Voting Systems | Simple Majority, Ranked Choice & Approval Voting. Plus, a guy called Condorcet.
“Democracy must be something more than two wolves and a sheep voting on what to have for dinner.”
― James Bovard, Lost Rights: The Destruction of American Liberty
Democracy means participating in polls. The main difference between a dictatorship and a free, liberal, and prosperous democracy is that the populace can decide who represents their interest by participating in open, fair and transparent elections.
Some democracies go a step further and allow citizens to vote on matters that directly affect their lives, like a new highway that would run through their village.
But how is the winner of such a poll determined? It may seem as simple as counting votes.
Turns out, it’s not as simple as it sounds. Many clever thinkers have spent their entire academic careers theorizing and categorizing voting systems. Few are as influential as Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet.
A French mathematician and philosopher born in 1743 and deceased in 1794, the Marquis made significant contributions to social choice and voting theory. He is often cited as one of the founding fathers of voting theory and perhaps best known for a rank-based voting system, which elects the candidate who would beat each other in a head-to-head election.
And right there, in this simple introduction of our hero, we have a door to a giant rabbit hole: “The Condorcet criterion is fulfilled when a voting system elects the candidate who would beat each of the others in a head-to-head election.”
What is a rank-based voting system?
Why is it important to select the candidate that would beat each of the others in a head-to-head election?
Ladies and Gentlemen, please follow us into the wonderful world of voting systems design. We’ll start simple and touch upon the intricacies of three popular systems. The simple majority, the ranked-choice, and the approval voting system. All have their strengths and their drawbacks.
DAO governance is well advised to keep all of them in their toolkit to design polls that reflect the overall preferences of the DAO best, and to ensure that minority voices are surfaced so that no lingering resentments are created or parts of the community are disenfranchised.
Simple majority
We promised you a rabbit-hole and we deliver, straight from the get go.
Simple majority voting can mean three things:
- Majority voting — where the candidate who gets more than half the votes wins.
- Plurality voting — where the candidate with the most votes wins, even if they receive less than half.
- First-past-the-post voting, which is a special form of a plurality vote where voters are divided into districts and each district has a distinct winner who then gets to vote on the districts behalf.
Majority voting is only guaranteed to succeed when there are two options (maybe with formally abstaining as a third). The candidate who gets more than half of all (non-abstaing votes) is the winner. Counting is straightforward and has very little complexity.
Formal Abstain, which means voting Abstain, instead of simply not showing up, is an interesting special option, because even if 95% of all voters formally abstain, the majority is still decided by the voters choosing the other two options. Voters usually abstain out of protest or to leave a message.
If there are more than two options, a majority poll might not be able to choose a winner. Let’s say candidate A gets 40%, candidate B gets 35% and candidate C gets 25%, then no candidate has attracted more than 50% of the voters. So there’s no (simple) majority. Another round between the top two candidates would be needed to come to a conclusion.
Plurality voting is the US American name for relative majority voting, as it’s called in the UK and the Commonwealth. Plurality voting always turns up a winner, even with multiple candidates. In the example above, a Plurality vote would return candidate A as a winner, because this candidate got the most approval, relative to others.
First-past-the-post (FPP or FPTP) voting breaks the constituency down into districts, with all the votes of one district going to the majority holder there. This system is used in the UK, the US and many other countries in the world. It works just the same as plurality voting and always returns a winner.
The voting system has received lots of criticism, mostly because of the way votes are assigned to districts in a winners-take-all fashion. This can lead to outcomes where a candidate with less than the majority of votes in absolute terms still wins the election because they won the right districts.
According to Wikipedia 19 of the 24 general elections since 1922 in the United Kingdom have produced a single-party majority government. In all but two of them (1931 and 1935), the leading party did not take a majority of the votes across the UK. Let us repeat. 17 of 24 general elections in the UK resulted in a candidate winning that had not taken the majority across the UK, simply because they were the first “past the post” in the right districts. Often districts are weighted in a haphazard and highly politicized fashion with huge historical debt. A district might still carry a lot of weight because it had more population than it has now, for instance.
In addition, all votes for candidates other than the one taking majority in a district are discarded, which leads to low voter participation and a lingering feeling of disenfranchisement and misrepresentations.
TL; DR: Both Plurality and First-past-the-post are simple majority voting systems that guarantee an outcome even with multiple candidates. Both also have their drawbacks. They do not guarantee that the result represents the overall preferences and interests of the voters.
We need more sophisticated systems that allow for multiple options and ensure a representative winner to the greatest degree of accuracy possible.
The next two systems we will discuss, Ranked Choice and Approval Voting, are designed to do just that.
Ranked Choice Voting — Condorcet strikes again
The hero of our journey into the voting system rabbit-hole, Jean Antoine Nicolas de Caritat, the Marquis of Condorcet, returns in this chapter. He and his followers developed the Ranked Choice Voting (RCV, or “Alternative vote” in the UK) as we know it today. His name is engraved into the history of social choice theory forever, thanks to the Condorcet criterion.
The Condorcet criterion is satisfied by voting systems, which select the candidate that would beat each other in a one on one election. This candidate is called the Condorcet Winner and is deemed the one appealing to the broadest set of the constituency.
Ranked choice voting systems are called Condorcet methods, in honor of our hero. Not all Ranked Choice voting systems satisfy the Condorcet criterion. There are numerous implementations of which Instant-Runoff-Voting (IRV) is probably the most widely used, and the one we will discuss here.
Voters also assign a rank to all their choices for Instant Runoff Voting. The least favorite candidate is then eliminated and the respective votes are attributed to their next favorite choice of each voter. One after the other candidates are eliminated and their votes redistributed until only two candidates are left, one of which now has the majority of votes.
Does this sound a bit confusing and weird? Let’s look at an example: Say there are three candidates: A, B and C.
- Voter 1 ranks A > B > C
- Voter 2 ranks B > A > C
- Voter 3 ranks B > C > A
To make calculation easy we assign 3 points to first preference, 2 to second and 1 to third. Candidate A has 6 points, B has 7 and C has 4. We eliminate C, and redistribute the second choice of Voter 3 to candidate A.
- Voter 1: A > B
- Voter 2: B > A
- Voter 3: B > A
Now A has 7 points, B has 8 points. And the winner is: Candidate B. Cheers, Champagne, wild riots in the streets celebrating their hero erupt… At least in our example.
The beauty of this system is that it allows voters to express their preferences in much more detail, and no vote is discarded. If a preferred candidate is eliminated, the votes are simply reassigned. Did we just enter the Valhalla of voting systems? Alas, no.
The Marquis of Condorcet was a diligent thinker and identified a special configuration that would send our system into an endless loop, making it unable to determine a winner. Imagine the following scenario, where the three candidates in our highly contested and very important election are ranked in this way:
- Voter 1 ranks A > B > C
- Voter 2 ranks B > C > A
- Voter 3 ranks C > A > B
As you can see every candidate has the same amount of points, no elimination is possible, or if you count pairwise matches you enter into a cycle, with no result. Named after our hero, the Condorcet Paradox shows that collective preferences can be cyclic.
The likelihood of a ranked choice vote entering into a cyclical loop can be calculated by the amount of voters and the number of candidates. The more candidates the likelier such an outcome becomes.
The paradox is not pure theory but actually appears in the real world, too. A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4%, which is on the high end of what can be expected, likely due to selection bias. Another analysis of 883 three-candidate elections extracted from 84 real-world ranked-ballot elections of the Electoral Reform Society found a Condorcet cycle likelihood of just 0.7%, according to Wikipedia.
Back in DAO Land, ENS DAO employed Instant Runoff Voting to choose the new manager of the ENS Endaoment on 23 November, 2022. A large percentage of voters chose “None of the Above’’ which led to surprising dynamics, because the election looked like it might fail in choosing a candidate, as none seemed acceptable to the widest possible community. Here’s a graphic representation.
Notice how Llama loses to Karpatkey in round one, and their votes get attributed to Karpatkey and to None of the Above as they get eliminated. The tiny fraction going to None of the Above are voters who chose Llama as their first choice and None as their second. We can deduct that the election had two basic camps: “We’ll chose a capable person as the chair”, vs “This is all bullshit, we don’t want any of these”. If voters had chosen a candidate as their first option they’ likely chose another candidate as their second, and no “None of the Above”-
This redistribution allowed Karpatkey to accumulate more votes than Avantgarde leading to the latter’s elimination in the third round. Now Karpatkey got assigned voters from Avantgarde, because no other candidate was available. They achieved majority support because they now command more votes than None of the Above. Notice how votes for Avantgarde got assigned to Karpatkey even if those voters did not choose Karpatkey as their second or even third choice. This is a prominent issue with rank-choiced voting, that voters sometimes feel cheated because their weight got assigned to a candidate that they had no preference for in the end.
If you’re intrigued about the dynamics, please read Alex Van de Sande’s excellent Twitter thread for more information.
Instant-Runoff-Voting can sometimes select the second-to-worst candidate, this is the one who would only win over the Condorcet loser, the one candidate who would lose against all others in a one on one election. Imagine, if someone voted for a strong candidate, and their 2nd and 3rd choices are eliminated before their first choice is eliminated, IRV gives their vote to their 4th choice candidate, not their 2nd choice.
Compared to First-Past-The-Post voting, discussed above, IRV performs better. FPTP has been shown to occasionally select the Condorcet loser, or the worst possible candidate as a result of how districts make their choices.
For now, we’ll leave the Marquis behind to engage in his various duties and turn to another voting system that proposes to eliminate negative campaigning (dirt campaigns) and incentivize centrist, reasonable behavior by candidates.
Approval voting
Approval voting is another voting system, and one that’s simpler than the Condorcet method. In Approval voting, the voter can select any number of the candidates.
Let’s say there are three candidates. The voter can select all three, two, one, or none of them. The candidate with the most total votes wins the election.
In our example with three voters and three candidates, let’s say that:
- Voter 1 chooses candidates A and B
- Voter 2 chooses candidate B only and
- Voter 3 chooses candidates A, B and C
A would receive two votes, B three and C one. Candidate B would be the elected winner.
Approval voting allows every vote to be counted, and since more than one vote can be cast, minority candidates are not suffering as much from tactical voting common in most other voting systems. Tactical voting is when voters select a candidate that is not their first choice, because they believe their vote would be discarded, if they select the one they’d prefer. Note: there are many other forms and strategies for both tactical voting and tactical nominations. If you’d be interested in reading an article on that, leave a comment or reach out to us on Twitter.
Approval voting is easier to understand and implement than ranked systems, but it has some drawbacks. Most importantly, it does not guarantee that the Condorcet Winner will be elected, because voters cannot express their preference between candidates. It also incentivizes gaming, as a voter may split their votes to prevent one single candidate from winning.
DAOs are built differently
All of the voting systems we discussed so far have been designed for closed ballots and private voting. When you vote in your country’s presidential elections, no one can see who you voted for, and you can’t change your vote once it’s in the ballot box.
Compare that to DAOs, where voting is mostly open and changeable. MakerDAO allows delegates and voters to change their choices up to the last minute and allows delegators to redelegate during active polls, changing the weight a particular delegate has. This has led to surprising outcomes and veritable thrillers during especially contested polls, like Luca Prosperi’s Lending Oversight Core Unit proposal.
Most of social choice theory can only be referred to with huge caveats, because DAOs are just built differently. We encourage governance to experiment with voting systems and also with transparent vs private polls, to find the sweet spot for their particular community. We think that one approach doesn’t fit everywhere and would love to see more experimentation in the real world as well.
Mixing things up once in a while keeps the constituency engaged and has a chance to surface minority opinions and valuable fringe strategies that would get buried otherwise.
Time to wrap it up
What a journey. We covered Simple Majority voting, which turned out to be quite intricate. We visited the Marquis of Condorcet and discovered his clever voting method; and we touched on Approval voting, a simple and powerful voting system designed to give minorities more say.
Funny enough when we searched for quotes by our hero, the Marquis, by far the most popular is his adage:
“Enjoy your life without comparing it to that of another!”
We’d like to paraphrase it: Enjoy your DAO without comparing it to another, and experiment with voting systems until you get a deep understanding of when to apply which system to get the best and most inclusive results possible.
We hope you enjoyed this article. We’re always open to chat about DAOs and governance, and are actively looking for delegations on a variety of protocols. If you want to entrust us with voting on your behalf, just click the links below to get to the respective delegation page:
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