 1, 2, 3, . . ., n. (How far you take the division varies. The more seats you have to allocate, the further you have to divide. For our purposes, 3 or 4 divisions should do the trick.)

Party A Party B Party C Party D

100 150 300 400
div. by 1 100 150 300 400
div. by 2 50 75 150 200
div. by 3 33 50 100 133
div. by 4 25 37.5 75 100

3. Pick the highest quotient in the list (including the quotients obtained by dividing the votes by 1). The highest quotient is “400” in the Party D column. We therefore award one seat to Party D.

4. Pick the next highest quotient in the list. The next highest quotient is “300” in the Party C column. We therefore award one seat to Party C.

5. Pick the next highest quotient in the list. The next highest quotient is “200” in the Party D column. We therefore award another seat to Party D. We have successfully filled all the seats available in this constituency.

The final results of the election are therefore:

Party C  1 seat  (or 33% of the total available seats)
Party D  2 seats  (or 66% of the total available seats)

Notice why we call this system “proportional representation:” Under a plurality system, Party D would have received 100% of the seats because that party received a plurality (40%) of the vote--even though 60% of voters voted against Party D by choosing other parties. Under PR, however, we are able to represent some of the interests of the other voters. Party D’s representation in parliament is reduced to 66% of seats, while Party C’s is increased to 33% of seats. The system yields a result that is clearly not perfectly proportional. But the distribution more closely approximates the actual percentage of votes that each party received than would a plurality or majority system.

The d’Hondt method is only one way of allocating seats in party list systems. Other methods include the Saint-Lague method where the divisor is the set of odd numbers (1, 3, 5, 7, 9, . . ., n) and the modified Saint-Lague method used in Denmark, Norway and Sweden, where the divisor is 1.4 plus the set of odd numbers (1.4, 3, 5, 7, 9, . . . , n). Other methods divide the votes by a mathematically derived quota, such as the Droop quota or the Hare quota (see below) 

One other feature of party list systems is called the vote threshold. Party list systems normally establish by law an arbitrary percentage of the vote that parties have to pass before they can be considered in the allocation of seats. The figure ranges from 0.67% in the Netherlands to 5% in Germany and Russia, or even more. Any party that does not reach the threshold is excluded from the calculation of seats. The vote threshold simplifies the process of seat allocation and discourages fringe parties (those that are likely to gain very few votes) from competing in the elections. Obviously, the higher the vote threshold, the fewer the parties that will be represented in parliament.

Single transferable vote (STV)

STV is another important form of proportional representation. In various forms, it is used widely in many countries, although only Ireland, Australia, and Malta have used it in major national elections. Other countries have used it in local elections, and even some communities in the United States (such as Cambridge, MA) use it today. Many student organizations in Europe also use this system for election to university student associations, because it yields an even more proportional result than party list systems, and certainly more proportional than plurality or majority voting. 

STV was originally developed by Thomas Hare (1806-1891), a British politician whose writings greatly influenced the views of the philosopher John Stuart Mill. Under STV, voters vote for individuals, not for parties as in the party list system. The key feature of STV is that individual voters rank candidates according to their 1st, 2nd, 3rd, . . ., nth choices. Rather than simply voting for a single candidate, voters have the opportunity to express a range of preferences for several candidates on the ballot. Like party list systems, though, STV depends on having multi-member constituencies.

The complicated part of STV is tabulating the seats to be awarded after the votes have been cast. As with party list systems, there are a number of mathematical formulas that one can use to accomplish this task. One of the most widely used methods is known as the Droop quota, named for the nineteenth-century thinker and mathematician H. R. Droop. The Droop quota is used to determine the minimal number of votes that an individual candidate must get in order to be awarded a seat. It is calculated using the formula:

 [V/(S+1)] +1

where V is the total number of valid votes cast in the constituency, and S is the total number seats up for election in the constituency. Hence, if we have 1,000 votes cast for 3 seats, the Droop quota is [ 1,000 / (3 + 1) ] + 1 = 251. That means that any candidate who is able to get at least 251 votes will be assured of winning a seat. Once the Droop quota has been calculated and all the votes collected, we still have to allocate the seats. 

In this example, assume that we have 5 candidates (A, B, C, D, E) for 3 seats. In accordance with STV, individual voters have ranked each of these candidates (1 to 5, with one being the first-choice candidate) on their ballots. The allocation of seats then proceeds according to the following steps--but remember that there are a variety of STV methods in use. We will try to keep things very simple here:

1. Pull each ballot out of the ballot box one at a time and place them in piles according to the first-choice candidate marked on the ballot (e.g., if a ballot indicates candidate C as the first choice, place it in a pile marked “C”).

2. As soon as one pile of ballots reaches 251, that candidate is awarded a seat. Let us assume that candidate C was the first to reach the Droop quota of 251 first-choice ballots.

3. Continue drawing ballots out of the ballot box and placing them in piles according to the first-choice candidate marked on the ballot. But since C has already been elected, place any ballots that indicate candidate C as first choice in the pile of the candidate indicated on that ballot as the voter’s second choice. For example, if you pull out a ballot that indicates candidate C as first choice and candidate A as second choice, place the ballot in the pile for candidate A, since candidate C has already been awarded a seat. In this way candidate C’s surplus votes (i.e., the votes beyond those needed to win a seat under the Droop quota) are “transferred” to the next-choice candidate--hence the name “single transferable vote.”

4. Continue with Step 3 until another candidate reaches the 251 mark. Then, continue carrying out Step 3 until you fill all the available seats. For example, let us assume that we have already elected candidate C on first-choice ballots alone, and that by combining second-choice ballots from candidate C with further first-choice ballots from the box, we have also been able to award a seat to candidate A. How do we fill the third seat? We continue in a similar manner as before. Any ballots that list candidate C as the first-choice will be transferred to the second-choice candidate; if the second-choice candidate turns out to be candidate A (who has also already been elected), then we will transfer them to the third-choice candidate. Similarly, all first-choice ballots for candidate A will be transferred to the second-choice candidate indicated on the ballot; if the second-choice candidate turns out to be candidate C (who has already been elected), the ballot is transferred to the third-choice candidate. And so on.

5. But what happens if, after distributing all first-choice ballots, no further candidates have reached the Droop quota and we still have empty seats to fill? In this case, simply eliminate the candidate with the lowest number of first-choice ballots and transfer those votes to the second-choice candidates. Repeat this step as many times as necessary (always eliminating the lowest vote-getter)  in order to reach the number of votes mandated by the Droop quota.

As with party list systems, there are a variety of ways of conducting an STV election. For example, instead of using the Droop quota, we might use the Hare quota (V / S) or the Imperial quota [V / (S + 2)]. A country’s choice of which system to use depends on its history and the degree to which policymakers value genuinely proportional representation.

STV can clearly be rather confusing. Some voters may feel that a plurality system is somehow more “natural,” or that STV and other forms of PR are simply “tinkering with the numbers.” But PR in general, and STV in particular, can yield results that are more truly representative of the choices of individual voters. There is a strong movement for PR in the United Kingdom, with some political leaders arguing that STV should replace the current plurality system for electing parliamentarians to the House of Commons. There is a similar movement in the United States, although since few Americans could even explain how the Electoral College works, they are probably not going to learn STV any time soon. 

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