How to do democracy — Voting Systems

If you like democracy, that’s great, go you! You’ll probably find this article interesting. Democracy is a political system whereby the will of the people is reflected in the decisions of their government. A big part of doing democracy is voting, where the people pick representatives who they think (or hope) will act in their best interests.

It sounds simple, but democracy gets complicated very quickly. For instance, if you’re a representative, does acting in the best interest of your constituents involve doing what they want you to do? Or is it what you think is best, since they presumably elected you to exercise your judgement? Or perhaps what your party thinks is best, because that’s the platform you were elected on? Maybe it’s something else, maybe you should compromise and negotiate with other representatives, even if nobody is 100% happy, because compromise might yield the best possible outcome in your particular situation.

There are many, many issues, but this article is going to cover voting systems, that is, how we elect our representatives in the first place. There are many democracies over the world which use a range of different systems. We’re going to focus on three main ones and show you the very different results that can come about from the same preferences in the same population.

We’re using data from the 2010 Australian election, and here’s why:

  • The election was a close one — no party won an outright majority in the Australian House of Representatives. This is not unprecedented, but certainly unusual, and this particular data set is illustrative for our purposes because a close election can have very different results depending on your voting system.
  • Australia uses a preferential voting system. This means that we have data about which parties people preferred in relation to other parties. More on this later, but it essentially means we have more data to play with.

You don’t have to know anything about Australian politics to understand this article. We’ll talk a tiny bit about the basics of the parties so you get a sense of the significance of different outcomes. The important thing to know is that we are not using hypothetical data, that voting systems really do change outcomes within a democracy.

Preferential Voting

Australia uses the system of Preferential Voting, so in this section we’re discussing the actual 2010 election outcome.

Preferential Voting is not used in many jurisdictions, but it is used throughout Australia. You get a list of candidates for your electorate, and you order them from most to least preferred, like so:

Image by Rspeer at the English language Wikipedia, CC BY-SA 3.0,

The major upside of preferential voting is that it means if your favourite candidate is unlikely to win, you don’t ‘throw your vote away’ by voting for them. Instead, your vote is redistributed down your preferences. This can be very important for showing your support for minor parties, even if they don’t end up winning.

This joke doesn’t make sense to Australians

There aren’t split votes either. So, for example, if an electorate has four right wing candidates and one left wing candidate, the right wing candidates aren’t disadvantaged if votes are spread amongst them. The votes for the weakest right wing candidates would presumably be reallocated to stronger ones.

There are downsides too. The main one is that it doesn’t pass the Condorcet rule, which states: ‘that an alternative that defeats every other by a simple majority is the socially optimal choice.’ In plain terms, a candidate who would beat any other candidate in a two-horse race won’t necessarily win under this system.

The general effect of Preferential Voting is that it tends to advantage less polarising candidates. Winners are the most acceptable candidates for the most people in an electorate.

In 2010, Preferential Voting yielded a ‘hung Parliament’ in Australia. This means that there was no clear majority in the House of Representatives. Here’s what it looked like:

There were 150 seats up for grabs, which means that a party needed 76 seats for a majority.

The Liberal and National Coalition (we will refer to them just as ‘the Coalition’ — they’re technically two parties but they run together in some states so we’ve decided to group them for convenience sake) won 73 seats. The Labor Party won 72 seats. The Greens won 1 seat and four independent candidates won seats. In effect, the Labor Party formed alliances with the Greens and three of the independents to form a ‘minority government’, even though they had one fewer seat than the Coalition. A Labor government was thus formed and Australia had a Labor Prime Minister.

This is an unusual result in Australia because of the preferential voting system. It’s a very, very simple summary (and some Aussies might not like it), but you can think of the Labor party as ‘centre-left’ and the Coalition as ‘centre-right’. Because preferential voting is big on voters’ lukewarm feelings, most of the Parliaments tend not to have many small party, extreme, or independent representatives, although electoral changes may mean that increasingly, the big parties will have to learn to form alliances. But we digress.

First Past The Post

First Past The Post gets its name from the proto-democratic institution that is horse racing.

First Past the Post is a very simple, and widely used voting system (including in the UK and USA) where voters get a list of candidates and choose their favourite, the end. Like this:

CC BY-SA 3.0,

It has the opposite problems and benefits of Preferential Voting, although, like Preferential Voting, it seems to favour big parties.

This is what the Australian 2010 election results would have looked like if First Past The Post was the voting system utilised:

It would not have been a ‘minority government’, and, in fact, there would have been a different government. The Coalition would have won the necessary 81 seats to win in their own right, the Labor party would have gotten 66 seats, and three (instead of four) seats would have gone to independents. The Greens wouldn’t have gotten any seats at all.

Note here that despite being a big party, Labor would suffer a lot under a First Past the Post system. They tend to rely a lot from preferences from the Greens Party (which you can think of as a progressive party with a strong emphasis on environmentalism) and other minor parties. In Australia, the left-wing vote is much more split than the right-wing vote, so right-wing governments would be much more common under First Past the Post.

Proportional Voting

Proportional Voting is widely used throughout the world, particularly in Europe. It is a system whereby the number of people voting for a party is directly reflected in the number of seats that party gets. So, for instance, if a party gets 13% of the vote, they should get 13% of the seats.

But, remember that the Australian House of Representatives has 150 seats. So, if your party gets 13% of the vote, that translates into 19.5 seats. Oh dear.

Via GKB Furniture

Do you have half a seat? Well no, of course not, there’s a number of different ways to deal with rounding problems — we look at two of them, the D’Hondt system and the Sainte-Laguë system. The different methods are complex and mathematical, so we won’t go into them here, but if you are interested in learning more, Wikipedia is a good place to start. As we found, there’s not too big of a difference between them (although if you’re a Coalition or independent candidate you may disagree), but the D’Hondt system slightly prefers larger parties.

The upside of Proportional Voting is elucidated pretty strongly by John Stuart Mill in Considerations on Representative Government:

In a representative body actually deliberating, the minority must of course be overruled; and in an equal democracy, the majority of the people, through their representatives, will outvote and prevail over the minority and their representatives. But does it follow that the minority should have no representatives at all? … Is it necessary that the minority should not even be heard?

In essence, just because your opinions aren’t shared by half of all voters, doesn’t mean your interests shouldn’t be represented.

Image via weheartit

On that note, however, one problem of Proportional Voting is that it chucks out the system of electorates. In Australia and many other countries, voters are divided into local groups who each choose one representative who looks out for their particular interests. Under a Proportional system, geographic interests are crowded out by the fact that all the views are mushed together to produce successful candidates. It could be a problem if there are disparities between people living in metropolitan and regional areas, or significant differences in where people live which correlates with racial, socioeconomic, or other kinds of inequalities; or for geographical areas that just have very specific needs.

As an imperfect solution, we’ve also looked at how Proportional Representation would have changed the 2010 Australian election outcome if each state were allocated seats based on their preferences. Note that bigger states (like New South Wales) get more seats than smaller ones. Some democracies like to reallocate the proportion of seats in order to over-represent minority regions — the EU does this, so does the US. Indeed, Australia does too in its Senate elections (which is distinct but related to the House of Representatives)— the smallest state, Tasmania, has the same number of seats as New South Wales. This is another digression, but certainly another factor to consider in doing democracy is asking whether it’s important for all geographic groups to be represented on an equal basis, regardless of their population size.

Madeline may be small, but she ought not be disadvantaged when pursuing diplomatic relations with Spain.


This method is used in Japan, the Netherlands, Denmark, Spain and many more. Here’s the election results under a national Proportional Voting system, using the D’Hondt method:

Under preferential voting systems, small parties really win out and there is even less chance of a major party nabbing a majority. The Coalition get 67 seats, Labor gets 58, the Greens get a whopping 18 seats (which is probably why they advocate for Proportional Voting!), the Family First Party (think of them as right wing) likewise get three seats and the Christian Democrats (likewise right wing) get one seat. Three independent candidates get seats, but note that this estimation is imperfect — we’ve lumped all the national independents, with often very different political views, into one category. The actual result would probably look a bit different.

Under such a system, alliances become very important, and they may be more liable to shift depending on issues under consideration. We’d probably see a Europe-style consensus politics, with more diplomacy, manoeuvering, and compromise, and fewer full-blown arguments. After all, you don’t want to burn bridges with the people who you might need to pass your next bit of important legislation.

This is not a good idea in a proportional system.

By way of musing, this style of voting may not only produce very different election outcomes, it may also call for different kinds of leaders with different skill sets, negotiation being key.

Here’s the outcome if we allocate seats based on state preferences instead of national ones:

The bigger parties do a bit better here, with the Coalition getting 69 seats, Labor getting 59. The Greens get 17, independent candidates get three, and Family First get two.


One alternative to the D’Hondt system is the Saint-Laguë system, used in Germany, Norway, Sweden and New Zealand, among others. As the Sainte-Laguë system isn’t vastly different from D’Hondt, the results aren’t vastly different either — but importantly, the system still changes the result enough to make a difference, especially in a parliament where there is no overwhelming party majority and so legislation can pass and fail as a result of only a few votes.

Here’s the national result:

Here, the independents are slightly favoured, picking up four seats instead of three. The Coalition loses a seat compared to the D’Hondt results.

And the state result:

As per the difference between the D’Hondt system’s national and state preferences, the Greens have one fewer seat, one of the independents loses their seat, and the Coalition pick up a couple. The Christian Democrats keep their seat this time.


We have our own views about what voting systems we like more than others, and unsurprisingly, we tend to prefer the ones that would yield outcomes that best reflect our individual views. But we don’t want to argue about which system is objectively the best — they all have pros and cons. What we do want to point out is that representing the will of the people isn’t straightforward, and with the same collection of data, we can get to a number of different outcomes. We get different Prime Ministers, very different collections of representatives, and likely different agendas, and even democratic styles. And we haven’t even tried out all the different possible voting systems yet! Nor have we got to how to draw electoral boundaries!

The Maryland, USA third congressional district doesn’t look all that democratically defined to us.

We wrote this to get you to start a conversation, and to think more about how the people’s will best translates to the way we vote. The way your nation does democracy is not inevitable, and each election outcome isn’t the only possible outcome, nor even the only ‘fair’ outcome.

On the other hand, perhaps there is no ‘fair’ outcome at all. Arrow’s Impossibility Theorem says that no voting system can satisfy all key democratic requirements. The requirements are:

  • If one party is universally preferred to another, that party should always do better.
  • Whether a party is preferred compared to another party should depend only on voters’ feelings about those two parties — not on their opinions about other parties.
  • No single voter can change the outcome of the election by themselves (this is called the ‘dictator’ rule, which rules out a North Korea-type ‘democracy’).

No system can meet these criteria. If everyone preferred Party A, Party A would win. If everyone started to switch their vote one-by-one to Party B, then eventually Party B would win. At some point when the votes are switching, Party A must stop winning. The person whose vote stops Party A from winning is a ‘dictator’ — someone who decides the result for everyone — even if you (or they!) don’t know who they are.

There is no fair voting system

Arguably this isn’t really a problem, most democracies have stipulations in place in case of a tie (usually the election’s Returning Officer will cast a vote, which they normally don’t get to do), but it is strange to think that the view of a single person can change how a country runs. It doesn’t really feel that democratic, even if we’re willing to accept that it is. See the 2008 film Swing Vote to see what a tie-breaker ‘dictator’ might look like.

By the way, ‘we’ consist of Erin Stewart, a writer; and Lindon Roberts, a mathematical modeller. We are a couple and, as you can see, take part in quite nerdy weekend activities.

An artist’s rendering of us

Our raw data is from the Australian Electoral Commission.