In the first installment of the CityDAO Journal, there was an excellent write-up on the general premise of quadratic voting, explaining how the voting system works, what its benefits are, and a few simple examples. This article will follow up by digging a little deeper into quadratic voting and what it might look like if implemented within CityDAO.

Quadratic Voting, also known as “QV” or “Plural Voting,” works by providing each citizen with a fixed number of “voice credits” that are available for the citizen to assign to the options within a vote. As discussed in the previous journal article, voting in this way allows a citizen to express not only their preferences amongst the choices presented but the strength of their preferences.

Contrast this with current voting systems where you can choose one option but cannot indicate whether you really really like that option or just barely like it. Or, contrast QV to a ranked-choice voting system where you can say “I like A more than B more than C”, but cannot indicate whether you like A ten times more than B, and B two times more than C. QV allows more information to be included in voting results, which in theory gets us to a more efficient and representative democracy.

Governance voting in CityDAO, as of this writing, consists of snapshot votes on City Improvement Proposals (CIPs). So far, almost every vote has been what snapshot calls a “Single Choice Vote,” where there are only two options, and citizens choose the one they like most. When we talk about converting CityDAO to QV, we suggest turning all of our CIP snapshot votes into votes using the QV system. So how would that actually work?

After refining the question to be put to vote through the CityDAO forum, CIP 43 “LFG: Parcel 0 Direct Drop for Citizens” offered citizens two choices on snapshot:

1. There will be a limited number of Parcel 0 NFT’s (1,000)

2. Every citizen will be eligible for an NFT

In the vote, which occurred between April 2 and April 9, 2022, every Citizen NFT had one vote, and they could assign that vote to either option. 1,531 Citizen NFTs chose option 2 (Parcels for Every Citizen), compared to 517 Citizen NFTs that chose option 1 (Limited Parcels).

Suppose the vote utilized QV. When opening Snapshot, the user would see 10 voice credits assigned for every Citizen NFT in the connected wallet; 1 Citizen NFT gets you 10 voice credits,  2 Citizens NFTs would get 20 voice credits, etc. You would then assign your voice credits to one or both choices (note, the system could also allow negative votes such that the absolute value of the voice credits you assign must equal the number of voice credits you have).

First, the voice credits are assigned:

Then, the voice credits are converted to their square root to determine actual votes, which are then totaled.

The choice with the highest positive value wins. In this example, the “Limited Parcels” option with 2.08 votes (compared to -3.24 votes for the other option) would go forward.

What is the point of “square rooting” the voice credits?

If the credit-to-votes conversion was linear, 10 credits would get you 10 votes, and 5 credits would get you 5 votes. In the quadratic conversion, 10 credits gets you 3.16 votes (square root of 10), and 5 credits gets you 2.24 votes (square root of 5). Because of the square rooting, you need to assign an increasing number of credits for each additional vote. If you are at 3 votes (9 credits) and you want to get to 4 votes (16 credits), you’ll need 5 more credits (16 - 9). To go from 4 votes (16 credits) to 5 votes (25 credits), you’ll need 9 more credits (25 - 16). And so on. This makes casting every additional vote more costly than the last.

For a full explanation of why that’s important, see chapter 2 in the book Radical Markets, but to sum it up; QV allows citizens who care deeply about one option to express that feeling by assigning more credits to that option. At the same time, square rooting the voice credits imposes enough of a cost so that a citizen would only assign as many voice credits as matches their true feelings. As described by The Economist; “instead of buying increasingly expensive votes for their number-one choice, voters are nudged to cast some relatively cheap votes for second- or third-choice options.” Or put another way in the book Radical Markets, QV helps to “induce those who care little to at least buy a few votes and restrain those who care a lot from buying too many.” The end result of all this is that in theory, the outcome of the votes in QV more accurately matches citizen sentiment than other systems.

When there are only two choices, QV is not very meaningful but still provides more information.

In the example above, where negative votes are allowed, it enables voters to say which option they like more while also expressing dislike of a particular choice. For example, Zach might be thinking, "I'm not sure what the best way to distribute parcels is, but I know for sure it needs to be more than 1,000." It's true that with only two choices, voting -10 credits for limited parcels is effectively the same as voting +10 credits for the other option, but there is a small additional benefit. If CityDAO studied the analytics of this vote, we could learn a little more about how Citizens really felt. Consider these plausible scenarios if 1,000 citizens with 1 NFT each voted on CIP 43. Here the winning option is highlighted in green.

In Scenario 1, all citizens assign 5 positive credits to the "Limited Parcels" option and 5 negative credits to the "Parcels for Every Citizen" option. In other words, they really liked the “Limited Parcels" option and really didn't like the "Parcels for Every Citizen" option. In Scenario 2, all citizens assign 10 negative credits to the "Parcels for Every Citizen" and 0 credits to the "Limited Parcels" option. In other words, citizens were ambivalent about the "Limited Parcels" option but truly hated the "Parcels for Every Citizen Option."

While the outcome is the same in terms of which option wins, QV provides more information that would drastically change how we interpret the results. For example, the presence of many negative votes in Scenario 2 could trigger some predefined formula indicating there was not enough positive sentiment for the winning option. This could necessitate CityDAO to go back to the drawing board and develop more options until more positive responses are generated. In other words, the results could be interpreted as a vote of no confidence. After the changes are made, the results might be totally different given the options will be different. Thus, even with only two choices in a vote, QV still provides more information compared to other voting systems.

QV does make voting more complicated

QV adds another level of complexity when casting your vote. Because of the square-rooting of voice credits, you have to strategize how to distribute your credits, not simply assign them all to the option you like most. For example, it actually makes more sense to do what Bob did, where his voting power was 4.48 by doing a +5, -5 split. In comparison, Zach reduced his overall voting power by putting all 10 of his credits on one option (for a voting power of 3.16). Requiring strategic thinking of your credit assignments is useful in QV because it forces you to think about and reveal the relative strength of your preferences. With two option votes, however, it doesn’t add much value and could potentially disenfranchise the casual voter if they are unaware of the strategies described above.

But, QV Can also Address the “Whale” Problem

The major benefit of QV in the context of two options is it can partially address the issue of individuals with multiple Citizen NFTs (a.k.a “whales”) having an outsized influence on election outcomes. CityDAO has recently passed CIP-48: CityDAO Conversion to Partial Quadratic Voting, which, once implemented, will take the number of NFTs in a wallet and square root them to determine how many votes that wallet can make. CIP 48 acknowledges this isn’t a full implementation of QV, and is somewhat of a temporary fix.

“Our version of QV is considered partial because it does not use the  full specification of QV authored by Vitalik and RadicalxChange's Glen Weyl.“

To explore this a little more, let's note what was passed in CIP 48 isn't quite QV (though closely related); it's more like quadratic apportionment. An analogy would be that CIP 48 changes the formula that determines how many seats each state gets in the U.S. House of Representatives (shift in apportionment), whereas implementing full QV would change the voting procedures used by the representatives (shift in voting method). CIP 48 doesn't quite get at the true benefits of QV, which is meant to allow voters to better express strength of preference instead of just reducing strength overall. In other words, QV is meant to give voters more options, not just reduce the influence of whales.

If full QV is implemented, it might not be necessary to also do quadratic apportionment (square rooting of a wallet's NFTs). While it's true that a single person with multiple NFTs would have more voice credits, they'd still be subject to the quadratic voting formula, which significantly diminishes the value of the additional NFTs.

For example, John owns 2 Citizen NFTs, which means he would have access to 20 voice credits compared to Alice's 10 voice credits with her 1 Citizen NFT. If Zach put all his credits on one option, he could cast 4.47 votes (√20), whereas Alice would cast 3.16 (√10) if she used all her credits on one option. John has twice the NFTs but can't exert twice as much voting power. It's true that if John spreads his 20 credits across multiple choices, his credits will go further (like Bob in the example above), but we greatly reduce John’s ability to determine the outcome, especially his ability to force the outcome towards a single choice.

Once you add a third option to Snapshot Votes, then the benefits of QV really start to shine.  This would require a shift in how proposals are formed - so that CIPs aren’t always boiled down into a Yes/No question, but it shouldn’t be that difficult. For example, CIP 43 discussed above actually started the forum discussion with multiple options before being edited to its current version. As another example let’s look at a thread that could become a CIP - the Comments on Parcel 1 Real Estate Guild Winnowing Process.  In this thread, CityDAO is narrowing down the options for Parcel 1 - the DAO’s next big move.

I’m speculating here, but let's just say this turns into CIP 98 and the choices are the following:

1. CityDAO Embassy in Denver

2. Kanye West Ranch

3. CityDAO Events and Education Center

4. CityDAO Township

In a single-choice vote, you’d have to choose just one option. If CityDAO decided to use a ranked-choice voting scheme, that would be better, but it assumes an even gradient between your first and fourth choice (i.e., you like #1 “one better” than #2, which you like “one better” than #3). Let’s say Alice can’t stand Kanye West and will rage quit CityDAO if the Kanye Ranch option wins. She simply won’t vote for the Kanye West Ranch in single-choice voting. She’ll still have to rank it as her fourth choice in ranked-choice voting, but even that is too generous for how Alice feels. With quadratic voting, she could allocate all 10 of her voice credits to vote against that option. Or alternatively, spread her 10 voice credits amongst the 3 other acceptable options. Either way, QV more accurately captures Alice’s preferences over the other available voting methods.

In the example above, what happens in one snapshot vote is irrelevant to the next vote.  Whether you don’t vote at all in CIP 98, or only use some of your credits, it has no effect on how many credits are available for you in CIP 99.  For a vote on CIP 99, everyone starts with a fresh slate and gets another 10 voice credits that can only be used on CIP 99.  But what if we allowed voice credits to carry over between votes?

Another way to apply QV to a DAO is to have the voice credits be replenished at a set time interval (like every year, after 10 snapshot votes, or at the beginning of every CityDAO season). These voice credits could then be used on any vote - regardless of the timeframe.  You may decide to spend a year’s worth of credits on one vote that is really important to you (even though you don’t know what other votes you’ll be missing out on, and the votes will still be subject to being square rooted - so you’ll exert less voting power overall).  Or you may watch a number of CIPs go up to vote and decide that you don’t have strong feelings and will save your credits for another day.

There would be a lot of logistics to work out. For example, if you sell your NFT do the voice credits stay with it? Do your credits go away if you are inactive for a really long time (as a way to incentivize voter participation)?  Regardless, extending the time frame in which voice credits can be used can allow for citizens to show the strength of their preference across multiple votes, rather than just within the choices of a single vote.

The carry-over option could be used in combination with ensuring every vote had more than two choices.  Alternatively, the carry-over option might make it less important to ensure every vote had at least 3 options. Even with two options in a single vote, citizens effectively have more than two options as they can decide on which votes (not just options within a vote) to use their credits.

Whether CityDAO chooses to use QV or not will ultimately be up to its citizens and likely require the development of a custom voting strategy in Snapshot. In the meantime, we hope this article can jumpstart the discussion and bring new ideas and questions forward.

Written by Nicholas Bonard (nicholas.bonard.eth) – This article was made possible with funding by the CityDAO Research and Education Guild