The Source of DAO democratic voting and Condorcet Paradox

原文标题：《 DAOrayaki ｜DAO 民主投票源头与孔多塞悖论 》

Original article by Raphael Spannocchi

DAOctor@DAOrayaki

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

Raphael Spanoki-James Bovard, Lost Rights: The Destruction of American Liberty

Democracy means voting. The main difference between a dictatorship and a free, free and prosperous democracy is that people can decide who represents their interests by participating in open, fair and transparent elections.

Some democracies go a step further, allowing citizens to vote on matters that directly affect their lives, such as a new highway through their village.

But how do you determine the winner of such a vote? It may seem as simple as counting votes.

As it turns out, that's not as simple as it sounds. Many bright thinkers have spent their entire academic careers theorizing and categorizing voting systems. Few have been as influential as the Marquis de Condorcet.

Marquis de Condorcet (1743-1794) was a French mathematician and philosopher who made major contributions to the theory of social choice and voting. He is often credited as one of the founders of voting theory. It is probably best known for its eponymous rank-based voting system, The Condorcet Criterion, which selects candidates who can beat others in head-to-head competition.

What is a rank-based voting system?

Why is it important to choose candidates who can beat others in a head-to-head election?

We will look at existing voting systems from the perspective of complexity and popularity: simple majority, ranked choice or Condorcet criteria, and approval voting.

For DAO governance, multiple voting mechanisms can be selected to design polls that best reflect the DAO's overall preferences, thus ensuring minority voices are heard and reducing community discontent.

Simple majority

A simple majority vote could mean three things:

Supermajority vote - the candidate who gets more than half of the votes wins. Relative majority voting - the candidate with the most votes wins, even if they get less than half the votes. Front-runner voting, which is a special form of majority voting, sees voters divided into districts, each with a unique winner, who then votes on behalf of the district.

A supermajority vote guarantees success only if there are two options (with the possibility of formal abstention as a third). The candidate who gets more than half (non-abstention) wins. Counting is simple, with very little complexity.

If you have 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 those who choose the other two options. Voters often abstain because of protests or messages.

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

Supermajority voting is the American term for a relative majority vote, and it is the same 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 of votes would make candidate A the winner because that candidate received the most support relative to the other candidates.

Front-runner is elected: (FPTP or FPP) : A single vote by way of a demarcated district, the front-runner will win. The result is a winner-takes-all situation. The United Kingdom, the United States and many other countries around the world use this system. It works the same way as relative majority voting and always produces a winner.

According to Wikipedia, 19 of the 24 general elections in the UK since 1922 have resulted in a single-party majority government. In all but two of these elections (1931 and 1935), the leading party did not win a majority across the UK. It has to do with district boundaries, political influence and history.

In addition, all votes cast for candidates except those who have a majority in the constituency are discarded, leading to low voter participation and a lingering sense of disenfranchisement and misrepresentation.

TL; DR: Both the relative majority and supermajority voting systems are simple majority voting systems that guarantee results even if there are multiple candidates. Both also have the disadvantage of not guaranteeing that the results represent the overall preferences and interests of the electorate.

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

Rank Choice Vote -- Condorcet

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

The Condorcet standard is met by the voting system, which selects candidates who beat each other in one-on-one elections. The candidate is known as the Condorcet Winner and is seen as appealing to the broadest constituency. There are many implementations, of which instant voting (IRV) is probably the most widely used and the one we'll discuss here.

In instant runoff voting, voters also assign a rating to all their choices. The least favorite candidate is then eliminated, and the respective number of votes is relegated to each voter's next favorite choice. One by one, candidates are eliminated and their votes are redistributed until there are only two candidates left, one of whom now has a majority.

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

Voter 1 ranked A > B > C

Voters ranked second. A > C

Voters 3 ranked B> C> A

For ease of calculation, we assign 3 points to our first choice, 2 points to our second choice, and 1 point to our third choice. Candidate A gets 6 points, candidate B gets 7 points and candidate C gets 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 seven points, B has eight, 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 discarded. If a preferred candidate is eliminated, the votes are simply redistributed.

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

Voter 1 ranked A > B > C

Voters ranked second. C > A

Voters ranked third. A > B

As you can see, each candidate has the same score, so it is impossible to eliminate, or if you calculate paired matches, you enter a loop with no result. The Condorcet paradox, named after our hero, shows that collective preferences can be circular.

The likelihood of rank-choice voting entering 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 Condorcet's paradox with an overall likelihood of 9.4 percent, which is at the high end of what can be expected, possibly due to selection bias. According to Wikipedia, another analysis of 883 three-candidate elections drawn from the Electoral Reform Institute's 84 real-world ranked voting races found that the likelihood of a Condorset cycle was just 0.7%.

Back at DAO Land, ENS DAO used an instant runoff vote on November 23, 2022 to choose a new administrator for ENS Endaoment. A large percentage of voters chose "none of the above," which led to a surprising dynamic as the election looked like it might not be able to choose a candidate because there seemed to be no one acceptable to the widest possible community. This is a graphical representation.

Note how Llama lost to Karpatkey in the first round, their votes were attributed to Karpatkey and were attributed to none of the above when they were eliminated. A small percentage of voters who voted for "None of the above" chose Llama as their first choice and None as their second. We can infer that there are two basic camps in this election: "We'll elect a competent person as chairman" and "This is all crap, we don't want any of this". If voters choose a candidate as their first choice, they may choose another candidate as their second choice instead of "none of the above."

This reallocation allowed Karpatkey to accumulate more votes than Avantgarde, resulting in the latter's elimination in the third round. Now Karpatkey gets Avantgarde's designated voters as there are no other candidates available. Note how Avantgarde's votes were allocated to Karpatkey even if 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 candidates they ultimately have no preference for.

Instant-Runoff Voting sometimes selects the second-worst candidate, which is the candidate who will only win the Condorcet loser, 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, and IRV gave their vote to their fourth choice candidate instead of their second choice.

IRV performs better than the leader preference voting discussed above. FPTP has been shown to occasionally pick Condorcet losers, or the worst candidates depending on how the region makes its choices.

Affirmative vote

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

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

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

Voter 1 chooses candidates A and B

Voter 2 only chooses candidate B, and

Voter 3 selects candidates A, B, and C

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

Approval voting allows every vote to be counted, and since more than one vote can be cast, minority candidates do not suffer as much as the tactical voting 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 discarded if they choose someone they prefer. Note: There are many other forms and strategies for strategic voting and nomination.

Approval voting is easier to understand and implement than a ranking system, but it has some drawbacks: it incentivizes, stimulates gaming, because voters may split their votes to prevent one candidate from winning.

DAO is constructed in a way that is different from existing democratic governance

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

Compare this to the 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 reauthorize during active voting, thereby changing the weighting of a particular delegate. In particularly contentious polls, this has led to surprising results and veritable throngs, such as Luca Prosperi's proposal for a core unit of loan supervision.

Most social choice theory can only be mentioned with plenty of caveats, because DAOs are simply constructed in different ways. We encourage governors to experiment with voting systems and transparent and private polls to find the best locations for their particular communities. One size does not fit all, and it would be nice to see more experimentation in the real world.

Mixing things up occasionally keeps voters engaged and gives them a chance to voice minority opinions and valuable brinkmanship that would otherwise be buried.

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