Contributor: DAOctor @DAOrayaki

Reviewed by: Shaun @DAOrayaki

原文：Voting Systems | Simple Majority, Ranked Choice & Approval Voting. Plus, a guy called Condorcet.

"Democracy has to be something more important than two wolves and a sheep voting on what to have for dinner."

Raphael Spanochi

--James Bovard, "Lost Rights: The Destruction of American Freedom"

Democracy means voting. The key difference between a dictatorship and a free, free and prosperous democracy is that the people can decide who represents their interests 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, such as a new highway through their villages.

But how is the winner of such a poll determined? It might look as simple as counting votes.

Turns out, this isn't as simple as it sounds. Many bright thinkers have spent their entire academic careers theorizing and classifying voting systems. Few men were as influential as the Marquis de Condorcet.

The Marquis of Condorcet (1743-1794) was a French mathematician and philosopher who made significant contributions to social choice and voting theory. He is often considered one of the founders of voting theory. Perhaps best known for his eponymous ranking-based voting system, The Condorcet Criterion, selects candidates who can beat others in head-to-head competition.

What is a ranking-based voting system?

Why is it important to choose candidates who can defeat others in head-to-head elections?

We will analyze existing voting mechanisms from the perspective of complexity and popularity: simple majority, ranked choice or Condorcet criterion, and yes voting.

For DAO governance, a variety of voting mechanisms can be selected to design polls that best reflect the overall preferences of the DAO, thereby ensuring that minority voices surface and reducing community dissatisfaction.

simple majority

A simple majority vote can mean three things:

Supermajority voting - the candidate who receives more than half of the votes wins. Majority voting - the candidate with the most votes wins, even if they get less than half the votes. Front-runner voting is a special form of plurality voting in which voters are divided into precincts, with each precinct having a unique winner who then votes on behalf of the precinct.

Supermajority voting is only guaranteed to succeed if there are two alternatives (with the possibility of formal abstention as a third). The candidate who obtains more than half of the votes (non-abstentions) wins. Counting is simple with minimal complexity.

If there are only two choices, you can always find a majority.

Formal Abstain is an interesting special option because even if 95% of voters formally abstain, the majority is decided by voters who choose the other two options. Voters often abstain due to protests or messages.

If there are more than two options, a majority vote may not select a winner. Assuming Candidate A gets 40%, Candidate B gets 35%, and Candidate C gets 25%, then no candidate attracts more than 50% of the electorate. So there is no relative majority. Another round between the top two candidates is needed to reach a conclusion.

Majority voting is the name given to relative majority voting in the United States, as well as in the United Kingdom and the Commonwealth. Supermajority voting always produces a winner, even if there are multiple candidates. In the example above, a relative majority vote would make candidate A the winner because that candidate received the most support relative to the other candidates.

First-past-the-post: (FPTP or FPP): A single vote based on gerrymandering, where the front-runner wins. The result is a winner-takes-all situation. This system is used in the UK, US and many other countries around the world. It works the same as plurality voting and always produces a winner.

According to Wikipedia, since 1922, 19 of the 24 general elections in the UK have produced one-party majority governments. In all but two of these elections (1931 and 1935) the leading party did not win a majority of the vote across the UK. This is related to the division of constituencies, political influence and historical factors.

Additionally, all votes cast for candidates other than those with a majority in the constituency are discarded, leading to low voter participation and a lingering sense of disenfranchisement and misrepresentation.

TL;DR: Majority and majority voting are both simple majority voting systems that guarantee an outcome even with multiple candidates. Both also have the disadvantage that there is no guarantee that the results represent the overall preferences and interests of voters.

We need more sophisticated systems that allow for multiple choices and ensure representative winners are as accurate as possible. Rank-based voting and approval voting are further solutions.

Ranked Choice Voting – Condorcet

Condorcet and his followers developed what we know today as ranked choice voting (RCV, or "alternative voting" in the UK). Because of Condorcet's criterion, its name is forever etched in the history of social choice theory.

The Condorcet criterion is met by a voting system that selects candidates who defeat each other in a one-on-one election. The candidate, known as the Condorcet Winner, is seen as appealing to the broadest possible constituency. There are many implementation methods, of which Instantaneous Voting (IRV) is probably the most widely used and the one we will discuss here.

In instant runoff voting, voters also assign a rank to all of their choices. The least-preferred candidates are then eliminated and their respective votes are lumped into each voter's next most-preferred choice. One by one, candidates are eliminated and their votes redistributed until only two candidates remain, one of whom now holds a majority of the votes.

Let's look at an example. Suppose there are three candidates. A, B and C.

Voter 1 ranks A > B > C

Voter 2 ranks B > A > C

Voter 3 Ranking B>C>A

To facilitate calculations, we allocate 3 points to the first choice, 2 points to the second choice, and 1 point to the third choice. Candidate A has 6 points, B has 7 points, and C has 4 points. We eliminate C and reassign voter 3's second choice 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 B is the winner.

The beauty of this system is that it allows voters to express their preferences in greater detail, and no ballots are thrown away. If a preferred candidate is eliminated, the votes are simply redistributed.

The Marquis of Condorcet was a diligent thinker who discovered a special configuration that would throw our system into an infinite loop, making it impossible to determine a winner. Imagine the following scenario, where the three candidates in our hotly contested and very important election rank like this:

Voter 1 Ranking A > B > C

Voter 2 Ranking B > C > A

Voter 3 Rank C > A > B

As you can see, each candidate has the same score, so elimination is not possible, or if you count pairwise matches, you enter a loop with no result. The Condorcet paradox, named after our hero, shows that collective preferences can be cyclical.

The probability that ranked-choice voting will enter a cycle can be calculated by the number of voters and the number of candidates. The more candidates there are, the more likely this outcome is.

This paradox is not purely theoretical, it actually occurs in the real world. A summary of 37 studies, covering a total of 265 real-world elections large and small, found 25 instances of the Condorcet Paradox, with an overall likelihood of 9.4%, which is on the high end of what can be expected. This may be due to selection bias. According to Wikipedia, another analysis of 883 three-candidate elections drawn from 84 real-world ranked-vote elections by the Electoral Reform Association found that the probability of a Condorcet cycle was only 0.7%.

Back in DAO Land, the ENS DAO elected a new manager for the ENS Endaoment using instant runoff voting on November 23, 2022. A large portion of the electorate chose "none of the above," which leads to a surprising dynamic as the election looks like it may fail to select a candidate because no one seems acceptable to the broadest possible community. This is a graphical representation.

Note how Llama lost to Karpatkey in the first round, their votes were attributed to Karpatkey and when they were eliminated they were attributed to none of the above. A small minority of voters who voted "none of the above" chose Llama as their first choice and None as their second choice. We can deduce that there are two basic camps in this election: "We will elect a competent person as president" and "This is all bullshit and we don't want any of this". If voters choose one candidate as their first choice, they are likely to choose another candidate as their second choice rather than "none of the above."

This redistribution allowed Karpatkey to accumulate more votes than Avantgarde, causing the latter to be eliminated in the third round. Karpatkey now gets Avantgarde's designated electorate because no other candidate is available. Notice how Avantgarde's votes were allocated to Karpatkey even though these voters did not choose Karpatkey as their second or even third choice. This is a prominent problem with ranked-choice voting, where voters sometimes feel cheated because their weight is assigned to a candidate they ultimately have no preference for.

Instant-Runoff-Voting can sometimes select the second-worst candidate, which is the candidate who will only win the Condorcet loser, i.e. the candidate who loses to all others in a one-on-one election.

Imagine if someone voted for a strong candidate, and their second and third choices were eliminated before their first choice was eliminated, IRV shifted their vote to their fourth choice candidate, while Not their second choice.

IRV performs better compared to leader-first voting discussed above. FPTP has been shown to occasionally select Condorcet losers, or the worst candidates based on how districts make their selections.

vote in favor

Yes voting is another voting system that is simpler than the Condorcet method. In an affirmative vote, voters can choose any number of candidates.

Suppose there are three candidates. Voters can choose all three, two, one or none. The candidate with the highest total wins the election.

In our example of three voters and three candidates, assume:

Voter 1 chooses candidates A and B

Voter 2 chooses only candidate B, and

Voter 3 chooses candidates A, B and C

A will get two votes, B will get three votes, and C will get one vote.

Yes voting allows every vote to be counted, and since more than one vote can be cast, minority candidates do not suffer as much as strategic voting is common in most other voting systems. Strategic voting is when voters choose a candidate who is not their first choice because they believe their vote will be thrown away if they choose someone they prefer. Note: There are many other forms and strategies of strategic voting and strategic nominating.

Approval voting is easier to understand and implement than a ranking system, but it has some disadvantages: incentives, incentivizing gaming, as voters may split votes to prevent one candidate from winning.

The way DAO is built is different from existing democratic governance

All the voting systems we have discussed so far are designed for closed voting and private voting. When you vote in your country's presidential election, no one can see who voted, and the vote cannot be changed once it's in the ballot box.

Compare this to a DAO, where voting is mostly open and variable. MakerDAO allows delegates and voters to change their choices at the last minute, and allows delegates to be re-delegated during active voting, thus changing the weight of a specific delegate. This has led to surprising results and veritable thrills in particularly contentious polls, such as Luca Prosperi’s proposal for a core ministry of loan supervision.

Most social choice theories can only be mentioned with a lot of caveats, because DAOs are just built in a different way. We encourage governors to experiment with voting systems as well as transparent and private polls to find the best position for their specific communities. One size does not fit all, and it would be nice to see more experimentation in the real world.

Mixing things up occasionally can keep voters engaged and give them a chance to express minority opinions and valuable fringe tactics that would otherwise be buried.