# Voting Systems Stuff By [mycelia 🌌](https://paragraph.com/@mycelias) · 2022-01-08 --- 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**](https://dune.xyz/0xoxox0/Unvoted-Voting-Power). With participation typically ranging from 7-15%, we can hardly call this “democratic”. Given that these proposals use [token-weighted](https://zefram.xyz/2019/02/why-voting-tokens-are-fking-horrible.html) 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**](https://en.wikipedia.org/wiki/May%27s_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](https://rangevoting.org/FeyMay.pdf) 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](https://en.wikipedia.org/wiki/1998_Minnesota_gubernatorial_election). 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](https://www.sun-sentinel.com/news/politics/os-ne-florida-ghost-candidate-scandal-explained-20211230-urxkvy52ybb4njkqvb5qek3nay-story.html)” 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](https://en.wikipedia.org/wiki/Condorcet_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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Cardinal_voting), 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](https://ballotpedia.org/Ranked-choice_voting_\(RCV\)) to [STAR](https://en.wikipedia.org/wiki/STAR_voting) to [Approval Voting,](https://en.wikipedia.org/wiki/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](https://vitalik.ca/general/2019/12/07/quadratic.html), 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](https://twitter.com/myceliastargaze) :) --- *Originally published on [mycelia 🌌](https://paragraph.com/@mycelias/voting-systems-stuff)*