In a representative democracy positions must be filled with positions using elections. The electoral system one chooses will have profound effects on the composition of positions, voter behaviour and the dynamics of democracy. This article is my examination of various proportional voting systems, including my personal notes on them for future reference. I was always fascinated by voting systems mainly because I noticed early on that they can result in radically different outcomes, even though the preferences of the group have not changed. I personally value proportionality (minority representation), fairness and preventing vote wasting the most. In the last Slovenian parliamentary elections I voted for the Pirate Party, but since the party didn’t make the threshold I effectively wasted my vote. Slovenia uses a national open party-list system with ensured regional proportionality and while the system is fair and proportional, resulting in a healthy multi-party democracy, it is not perfect and many people strategically vote for larger parties (or those likely to clear the threshold). It is thus fun to theorise and examine the many different kinds of proportional voting systems around the world.
Cluster theory#
Voting isn’t just a collective choice mechanism, it is a method to select the best choice among many given very diverse preferences. A group’s preferences, where individual preferences are points, are multidimensional, where the dimensions are some metric like the position on some issue, candidate quality or general attiude (such as the authoritarian-libertarian axis in the famous political compass). A voting system is thus set to achieve a successful identification of preference clusters and their sizes.
Of course there is great variance with what to do with this information. Some voting systems select a single large cluster, whereas others try to select multiple clusters and grant them the number of seats proportional to cluster size. It is possible to see elections as a machine learning-like classification problem, except that it is the voters themselves that do the identifying and that preferences may change over time.
The distance between a voter and a candidate determines the strength of preference.
Criteria for evaluating proportional voting systems#
There are many criteria which can be used to evaluate voting systems. I won’t cover them all, but the most important ones will be listed here.
- Resistance to strategic voting
In many voting systems you have to vote strategically to improve the odds of a desired outcome. The spoiler effect means that voting for your favourite choice, which does not win, negatively affects your other choice. This effect is related to the clone effect where introducing a candidate similar to another one can lead to both losing. In voting systems that are susceptible to such effects, you have to vote for the lesser evil or fill out the ballot dishonestly. A voting system that is well designed should be as strategy-proof as possible and prevent favourite betrayal. You should be able to vote for whom you want to vote for independent of other’s preferences without compromising the outcome.
Note that no voting system is immune to strategic voting because of Gibbard’s theorem.
- Minimisation of vote wasting
A good voting system should prevent vote wasting.
- Proportionality
A voting system is proportional if its result accurately represents the factional composition of the electorate. Only multi-member voting systems can be proportional. By becoming more representative, at least in theory, the policies passed will more closely reflect the will of the people. This is in my opinion one of the most important features.
- Prevent excessive fragmentation
Proportionality can, however, lead to an obscene amount of political parties making it into a legislature, making governing hard, so finding the optimal cluster resolution is hotly debated.
Map#
Below is a simple map showcasing the families of the various voting systems. I bet there are a lot more.
Single-winner voting systems#
I want to briefly go over single-winner elections. In some countries these are used for positions like presidents or mayors, but in others they may be used to elect representatives. All in all, single-winner systems will always produce suboptimal results . A great overview of single winner systems is here:
First past the post (FPTP)#
The UK and other former British colonies such as Canada, the US and India all use first-past-the-post. In this voting system the candidate is selected which has the most votes. This is a terrible system and will inevitably result in two-party systems. It’s not really democratic nor fair, because it is possible to win elections with minority support (say 30%) leading to extreme tactical voting, limiting voter control and maximising vote waste. In a FPTP-election, an honest voter is effectively handing the win to his opposite preference due to the spoiler effect. Such systems are extremely susceptible to gerrymandering, where constituencies are drawn in a way that increases the odds of a party winning, effectively deciding the outcome of the election before the election itself. The UK is notorious for having extremely unrepresentative elections. In the 2024 British parliamentary election Labour only won 33% of the vote, but a very large majority in the House of Commons and 100% of the control.
Some countries tried alleviating FPTP such as France which uses run-off elections for the top two candidates in each constituency, but it doesn’t fix elections at all. FPTP systems simply select the largest cluster cluster among the clusters (the plurality).
Ordinal voting systems#
Ordinal voting systems use ranked ballots to capture the voters’ preferences. All ranked choice ballots all suffer from Arrow’s theorem meaning that all have some flaw, though since most elections are roughly on a left-right axis, there exists a median voter thus weakening the negative effects from Arrow’s theorem.
Ranked choice voting#
Australia notably uses ranked choice voting for their parliament. It improves FPTP by introducing ranked ballots and it fixes the spoiler effect. Unfortunately, however, it doesn’t make the result proportional at all, tends to produce centre-squeeze (a phenomenon where centrist candidates have close to no chance of winning) and the election outcome can be radically different depending on the order of candidate elimination. If the voters have access to polls they can vote tactically to change their rankings and thus change the outcome. It also doesn’t minimise vote waste, as slightly less than half of a constituency will be left with a representative they do not like.
Ranked choice voting terminates smaller clusters and adds them up until there are only two clusters and the biggest one wins.
Condorcet methods#
I personally find the Condorcet condition to be the most desireable feature among ranked choice systems
A candidate is a condorcet winner if they would receive more support than their opponent in a one-on-one race against any one of their opponents. This idea can also be expanded to a multi-winner context, where the optimal combination of candidates is referred to as the Condorcet committee. The condorcet criterion is the gold standard of single-choice voting criteria, in terms of cluster theory, it ensures that the outcome is truly prefered by the majority of voters and not just a plurality. A condorcet winner does not always exist, asides from ties, it is possible that there are preference cycles. If A prefers B and B prefers C, while C prefers A, then no majority winner exists. This is called the condorcet paradox, but it is quite rare in real politics. When candidates are tied or in a Condorcet cycle they are said to be in the smith set.
The reason why I like Condorcet methods is that they’re more strategy-proof. One of the voting systems that achieve this are Ranked pairs and Schulze’s method. Voters still rank their choices. However, in regular ranked voting, candidates are eliminated based on first-preference votes and transfers. Where a candidate is placed is completely arbitrary and a candidate who would be most liked by all people (i.e. a condorcet winner) could be eliminated fairly early on. Imagine a scenario where a person X is ranked 2nd by everyone, but even though everyone likes the person, they have no first votes and are immediately eliminated, but this doesn’t happen under Condorcet. In fact in Condorcet the most liked, typically consensus, candidate is likely to win. The image below by Closed Limelike Curves shows the ideological distribution of winners on a simulated political compass in different electoral systems, showing that Condorcet does not lead to centre-squeeze:
Ranked pairs has been shown to be even more resistant to strategy than Schulze as given by David C. Parkes and Lirong Xia’s paper and satisfies more theoretical properties, such as local independence of irrelevant alternatives. schulze is beautiful though much more complicated to understand:
Awesome, so we should just elect legislatives using ranked pairs in single-winner constituencies? Not so fast. If you think about it, such a system would produce a dictatorship of the centre. Condorcet winners would always be very lukewarm, appeal-to-all candidates which would be members of the political centre. I find this very unfair, because it would make a system very unaccountable and unrepresentative. Why should 10% of the population get 100% of the control? Centrism is not inherently more right just because it is a median ideology and we do not need to give up proportionality to have good continuity, political coalitions in parliamentary systems ensure more moderate policies, while staying very representative.
Cardinal voting systems#
Cardinal voting systems obtain the most data from the voter. We get to evaluate our preference of the candidates on a range (for example from 1 to 10 or just from 0 to 1). Cardinal voting systes are most notably not subject to Arrow’s theorem.
Approval voting#
Approval voting is an easy-to-count system that produces very good results. People are allowed to vote for any candidate they approve of. This system is great for groups quickly deciding on what to do. It is super easy to count, very fair and it is bound to find a candidate that is approved by as many people as possible. Tactical voting is still possible and it does not always ensure a condorcet winner. However, electing assemblies with it would also result in a dictatorship of the centre. Approval voting effectively hands the victory to the candidate who sits at the greatest intersection of preferences.
venn-beta set A[“Left”]:30 set B[“Right”]:28 union A,B[“Centre”]:42
Approval voting is used in my company for example to decide what to eat during lunch and it’s a great way of choosing a meal. However, when you repeat the same decision many times, it results in most of the time selecting pasta with vegetarian sauce. Neither pasta nor the vegetarian sauce is the problem. The issue with the system is that it leads to a lack of variety, assuming everybody votes honestly.
However, approval voting is not immune to strategic voting as shown in the simulation below:
Approval voting does not suffer from favourite betrayal, but if too many people vote strategically it turns into FPTP.
Score voting#
Score voting is the more expressive version of cardinal systems and it gives the most power to the voter by allowing us to express our preferences with scores. I really like this idea because when we rank candidates like A>B>C>D we do not really express the strength of our preference, we may like A and B roughly equally, while we strongly dislike C and D. It will also produce the most liked candidate with even more accuracy than with approval voting thus maximising happiness. Other than that it would still lead to a centrist dictatorship.
It is not free from strategy, however. Just like with approval voters are incentivised to vote dishonestly to make their candidate win by giving similar candidates a lower score than their first preference. If this is the optimal strategy then you can’t really support your second choices well in case your first choice does not make it, thus from a stratigic point of view leading to vote waste. In fact in the video above score voting does not necessarily produce the highest happiness for society. Here’s a scenario: Voters for the condorcet/majority winner (60%) vote honestly, while a radical minority (40%) votes dishonestly. This would sway the election producing minority rule. For this reason STAR voting poses an interesting alternative. It takes score voting but then puts the top two candidates to a runoff, but this takes its elegance away and does not really change that much, you get the French run-off FPTP system from a strategic point of view.
But assuming everyone votes honestly, the difference between the utilitarian and condorcet winner gets interesing. Suppose three friends want to go out and eat a Pizza together. One of the three is a vegetarian and of course the group is going to order a vegetarian Pizza, otherwise you’d be bad friends. The condorcet choice would not reflect that and would lead to salami pizza being selected (afterall 2 prefer it), whereas with score voting, a vegetarian pizza would be selected. Though this depends on the framing because most of the time we would get the same result, suppose there are multiple pizzas, then the rankings could look like this:
Salami > Vegetable+Cheese > Chocolate Pizza for the two omnivores and
Vegetable+Cheese > Chocolate Pizza > Salami for the vegetarian
The Condorcet winner then would be Vegetable+Cheese because it wins 4 times, while salami only wins 2 times and chocolate pizza 1 time in head-on-head comparisons. In fact the paper by Marcus Pivato shows that if the statistical distribution of utility functions in a population satisfies a certain condition (which it highly likely does in large populations), then a Condorcet winner will not only exist, but will also maximize the utilitarian social welfare function.
Opinion on single winner systems#
In my opinion if a group of friends needs to make a one-time decision, for example where to eat, approval voting is the best. For more serious single-winner elections, for example for the mayor, however, I would pick ranked pairs. Range voting has the nice property that Bayesian regret is minimised in a range-based election, but you can’t truly vote honestly, ranked pairs is less likely to lead to vote waste and is significantly more strategy-proof.
No matter how we spin it, single-winner systems will never be truly representative in legislative elections nor maximise happiness. No matter how optimised the voting system may be, the preference of a majority might be very different to that of a minority.
The Analysis#
We’ve had three homework assignments and the third one was optional. It’s quite clear from the charts that most students visited us right before the deadlines.
This is interesting. Most students visited us on Tuesdays and Wednesdays. I believe there are two reasons for this. Firstly, most lectures at the Hauptcampus are between Tuesday and Thursday and, secondly, the deadlines were on Wednesdays.
The average visit count by tutor was 8.25 students, I got 4 in total, which is on the low side. I think that’s mainly because the distribution of students mostly correlated with deadlines and my tutorial on Friday was always quite away. All in all, the results weren’t too surprising
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