Web3 has succeeded in making “governance” cool again. Whether through air responsibility drops or treasury bribes, active participation has been brought to the crypto masses. Or has it?

Looking at DeFi protocol turnout rates, they’re abysmal. With participation typically ranging from 7-15%, we can hardly call this “democratic”. Given that these proposals use token-weighted voting, this is little more than an apathetic plutocracy.

Thankfully, I’m not here to talk about that since it’s not my area of expertise. My final college paper was on the Mathematics of Voting Systems and the various ways they fail us. I’d like to run through a quick overview of favorable properties we’d like our voting systems to have, as well as the ways these break down.

Voting

This assumes a 2 option vote (with an odd number of voters)

“Fairness” Criterion

I say fairness in quotes because the users should be defining what’s “fair”. Different projects, protocols, and ecosystems will have their own way of structuring elections. These conditions are simple axioms that guarantee some sort of equality between participants. They’re stated as follows:

• Anonymity –– All voters are treated equally, i.e. if any two voters switched preferences, results would remain unchanged.
• Neutrality –– All candidates are treated equally, i.e. if every voter switched preferences between two options, the position of these options would be switched in the final results.
• Monotonicity –– An individual voter preference increase for an option must result in an overall increase for that option in the final results, i.e. if a single voter switched preferences for two options, updated final results must place the new option higher than before.

Already, we see token-weighted voting breaks anonymity, as certain voters have higher voting power than others. Switching a voter with 100 tokens for one with 1000 tokens will definitely change results.

This is good and all, but what kind of voting systems guarantees these properties.

May’s Theorem

• May’s Theorem –– Majority rule is the only system that is anonymous, neutral, and monotone, while avoiding ties.

Majority Rule

Unsurprisingly it’s exactly what it sounds like. This is an obvious one that should be implemented by default.

It states:

• Majority Rule –– An option must receive more than half the votes to be declared the winner. If no option receives more than half the votes, no winner is selected.

Cool, so you need >50% of participants to cast their vote for a proposal option for it to be adopted. Originally restricted to an odd number of voters, this has since been generalized to infinite populations.

Given a vote between two options, the “fairest” is a simple majority vote

In smaller operations, like a cooperative, democratic votes should implement majority rule as the default.

Now, onto > 2 options per vote

Given more than 2 options, additional desirable properties –– such as Condorcet Outcomes –– pop-up for us to integrate.

Condorcet’s Outcome

As decided by majority rule, they state:

• A candidate in an election who would defeat every other candidate in a head-to-head contest is said to be the Condorcet winner
• A candidate in an election who would lose to every other candidate in a head-to-head is said to be the Condorcet loser
• A voting system that will always elect a Condorcet winner, when one exists, is said to satisfy the Condorcet winner criterion (CWC)
• A voting system that will never elect a Condorcet loser, when one exists, is said to satisfy the Condorcet loser criterion (CLC)

Plurality Voting

Plurality voting (first-past-the-post) –– what we use in the US for federal elections –– is the worst way to run elections.

Plurality systems do NOT require you to have a majority of votes, simply the most votes. We’re familiar with the 2000 and 2016 presidential elections, but the 1998 Minnesota Governor’s race is a perfect example. Jesse Ventura, the reform candidate, won with 37% of the vote, with republican and democrat candidates coming in at 34% and 28% respectively. I’m finding conflicting studies, but my instinct is Jesse would have been the Condorcet Loser had he been matched head-to-head against the other two.

Not only does Plurality not satisfy the CWC, but it doesn’t even satisfy the CLC!!! Pitiful honestly. We can do better.

Improved Framework

First, we’d like our decision to remain unaffected by option “noise”, call this:

• Independence of Irrelevant Alternatives (IIA) –– The resulting relationship between option A and option B must depend solely on individuals preferences between A and B, i.e. if every voter altered their preference for X without altering their preferences for A and B, then the final results between A and B must remain unchanged

This is kind of a mouthful, but in the case of plurality, it prevents a “spoiler candidate” from altering the outcome of elections. Some truly wild stuff there you should check out. This would also mitigate DAO2DAO takeovers through malicious proposals.

Quickly, transitivity states that if overall preference looks like A > B > C, then it must be that A > C. This helps us avoid Condorcet’s Paradox.

Arrow’s Conditions and Theorem

These can be seen as a more generalized, slightly weaker set of “fairness” conditions. They go as follows:

• Independence of Irrelevant Alternatives (IIA) –– Defined above
• Universality –– No restriction (other than transitivity) must be placed on the voters, i.e. systems should not dictate that some preference orders are acceptable while others are not; every possible collection of transitive preference ballots must yield a transitive societal preference order.
• Non-dictatorship –– There should exist no voter v such that if v prefers A over B, then society will also prefer A over B.
• Pareto Condition / Unanimity –– If there’s a pair of options in a vote such that every voter prefers A over B, then A must be ranked higher than B in the resulting preference order.

This is good and all, but Nobel laureate Kenneth Arrow isn’t done with us yet:

• Arrow’s Theorem –– Given more than two options, it’s impossible for a voting system to satisfy IIA, Universality, Unanimity, and not be a dictatorship.

Yep! Even if everyone voted according to their preference, there’s a pivot voter whose ballot necessarily decides the outcome of the election. Scary stuff. No matter how hard we try, it’s impossible for us to construct a ranking system –– Arrow’s only applies to ranked voting systems, as opposed to cardinal voting systems, since they violate universality –– that meets every property we want. As Arrow himself says, “Most systems are not going to work badly all of the time. All I proved is that all can work badly at times”.

For us, it’s a matter of choosing which condition we’re willing to forgo in order to meet the rest. From Ranked Choice Voting to STAR to Approval Voting, electoral systems are a deep and fascinating rabbithole to fall down.

There’s so much more nuance and discussion to have here; more conversations on this topic would help educate new designers on the best way to structure their systems.

An area I haven’t explored enough yet is quadratic voting, which claims to address some of these problems. I’d like to actually get into the math of it and flex some proofs, but we’ll see how my motivation is doing. I’m looking forward to reading the original paper and seeing what it has to say.

Update: Conviction Voting is also an interesting development which could be combined with Quadratic Voting

If you’re into voting system design, and potentially voting minimization as well, I’d love to hear from you on my twitter :)