Explainer: How your vote elects a government

This was originally published on 1st June, 2019, and republished here on 4th January, 2022.


How does one best discover the democratic will of the people? Over the many years that democracy has existed in some form or another, all kinds of electoral systems have been used. Some are formal, others are informal; some are simple, others are hugely complex and impossible without a calculator. But regardless of their nature, all electoral systems are attempts to find out what the people really want. For this reason, it’s important that the people themselves know how their own electoral system works, and this article will hopefully be of assistance for that purpose. Rather than using votes from a real nation, we’ll be using a fictional one based on a real election held recently.


The candidates in today’s election

Our subject today is the Commonwealth of Arnott’s Land, a nation with a long and storied history. It is divided into multiple states, the largest and most well-known of which is Assorted Cremaria. Over time, a number of factions have arisen within this great state, and so the electoral commission has been trying to figure out how best to decide which faction should rule the land. There are five groups in total, three of whom can be considered major players (the Kingstonians [KNG], the Monte Carlites [MCL] and the Shortbread Cremarians [SCM]), while the other two are less popular minor factions (the Delta Cremarians [DCM] and the Orange Slicers [ORS]).


The electoral commission held a vote among the people of Assorted Cremaria, and is running the received votes through a number of different systems, both as a demonstration of the way the different systems work, but also in order to determine the winner of winners across them all. There are two groups of systems in place, one of which determines a single winner (like a presidential vote, or a vote for members of a single-member district), and the other which determines multiple winners (mostly used for proportional parliaments).


Assorted Cremaria has chosen to have an American-style system of government, with an elected governor (chosen through single-member elections) and a parliament (chosen through multi-member elections). Each voter has been given one ballot paper, and has been told to number all five parties on the ballot in order of preference. With the votes tallied, the electoral commission can now sort through them using a collection of different electoral systems to determine the winner(s).


(These votes are based on a real poll I did, but I'm lifting the numbers up by 10,000 per vote to make it look a little more 'realistic')


Single-member elections

First-past-the-post This system is the oldest, simplest, most straight-forward of all electoral systems: most votes wins (known as a plurality). In this case, the winner is the party who has the most votes with a number 1. The results are as follows:


MCL: 210,000 (36.2%) KNG: 200,000 (34.5%) SCM: 130,000 (22.4%) DCM: 20,000 (3.4%) ORS: 20,000 (3.4%)


Even though the Monte Carlites received a little over a third of the vote, and only one vote more than the Kingstonians, they win everything by having a plurality of votes. This voting system is very simple, and comes from a time when there was usually only two options on the ballot paper, so there was no reason to express preferences for multiple candidates in order of how acceptable they are.


The ease of counting votes in a FPTP system are one of the few positives it has. The other is that is often results in a majority government in the places that use it, as voters are forced to pick the viable party that they would most want in government, rather than the party that most closely aligns with their beliefs. This can be important in any Westminster system, where the government comes from the majority party in the parliament, which is why Britain — the birthplace of modern parliamentary democracy — still uses this system in its House of Commons, which contains 650 constituencies.


(Image: Constituencies of the British House of Commons by 2017 general election results — By RaviC — removed residual colour code from 2010 election from 2015 base map, Brythones, recoloured by Cryptographic.2014 — This file was derived from: 2015UKElectionMap.svg, CC BY-SA 4.0)

But, as the number of parties contesting elections continues to grow — a result of dissatisfaction with traditional major parties — this system becomes increasingly outmoded and unable to cope. For example, imagine the votes had looked like this:


MCL: 160,000 (27.6%) KNG: 150,000 (25.9%) SCM: 130,000 (22.4%) DCM: 120,000 (20.7%) ORS: 20,000 (3.4%)


With a mere 27.6% of the vote, the Monte Carlites still win, despite nearly three-quarters of voters wanting a different party. In real life, FPTP voting has caused all kinds of problems, some of which remain unresolved issues to this day. At the 1983 British general election, the Labour Party received 27.6% of the vote, and won 209 seats in the House of Commons. Meanwhile, the SDP-Liberal Alliance won 25.4% of the vote, and only 23 seats — 2% less of the vote for 10% of the seats. At the 2015 British general election, UKIP won 12.6% of the vote — comfortably in third place — for a grand total of 1 seat out of 650.


Some countries who appreciate having single-member districts (and the majority governments they often produce), or have multiple presidential candidates to pick from, have sought to solve this by getting voters to express preferences for a number of candidates, which can be done in a few different ways.


First-past-the-post winner: Monte Carlo


Two-round/runoff Two-round (or ‘runoff’) voting is generally used in presidential elections around the world, and is really a pretty simple idea: run two FPTP elections instead of one.


In the first vote, the top two are the winners. Voters then come back a week/month later and vote for one of those top two, and the winner is whichever candidate of those two gets the most votes in the second round.


The first round results would be the same as with FPTP, but with two winners:


MCL: 210,000 (36.2%) KNG: 200,000 (34.5%) SCM: 130,000 (22.4%) DCM: 20,000 (3.4%) ORS: 20,000 (3.4%)


Because our voters have expressed preferences, we can use those preferences to show where voters would put their votes in the second round, with the 170,000 voters for Shortbread, Delta and Orange having to choose between Monte Carlo and Kingston. Here are the results:


MCL: 350,000 (60.3%) KNG: 230,000 (39.7%)


As you can see, the vast majority of voters sided with the Monte Carlites when forced to pick between them and the Kingstonians. This actually reflects a recent real-life result under this system, that being France in 2017. The French really use this system quite well, as they avoid falling into the psychological trap that is still apparent with this voting system: herding.


Voters have just as much reason to vote tactically in a system where FPTP is used twice as they do in one where it is used once. If their most preferred candidate is either very likely or very unlikely to finish in the top two, a voter in a two-round system may choose to vote for someone else, to make sure that they get to the final two instead of a third candidate that they really dislike. This is psychological because the only way to ‘know’ the likelihoods is by either watching opinion polls and hoping they’re right, or asking your neighbours, work colleagues and the like, and hoping they’re right.


Presumably, the French avoid this because they’re French, and so just vote however they want without outside pressures influencing them. The first round in 2017 looked like this:


(Image: Winners of the first round of the 2017 French presidential election by commune — Starus — File:2017 French presidential election — First round — Majority vote (Metropolitan France, communes).svg, CC BY-SA 4.0)

Emmanuel Macron (En Marche!): 24.01% Marine Le Pen (National Front): 21.30% François Fillon (The Republicans): 20.01% Jean-Luc Mélenchon (La France Insoumise): 19.58% Benoît Hamon (Socialist Party): 6.36% Nicolas Dupont-Aignan (Debout la France): 4.70%


Macron and Le Pen went through to the second round, which had a drop in turnout and valid votes cast, but which nevertheless produced a fairly comprehensive win for Macron:


(Image: Winners of the second round of the 2017 French presidential election by commune — Starus — File:2017 French presidential election — Second round — Majority vote (Metropolitan France, communes).svg, CC BY-SA 4.0,)

Emmanuel Macron (En Marche!): 66.10% Marine Le Pen (National Front): 33.90%


The question is, though, how many people voted tactically, to ensure that either Macron or Le Pen was there, knowing that Macron was in a position to unite disparate groups against Le Pen in a way that other candidates couldn’t?


Psychology isn’t the only, or even the main, reason why two-round voting is only really used in certain presidential system. Logistically, it’s also time-consuming and costly, and in other situations (such as parliamentary elections) just needlessly wasteful. For this reason, a similar-but-different system is used elsewhere.


Two-round winner: Monte Carlo


Instant-runoff/preferential Annoyingly, the votes cast for the 2019 Assorted Cremaria election don’t actually produce a different result to the two-round system, so the Monte Carlites win again. However, it’s worth creating a hypothetical situation to show the difference between the two.


Unlike in the previous systems, instant-runoff voting—also known as preferential voting or the alternative vote — asks voters to number candidates in order of preference (this can be optional, allowing voters to number only some candidates, or mandatory, forcing them to number all of the boxes). When these votes are counted, candidates are eliminated in order of the least number of votes until one candidate has at least 50%+1 of all votes. So, let’s make a slightly adjusted scenario like this to start with:


First round MCL: 210,000 (36.2%) KNG: 160,000 (27.6%) SCM: 130,000 (22.4%) DCM: 60,000 (10.3%) ORS: 20,000 (3.4%)


These are called ‘first preferences’. With all the first preferences distributed, the Orange Slicers are eliminated from the count, and their two votes go to whichever candidate is listed second on those two ballot papers.


Second round MCL: 210,000 (36.2%) KNG: 160,000 (27.6%) SCM: 140,000 (24.1%) DCM: 70,000 (12.1%)


As you can see, one vote was given to Shortbread and Delta, but that means that the Delta Creamers now has the least number of votes, and is eliminated.


Third round MCL: 210,000 (36.2%) SCM: 190,000 (32.8%) KNG: 180,000 (31.0%)


Here, crucially, the Delta Cream preferences (along with the Orange Slice vote they initially received) predominantly went to their fellow Creams, rather than to Kingston, meaning that the Kingstonians are eliminated from the count, and their preferences are distributed thusly:


Fourth round MCL: 34 (58.6%) SCM: 24 (41.4%)


With the Kingstonian preferences distributed, the Monte Carlites still finish on top, although the preferences haven’t flowed quite as strongly in their direction compared to when the Shortbread Creamers were eliminated in the two-round system. Theoretically, instant-runoff voting made it possible for the Shortbread Creamers to win after finishing third on the first count, something that wouldn’t have been possible in two-round voting. This is a big advantage for IRV, as it means that voters can preference the collective ‘least worst’ candidate, ie. the candidate who is most agreeable to the most people, and be fairly certain that they’ll end up with that candidate winning at the end of the count.


A typical IRV ballot paper

This system came to prominence first in Australia, where it has been used in lower house elections since 1918. The reason for its introduction was that conservative parties (the Nationalists and the Country Party) were splitting their vote in rural constituencies, which meant that the Labor Party was able to get a plurality under FPTP and win rural seats with 35% of the vote. In 1919, the first election after IRV was introduced, the organisations that would in time become the Country Party won 11 seats, all on the back of IRV.


Even so, instant-runoff voting doesn’t guarantee that the ‘least worst’ candidate in everyone’s eyes gets elected. After all, if a candidate receives 100% of second preferences, they’ll get eliminated first because they didn’t have any first preferences. For this reason, another inventive method of voting has been created.


Instant-runoff winner: Monte Carlo


Borda Under the Borda counts (there are multiple systems, but they all use the same idea), candidates receive a certain number of points based on which preference they receive.

We will use the original Borda count, which designates 5 points for the first preference, then 4, 3, 2, and 1 for last place. I have used this for a similar poll in the past, as I suspect it best represents the overall feeling of an ‘electorate’ in picking the candidate they find most agreeable, collectively.


Under this counting system, the Assorted Cremaria vote would be as follows (please note that four ballots have been excluded on account of not giving full preferences, which makes them impossible to use in this system):


MCL: 18x5, 19x4, 11x3, 5x2, 1x1 = 210 KNG: 20x5, 9x4, 14x3, 4x2, 7x1 = 193 SCM: 12x5, 17x4, 13x3, 7x2, 5x1 = 186 DCM: 2x5, 7x4, 12x3, 26x2, 7x1 = 133 ORS: 2x5, 2x4, 4x3, 13x2, 33x1 = 89


While at first glance the order of these results looks the same as the other systems, there are two things worth nothing. Firstly, the gap between the Delta Creamers and the Orange Slicers is made abundantly clear in this system, in a way it hadn’t been previously. Despite being comfortably behind third place, the Deltas were under no threat whatsoever from the Oranges, even though all the previous counts had them on level pegging on account of first preferences.


Secondly, had the excluded ballots been included, the Shortbread Creamers would’ve overtaken the Kingstonians for second place, and the Monte Carlites lead would be larger. Nonetheless, the winner remains the same.


Borda count winner: Monte Carlo


Condorcet method While we’re using the Borda count, it’s worth spending a little bit of time on the Condorcet method, which isn’t used much in elections but was first theorised (along with the Borda count) by Catalan polymath Ramon Llull back in 1299, and, much like Borda, seeks the most optimal result through preferences. However, the Condorcet method does this through counting the comparative results of different pairings, with the winner being the candidate with the most wins among the pairs (though this varies depending on the specific method used; we will be using Llull’s method).


Nicolas de Condorcet, mathematician after whom the Condorcet method is named. He created (or, rather, recreated) the method after disagreeing strongly with Borda.

In our example, there are 10 different pairings. The winner is whichever candidate has the most higher preferences on a ballot. The results as follows:


Delta Cream — 12 vs 42 — Kingston Delta Cream — 9 vs 49 — Monte Carlo Delta Cream — 45 vs 9 — Orange Slice Delta Cream — 14 vs 42 — Shortbread Cream Kingston — 23 vs 35 — Monte Carlo Kingston — 44 vs 10 — Orange Slice Kingston — 30 vs 26 — Shortbread Cream Monte Carlo — 53 vs 5 — Orange Slice Monte Carlo — 35 vs 23 — Shortbread Cream Orange Slice — 8 vs 48 — Shortbread Cream


While not really telling us necessarily anything we don’t already know, in that the final placings would be Monte Carlo (4 wins, 0 losses), Kingston (3W 1L), Shortbread Cream (2W 2L), Delta Cream (1W 3L) and Orange Slice (0W 4L), it does also draw out how different candidates are perceived — those who prefer Kingstons really like them, but Shortbread Creams are a slightly more well-rounded option (to the extent that only against Kingstonians do the Orange Slicers get into double digits). Delta Creams are unpopular, but still magnitudes more popular than the Orange Slice.


All these single-member votes have taken different routes to come to the same conclusion. We can take that as an indication that, if these results were to be collected together as a presidential/gubernatorial election, there is no doubt which faction the Governor of Assorted Cremaria would come from. However, we still need a parliament, so we’re going to move to a second set of electoral systems to create one.


Condorcet method winner: Monte Carlo


The new Governor of Assorted Cremaria

Multi-member elections

The parliament of Assorted Cremaria is a 40-seat chamber, elected through a variety of proportional and plurality voting systems.


Plurality-at-large/block voting Block voting refers to a number of similar systems that elect multiple candidates from a FPTP system, essentially using the two-round system we used earlier, but finishing after the first round. This can be done in a couple of ways.


Plurality-at-large voting is exactly like what I just described — the candidates with a plurality of votes are the winners. In our example, we will be electing two candidates, which means that the Monte Carlites and Kingstonians both get voted in.


MCL: 210,000 (36.2%) KNG: 200,000 (34.5%) SCM: 130,000 (22.4%) DCM: 20,000 (3.4%) ORS: 20,000 (3.4%)


However, we could also use the majority-at-large method. We will again be electing two candidates, but this time we will add up the total number of first or second votes received by each candidate. In the MAL method, voters get to pick multiple candidates on their ballot, rather than the one in PAL. Using this method, we end up with the following:


MCL: 410,000 (70.7%) SCM: 310,000 (53.4%) KNG: 290,000 (50%) DCM: 90,000 (15.5%) ORS: 40,000 (6.9%)


This is a fairly simple method, generally used by local council to elect councillors, but like its single-method counterpart, it can fall victim to tactical voting quite easily, especially when there are many candidates running.


The first four seats of the parliament are filled (Monte Carlites in red, Kingstonians in Blue, Shortbread Creamers in yellow)

Block voting winners: Monte Carlo & Kingston; Monte Carlo & Shortbread Cream


Single transferable vote From a very simple method to a very complex method, the single transferable vote is one that maths nerds should appreciate, but which also ends up giving a fairly representative result once completed. It is essentially an attempt to recreate the ‘least worst’ method of instant-runoff voting, but with the intention of electing multiple candidates instead of one.


The Australian Senate, the largest parliamentary body in the world to use single transferable vote. (Image: AAP/Alan Porrit)

For this reason, a mathematical equation is used to create a ‘quota’ of votes that candidates much reach in order to get one of the seats on offer. The most common quota is (votes/[seats+1])+1. This method means that the less votes there are, the tougher the quota is to each, due to impact of that final +1. For the sake of mathematics, in our election there shall now be 580,000 votes, and there will be nine seats, so the quota is (580,000/[9+1])+1, which comes out to 58,001. Normally the calculation results in a decimal point — if that occurs, the quota is rounded down to the nearest whole number.

To begin with, we start at the start:


MCL: 210,000 KNG: 200,000 SCM: 130,000 DCM: 20,000 ORS: 20,000


Already, we can see that three parties are well over quota, and two of them have three quotas already. Each party with at least a quota is given a seat in order of votes, and the surplus are transferred to the next candidate in the party after calculating the the ‘transfer value’, which uses the sum (surplus/number of votes for candidate)=transfer value.


With the Monte Carlites having the most first preferences, their first candidate’s votes are transferred to the next Monte Carlite candidate, at a value of 151,999/210,000 which is 0.724. Because all their votes are staying within the party, the number of votes ‘transferring’ is 151,999! So, the Monte Carlites now have one seat, and the vote tally looks like this:


KNG: 200,000 MCL: 151,999 SCM: 130,000 DCM: 20,000 ORS: 20,000


The Kingstonians now have the most votes, and receive a seat, with their votes ‘transferred’ to the next Kingstonian. This continues on until there are no parties with a full quota, which would look like this, with MCL and KNG getting three seats and SCM getting two, leaving one seat left to fill:


MCL: 35,997 KNG: 25,997 DCM: 20,000 ORS: 20,000 SCM: 13,998


At this point, we must begin excluding parties from the ballot, from the bottom. The Shortbreads are the first to be excluded, with their votes redistributed at a value of 13,998/130,000.


MCL: 46,764.692 KNG: 27,073.769 DCM: 21,153.538 ORS: 20,000


As no party has reached a quota, the Orange Slicers are eliminated, and their votes are redistributed at full value as they have not been used to elect any candidate yet:


MCL: 56,764.692 DCM: 31,153.538 KNG: 27,073.769


The Monte Carlites are now very close to a quota, but not close enough to gain the final seat. This means the Kingstonians must be eliminated and their preferences distributed at a value of 25,997/200,000, while the Shortbread preferences move at their previously stated value of 13,998/130,000.


MCL: 77,339.211 DCM: 37,652.788


On account of a very strong preference flow towards them, the Monte Carlites go over a quota and win the final seat. Of the nine seats, the Monte Carlites received four, the Kingstonians three and the Shortbreads two. This reflects the preference flow quite well, as the Monte Carlites were a popular choice for second preferences — even without the 15,000 vote advantage they had once the first eight seats were distributed, they still would have won the final seat. The Delta Creams were also somewhat popular, but their inability to get more first preferences cost them dearly.



Single transferable vote winners: Monte Carlo, Kingston and Shortbread Cream


Party-list proportional Party-list proportional is often named by its proponents as the most ‘fair’ way of determining parliamentary seats, as the seats are distributed purely according to the percentage of votes received. The more seats on offer, the more likely it is that minor parties can get a seat, although they will still be overwhelmed by parties with much larger shares of the vote.


There are a whole host of different counting methods used to distribute seats, but for our election we will be using the d’Hondt method, which is the most oldest and most widely used (and is also known as the Jefferson method in the United States, although it used a different calculation to reach the same result, without fail). For anyone wanting to calculate at home, the method used relies on each party having a quotient of votes. The party with the largest quotient wins a seat, and has its votes divided by the number of seats it has won so far, plus one ie. votes/(seats+1).


This calculation continues until every seat is filled. We won’t go through every single count division here, as there are 29 seats to fill (to give us a 42 seat chamber), but the final result of this system would be:


MCL: 11 seats KNG: 10 seats SCM: 7 seats DCM: 1 seat ORS: 1 seat


It would take until the 27th seat before either the Deltas or the Oranges get a seat, but even with the overwhelming majority of seats being spread between three parties, it’s easy to see the biggest challenge presented by purely proportional parliaments: they rarely create a majority government. The Shortbreads hold the balance of power, as neither the Monte Carlites nor the Kingstonians have enough seats to command a majority.


Typically, this system results in coalition governments. Even the typical arch-shape of proportional parliaments (as opposed to the adversarial, directly facing Westminster-style of parliaments) lends itself to cooperation and coalition. For an example of what happens when this system is taken to its uppermost limit, check out this election preview of the 2017 Dutch general election.


The more proportional the system, the more likely it is that small parties can gain a foothold and stay, making it nearly impossible for a majority government to be elected. Over the past few years, as minor parties have gained a foothold across Europe, there’s been an increase in ‘grand coalition’ governments, comprised of the two historical major parties in those countries. In our example, it would be like the Monte Carlites and the Kingstonians forming a government, leaving the Shortbreads to be the only real opposition.


The filled parliament, using all the above methods

This tends to backfire in the long-run, as the minor parties gain notability from becoming the main opposition, and one or both of the coalition partners suffers from governing with the party that’s meant to be their main rival. For example, current opinion polling in Germany has seen The Greens overtake the century-old Social Democrats, who are in their second term of ‘grand coalition’ with Angela Merkel’s Christian Democrats.


Party-list proportional systems also struggle with local representation, as candidates from a nationwide party can theoretically all come from the same city. However, it is also the best method to ensure ideological representation, as seen in the Netherlands example linked above. Some nations, most notably Germany and New Zealand, attempt to combat this by having a mixed-member proportional chamber, whereby members of parliament can be elected as local representatives or on a party list if they were able to win a single-member district.


Party-list proportional winners: Everyone, but some win more than others


The result There are no perfect electoral systems. Without a direct connection to the brain of every voter, all the time, it is impossible to truly know the ‘democratic will of the people’. Elections are the best method we have available, but all the systems used to count them have positives and negatives. Some are great at ensuring local representation, but fail to adequate demonstrate a full range of ideological beliefs, while others do the opposite. Some over-emphasise the gap between vote totals, but help create a stable majority government that can be thrown out at the next election. Others accurately represent the size of the voting margins, but almost guarantee coalition governments that never really throw a government out entirely.


In our own election-of-the-biscuits, every single system produced the same final result. The winner in first-post-the-post ended up being the same winner in every other single-member electoral system used, even though that was no guarantee. Similarly, using other multi-member systems also resulted in Monte Carlo getting ahead of the others and winning the most seats even though, again, this is no guarantee.


This is a fairly common phenomenon. Most elections will end up with the same winners, regardless of the system used. The question of which system is better is all about the grey area — the first-past-the-post results that are really tight, the proportional results that end up with a minority government where there probably needs to be a majority, the theoretical Condorcet method that would be far too hard to use practically, the single transferable vote that is used and is still difficult to understand at first.


Whichever system is used in your country, remember that — for the most part — the end result in an election would happen under another system to. Weigh up what you think your parliaments should be representation, and try to understand how your current system does and doesn’t do that. Appreciate what it does well. Criticise what it does poorly. Take part in the democratic process when you have the option.


And, of course, all hail Monte Carlo.