Introduction to Voting Methods, by James Green-Armytage

Contents...

Introduction: plurality

Single-winner: majority rule, two-round runoff, ranked ballots, instant runoff voting, pairwise comparison

Multiple-winner: proportional representation, party list, single transferable vote

Voting methods matter; a democracy is only as good as its voting methods!

The main voting system in the USA is known as plurality, which is not the same as majority rule. Plurality means that whoever gets the most votes wins, whether they have a majority or not.

For example, imagine that there is one Republican candidate with 30% of the vote, and 7 Democrats with 10% each. The Republican wins with a plurality, although he is nowhere near to a majority (50%+).

The plurality method is the main reason that we are stuck with a two party system. As long as we use plurality, it will be extremely hard for third party and independent candidates to run as anything other than “spoilers” or protest candidates. That is, in the long run, the plurality system only supports the existence of two viable candidates. Once those two candidates have been established in the minds of the voters, any additional candidate will only serve to take votes away from whichever candidate he is most similar to, thus acting against the interests of his would-be supporters... that is, to be a spoiler. As long as enough people care about the race between the two front-runners, it’s close to impossible for third party candidates to become competitive enough to actually win.

Instead of plurality, we need a truly competitive voting system that allows many independent candidates with diverse viewpoints to run for office without worrying about being spoilers, where the politicians we elect aren’t considered to be the lesser of two evils, but rather the greatest of many goods. We need politicians who are accountable and responsive to the will of the people, rather than being insulated by the conservative tendencies of the plurality system.

So, if we don’t want plurality, what do we want?

Actually, this question should break down into a few separate questions:

1. What methods to use to elect executives?

2. What methods to use to elect legislatures?

3. What methods to use for direct-issue voting?

1. Executives / single-winner systems

Let’s start with the question of executives, since that’s the simplest. Here we’re talking about just picking one person to be a president, governor, mayor, or something like that. A city has lots of people on its city council, but it only has one mayor, which is the difference between legislative and executive elections.

When you’re electing an executive, and voters preferences are strong enough for them to vote strategically, the best thing you can hope for is majority rule. But what is majority rule, exactly? It turns out to be trickier than you might expect...

To put this in more concrete terms, let’s imagine a group of 100 voters, where 36 of the 100 most prefer the right wing candidate, 34 most prefer the left wing candidate, and 30 most prefer the centrist candidate. Let’s imagine that everyone who likes the right wing candidate best would rather have the centrist than the left wing candidate, and likewise, everyone who likes the left wing candidate best would rather have the centrist than the right wing candidate. The centrist voters are evenly split between preferring right over left or vice versa. Here are the same numbers in an easier-to-read format:

36 voters: prefer Right, then Center, then Left

15 voters: prefer Center, then Right, then Left

15 voters: prefer Center, then Left, then Right

34 voters: prefer Left, then Center, then Right

What constitutes majority rule in this situation? Let’s look at a few different voting methods which aim for majority rule, to see how they answer this question.

The simplest majoritarian system is the two-round runoff. In the first round, everyone votes for one candidate. If one candidate already has a majority, then they are elected. If not, then you have a runoff election between the top two candidates in the first round.

In our election here, if everyone voted for their first choice, no one would have a majority right away, and the two candidates in the runoff election would be Right and Left. Right would then win the runoff, with 51 votes to 49. In this case, Right is preferred by a majority over Left. So has majority rule been satisfied?...

Before I answer that, I want to jump to another voting system, known as instant runoff voting, or IRV. As the name suggests, it is similar to the two-round runoff, except that it only uses one round of voting.

The cool thing about IRV is that it uses ranked ballots. Ranked ballots are really great! Instead of forcing voters to just vote for one candidate, ranked ballots allow voters to rank all the candidates in order of preference. So our leftist voters above can say, “Yes, my first choice is Left, my second choice is Center, and my third choice is Right!” This is really a big step forward in terms of voters being able to express themselves fully.

So, how does IRV count the ranked ballots?

1. First, it assigns each ballot to whoever is listed as the first choice on that ballot. If anyone already has a majority, great, they win.

2. If not, you eliminate the candidate with the fewest ballots, and count again. If that candidate was my first choice, then my ballot is now assigned to my second choice. If anyone has a majority now, great, they win. If not, you just keep doing this until someone has a majority, eliminating the candidates with the fewest votes, and assigning their votes to the next choice on each ballot.

So, it's a lot like a multiple-round runoff, where you eliminate one candidate at the end of every round, until someone gets a majority. Except, instead of actually having all those rounds of voting, you simulate them using ranked ballots.

In our imaginary election...

36 voters: 1st choice is Right, 2nd is Center, 3rd is Left

15 voters: 1st choice is Center, 2nd is Right, 3rd is Left

15 voters: 1st choice is Center, 2nd is Left, 3rd is Right

34 voters: 1st choice is Left, 2nd is Center, then Right

... the first round of counting shows Right with 36 votes, Center with 30, and Left with 34. So Center is eliminated, and the centrists’ ballots are then assigned to their second choices. Like the two-round runoff, Right now has 51, and Left has 49, so Right wins.

Now I’d like to return to the question of majority rule.

We have seen that Right is preferred by a 51-49 majority over Left. But what about Center? How many voters prefer Right to Center, and how many voters prefer Center to Right? The answer is that only 36 voters (just the right wing voters) prefer Right over Center, while 64 voters (the centrist voters and the left wing voters) prefer Center over Right. Furthermore, Center is also preferred over Left by a 66-34 majority.

So I would say that the real majority winner is Center rather than Right, since he is preferred by a majority over Right, and he is also preferred by a majority over Left. Right, on the other hand, is only preferred by a majority over Left.

So, IRV is pretty good, and it's better than plurality, but it’s arguably not quite the best. (see postscript) What is the best method for achieving majority rule? It's Condorcet’s method of pairwise comparison!!!

What is a pairwise comparison? Actually, I have already come close to describing it... To do a pairwise comparison between two candidates is basically to look at the ranked ballots and ask how many people prefer one candidate over the other, and vice versa. For example, we can tell by looking at the ranked preferences above that 64 voters prefer Center to Right, and 36 prefer Right to Center. That’s actually our pairwise comparison between Center and Right. Not very complicated!

Condorcet’s method of pairwise comparison simulates a series of elections between every candidate and every other candidate, like a round robin tournament. If there is one candidate who wins all of their pairwise comparisons, they are called a Condorcet winner, and they are elected by Condorcet’s method. Since a Condorcet winner is preferred by majorities over all other candidates in the election, how could you call anyone else a majority winner?

Here is the example again...

36 voters: 1st choice is Right, 2nd is Center, 3rd is Left

15 voters: 1st choice is Center, 2nd is Right, 3rd is Left

15 voters: 1st choice is Center, 2nd is Left, 3rd is Right

34 voters: 1st choice is Left, 2nd is Center, then Right

And here are the pairwise comparisons for the example:

Right vs. Center = 36-64, Center wins

Right vs. Left = 51-49, Right wins

Center vs. Left = 66-34, Center wins

Actually, the existence of a Condorcet winner is fairly likely. However, it is still possible that there will be a majority rule cycle. That is, for example, a situation where A is preferred by a majority over B, B is preferred by a majority over C, and C is preferred by a majority over A. This isn’t a flaw in the pairwise concept, but is rather a natural paradox of voting that cannot be avoided. There are different pairwise methods, which always elect a Condorcet winner when one exists, but do different things to resolve majority rule cycles. The simplest approach is to drop the weakest defeat in a cycle until a candidate becomes unbeaten. Most of the better pairwise methods are variations on this principle.

If you’d like to know more about the different pairwise methods, I have examined many of them in my voting methods survey. In short, three of the best pairwise methods are beatpath, ranked pairs, and the river method, and it is generally preferable to use winning votes rather than margins. I have also proposed my own pairwise method, called cardinal pairwise, which uses ratings ballots (voters can rank the candidates from 0 to 100) as well as ranked ballots. This one is a bit more complex than the others, but it may be worth it in some cases. It takes voters' strength of preference into account when resolving majority rule cycles, and it may reduce incentives for strategic manipulation, which would help to protect the sincere Condorcet winner when there is one.

By the way, Condorcet's method of pairwise comparison is named after the Marquis de Condorcet, who lived from 1743 to 1794, and proposed the pairwise method in 1785. 'Condorcet' is pronounced 'con-door-say'. He is a very interesting historical figure in his own right, and he played a significant role in the French revolution, but I won't go into that here.

Besides those mentioned above, other single-winner methods include approval voting, cardinal ratings, Bucklin, and the Borda count. I talk a bit more about approval voting in the postscript.

2. Legislatures / multiple-winner systems

Again, with executives, majority rule is the best that you can hope for. But when you are voting for more than one position at a time, then you can actually do something better. And that something is... proportional representation!

For example, let's say that you have a legislature with 100 seats. Let's say that in your country, 47% of the people want to vote for the Republican party, 43% are for the Democratic party, and 10% are for the Green party. How many seats does each party get?

Well that depends on whether you are using a winner-take-all system, or a proportional representation system. In a winner-take-all system, you would divide the country into 100 separate districts, with each district electing one legislator by majority or plurality rule. How many seats would each party get? Who knows... it largely depends on how the district boundaries are drawn. Hence with winner take all you have a perennial problem with gerrymandering, that is drawing district lines to give your own party an advantage. Anyway, it's doubtful that the Green party will get any seats at all, unless they are very heavily concentrated in one particular area. It's unclear whether the Democrats or Republicans will get more seats; again, it just depends on the district boundaries.

However, with proportional representation, you use multi-member districts, and then you allocate the seats within the districts proportionally. The simplest example would be if you decided all of the 100 seats in a single nation-wide election. With proportional representation, the Republicans should get 47 seats, the Democrats should get 43 seats, and the Greens should get 10 seats.

It is also possible to compose a legislature of more than one multi-member district. For example, for our 100 seats legislature, instead of having 1 national district with 100 members, we could have 10 districts with 10 members each, or 5 districts with 20 members each. As long as you have multi-member districts, proportional representation is possible, although the degree of proportionality increases with the number of seats that each district elects. (This number is called the district magnitude).

With winner-take all, it is not too uncommon for one party to get 90% of the seats in a legislature with only 60% of the vote, or to get no seats at all with 20% of the vote. This sort of effect is known as disproportionality. The goal of proportional representation is that a group should get a number of seats in the legislature that is proportional to its share of the vote.

Proportional representation goes further than majoritarian systems in providing minority viewpoints with a voice in government. Thus it allows for a much fuller political discourse. Also, proportional representation means that people are more likely to get a representative whom they strongly favor, someone who more closely represents their own point of view. Proportional representation should also facilitate changes in the balance of power which are more fluid, more gradual, less chaotic and less fraught with uncertainty. All of this goes toward creating a more fair, stable, and effective democracy, with a high level of accountability for those in power.

There are two basic kinds of proportional representation; these are party list, and single transferable vote (STV). Party list is a bit simpler, but single transferable vote is a significantly better system in the long run.

The basic principle of party list is simply that each voter casts their vote for one party, and then the result is tallied such that each party is awarded a number of seats in the legislature that is proportional to the number of seats.

Party lists can be either closed or open. With closed lists, the party leadership makes a list of their candidates, and puts it on the ballot in a certain order. If they end up getting 5 seats, then the first 5 candidates on their list get the seats, and so on. With open lists, voters have an opportunity to change the order of the lists, so that the candidates who gain seats are the ones who are more popular with the voters, rather than simply being chosen by the party leadership.

However, both of these methods suffer from the fact that they rely too heavily on parties as the unit of political representation. They assume that voters will always like all of the candidates from their favorite party more than they like any other candidates, which is not always the case. Thus they can submerge the differences between individual candidates into a general party identity, which can reduce the flexibility of the system. Also, there is still a chance that an emergent party will be too small to gain a seat, and that this too-smallness will be self-fulfilling because people will not want to waste their vote.

Single transferable vote (STV) addresses both of these problems. Like IRV and pairwise, STV uses ranked ballots. Also, unlike party list, people vote for individual candidates rather than political parties. So I could put someone from the Green party as my first choice, then a couple Democrats at second and third, then another Green at fourth, maybe an independent at fifth, and so on. So STV gives voters much more freedom than party list, and yet it takes all the ranked ballots and makes a result that is every bit as proportional. Wow! How does it work?

Well, I'll give you the short version here, without going into all the gory details. In general, STV is similar to IRV, in that people vote with ranked ballots, and there are successive rounds of counting where ballots are assigned to different candidates.

First of all, STV establishes a quota, which is the absolute minimum number of votes a candidate needs to secure a seat.

If a candidate has more votes than she needs for the quota, then you say she has a surplus. When a candidate has a surplus, a fraction of all the votes cast for her will be transferred to whoever is the next choice on each individual ballot. The fraction is worked out just right so that the candidate is left with exactly one quota's worth of votes.

So, if my first choice for Senator is Tony, but Tony is extremely popular and already has more votes than he needs, a fraction of my ballot skips along to my next choice, Stan. And if Stan has a surplus too, then a fraction of the fraction of my ballot is transferred to my third choice, Paulie. And so on.

In this way, STV doesn't punish people who vote for candidates who have more support than they need to get a seat. Therefore, people don't have to second-guess about whether voting for a popular candidate will be a waste.

STV also doesn't punish voters who vote for candidates who don't have enough support to gain a seat. In this way, it works just like IRV. That is, after all the surpluses have been transferred, if there are still unfilled seats and more unelected candidates than unfilled seats, STV eliminates the candidate who has the lowest vote count at that stage, and transfers their votes to whoever is the next choice on each individual ballot.

For example, let's say that my first choice is Carmela, but even after all the surpluses are transferred, Carmela still doesn't have a quota. Let's say that there are just a few seats left to fill, lots of candidates left who want to fill them, and of those candidates, Carmela currently has the lowest total of votes assigned to her. What happens is that Carmela is eliminated, and my vote is now counted, in whole, for my second choice instead.

So, to sum up, STV elects candidates who have more than a quota, and transfers a fraction of their votes to subsequent choices. STV also eliminates candidates who have the fewest votes, and transfers their votes to subsequent choices. There might be several alternating rounds of surplus transfer and elimination, with eliminations leading to more surplus transfers. The rule is that surplus transfers always take place first, so that no candidates are eliminated while there are still surpluses left to be transferred.

Again, STV is fully proportional. It doesn't allow a bare majority of the electorate to control all of the seats in a multi-member district. Let's say that there are 10 seats in a district, and Democratic candidates get about 55% of the vote. About 5 or 6 of the 10 candidates will probably be Democrats, but the strength of the Democratic voters' ballots is absorbed in the process of electing those candidates, leaving the remaining seats to be decided by the remaining 45% of the electorate.

So, that's the short explanation of STV. For a more detailed description, including some of the different kinds of STV, you can visit my voting methods survey. The Electoral Reform Society and the Center for Voting and Democracy also have loads of information about STV.

Other multiple-winner systems include block voting, cumulative voting, limiting voting, single non-transferable vote, parallel systems, and mixed member proportional.

3. Direct-issue voting

Although it is important to use better methods elect representatives, I don't think that a better democracy should end there. I would also like a system that enables citizens to vote directly on the the critical issues of the day. That is, a form of direct democracy.

People often dismiss direct democracy as impractical, because average people will not have enough time to follow the issues closely enough to take an informed vote. However, I think that those people have probably not considered a proxy system!

A proxy system of direct democracy means that voters have the option of voting directly on issues, but they also have the option of deferring their vote to someone else whom they trust; this person is their proxy. If I designate a proxy for a particular vote, then my vote is automatically cast the same way as my proxy's vote. Or, my proxy can designate someone else as a proxy in turn, which means that this person is casting both my vote and my proxy's vote. There are lots of possibilities here to play with, but this is the basic idea. I discuss the matter in greater detail in my delegable proxy paper.

Conclusion

There are lots and lots of voting methods which I didn't talk about in this short introduction. The idea here was just to get your feet wet, not to drown you. However, if you are interested in learning more, I've tried to make some resources available. My survey of basic voting methods attempts to fill in a lot of the gaps that I have left open here. There are also a bunch of essays and proposals on my main voting page which you might want to look at. The links page there should start you off on the road to other web-based resources, including the venerable election methods mailing list.

By the way, by promoting alternative voting systems, I'm not saying that better voting systems necessarily make for a perfect democracy. There are lots and lots of factors which go into a really healthy democracy, such as a media that is free from both government censorship and corporate oligopoly, a strong education system which trains people as creative and free-thinking citizens rather than downtrodden drones or narrow technocrats, a campaign finance system which does not put politicians on a leash held by the wealthy and powerful, and so on. All of these factors are terribly important, and very much worth fighting for.

I don't think that alternative voting methods are a cure-all for our society, but I do think that they are much more promising than most people realize. So what I would like is for people to become aware of alternative voting methods and their potential, to study and debate the different methods, to use them to make group decisions (e.g. in schools, unions, churches, businesses), and to push for their adoption in government.

Postscript on IRV, approval voting, and pairwise in the USA

IRV is becoming relatively popular with certain political groups in the USA. It is believed to be a means for alternative parties to participate in elections without 'spoiling'. While this is true up to a point, there may be a stage where third parties could gain enough of the vote to cause the early elimination of the more similar major party candidate, but not enough to win, thus once again throwing the race to the opposite party. This could cause something of a nasty surprise for voters, or, perhaps, voters would see it coming and be deterred from voting their sincere favorite in first place in order to avoid it.

(In the example above, the voters for the "Left" candidate will regret their vote if IRV gives the election to "Right", and wish that they had insincerely voted Center > Left > Right instead of Left > Center > Right, thus electing "Center" rather than "Right", which they would have preferred. But if they do anticipate the result and make this strategic reversal, then the true support for "Left" is concealed, and "Left" is put at an unfair disadvantage.)

For this latter reason, IRV probably has more of a conservative tendency than pairwise. Also, it is likely to maintain some of the polarization that we experience in the plurality system, whereas pairwise is more powerful tool for promoting stable compromises.

However, with all that said, IRV may still be a significant step in the right direction. For one, it introduces the use of ranked ballots, which could open things up considerably, while possibly paving the way for pairwise and single transferable vote proportional representation. Also, it should at least give alternative parties a sort of political toehold to build a base; once a party's first choice support is known to be large enough to threaten a major party with early elimination in elections, then the alternative party might have an increased amount of bargaining leverage to enact further reform.

There are versions of IRV which allow voters to give more than one candidate the same ranking. I think that these versions of IRV are preferable to versions that disallow equal rankings. Unfortunately, the versions which disallow equal rankings are much more widely-promoted at this time.

For a pro-IRV viewpoint with lots of information on the current status of IRV in the political system, visit fairvote.org. For a vitriolic criticism of IRV relative to pairwise, visit electionmethods.org. For more discussion of this and other voting methods issues, visit the election methods mailing list at electorama.com. I also wrote a letter about the IRV-pairwise debate that goes into some of these issues.

Approval voting is another method which is often discussed as an alternative to plurality. This system doesn't use ranked ballots, but it allows you to cast one vote each for as many or as few candidates as you like. The winner is simply the candidate with the most votes. Assessing approval voting as a possible method for contentious public elections can be somewhat difficult. The criteria that have been developed for ranked ballot methods often don't apply, and one's assessment depends on complex assumptions about voter psychology, voter strategy, campaign dynamics, etc. However, it does seem quite clear that approval voting is superior to the plurality system, without the complexity or new voting technology needed for IRV or pairwise. So, I hope to see approval voting used more often.

Why is approval voting superior to plurality? For one thing, where plurality often forces people to choose between voting for a candidate whom they perceive to be viable and a candidate whom they strongly agree with, approval allows people to do both. Thus, it arguably gives candidates a fair chance to prove themselves on election day even if they are not expected to be one of the top two contenders. Some also argue that approval would do fairly well at choosing Condorcet winners. They make an interesting case, but I would like to see this happen in practice before I accept it as being true.

Pairwise is also lagging behind IRV in terms of making political waves in the USA. Probably IRV has an advantage because it is already been used in other countries like Australia and Ireland, whereas pairwise has not been established in large-scale public elections. Perhaps it has an advantage because it is less unfamiliar than pairwise, being relatively similar to the two-round runoff which we do use in some parts of the USA. Perhaps some people who are aware of IRV's conservative tendencies find them comforting. Perhaps it is easier to explain than pairwise (although I'm still not totally convinced of this). Or maybe IRV has just been lucky to find relatively well-organized supporters.

Anyway, I hope that organized support for and use of pairwise will increase in the coming years. It does need to be tried more on smaller scales before it can become a serious proposal for federal elections, and so I hope that more people will begin using it as a tool for compromise in councils and legislatures, in schools, churches, NGO's, corporations, unions, and so on. Aside from contributing to a national democracy movement, I think that groups like these can derive immediate benefit from the use of the pairwise comparison method.

As for voting methods outside the USA, things are a bit further along. Actually, more advanced-industrial nations use proportional representation than plurality as their primary election system. Party list is still the most common form of proportional representation, but there are already a few nations which use STV. You can click here for a map of which countries use which voting systems, and you can click here for some more proportional-representation-around-the-world resources.

If you have questions or comments, you can e-mail me: jarmyta(at)antioch-college.edu

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