The CityDAO Journal #1 - Quadratic Voting as a Roadmap to a Pluralistic Society
ScottA
0x5b80
May 24th, 2022

Motivation

Voting is the bedrock of democracy. We usually think of democracy as the opposite of dictatorship. Yet, democracy can tend toward a de facto dictatorship without additional checks on power, clearly defeating the purpose. It seems self-evident voting to reach public consensus is often ideal. However, it is easy to lose sight of the fact that voting is a nebulous and complex topic. Many ill-liberal and anti-democratic consequences may result depending on the mechanism chosen to conduct voting.

This article outlines the advantages of a system known as quadratic voting (QV), also referred to as plural voting. Broadly QV is a voting system designed to reflect the intensity of people's preferences in collective decisions. For important context, we can juxtapose QV with common mechanisms used to conduct voting to understand QV and its benefits better. By focusing on the limitation of traditional voting, QV's strong points will become transparent.

One Person One Vote and Majority (Tyranny) Rule

Voting in its purest form is an expression of preferences; a vote for x over y is expressing a preference for x over y. The most common type of voting is one-person-one-vote (1p1v), in which each voter expresses a single preference given a multitude of options. These expressions, or votes, are equally weighted. To reach a collective decision, we ask everyone to vote and tally them up. If option x has more votes than y, x wins, and vice versa; this is commonly referred to as first past the post. A common standard is majority rule, where the winner is the first to have over half the votes. The above-noted mechanism has many strengths but also has many flaws.

For example, under 1p1v 51% could vote to disenfranchise the remaining 49%; see the tyranny of the majority. Solving this problem comes with the realization that voting is not enough to protect the rights of citizens. In addition to voting, a democracy requires constitutional checks on power – "We hold these truths to be self-evidence that all men (people)... are endowed... with certain unalienable rights"; see American Declaration of Independence.

The provision of checks on power solves some problems but not others. Another major flaw can be characterized by the Median Voter Theorem, which states that "in a representative democracy, politicians will converge to the viewpoint of the median voter". A famous illustration of why this is true involves a hypothetical scenario of two ice-cream vendors on a beach competing for customers. If people buy from the nearest vendor, logic dictates the vendors will end up side by side in the middle of the beach. To reason why this is true, realize that the vendors will get everyone to the side the other vendor isn't on. Thus both vendors want to move closer to each other, capturing the customers in between (see link for a visual).

Think of the vendors as political parties and the ice cream as votes. If we think of politics as a left vs. right divide, voters will vote for the party that aligns the closest. Given parties can generally count on votes to the far right or left, they move to the middle. Such a process can begin to mimic a populist dictatorship.

What would the model say about a 3rd vendor? The same logic posits they will move to the center. The result is three vendors in the middle. While they can jockey for position ad infinitum, the logical conclusion is to stop wasting resources and merge. The simple model captures a common phenomenon, 3rd parties tend to merge with the incumbent parties leaving only two major parties, think Democrats and Republicans (note this type of merger recently happened in Canada).

To understand this phenomenon, consider what occurs if a 3rd party doesn’t merge with one of the incumbent parties. The result is strategic or tactical voting; see also vote splitting. A famous example of this was the 2000 US presidential election that saw the right-wing candidate George W. Bush win a narrow victory over the left-wing candidate Al Gore. Many suggested that the left-wing 3rd party candidate Ralph Nader cost Gore the election. Those casting votes for Nader were blamed for “throwing their vote away” and allowing Bush to win even though the left had more votes combined.

Under such an incentive structure, a two-party structure with highly polarized political discourse is likely, if not inevitable. It is common to observe this in many democratic countries, and evidence suggests this tendency is worsening. While voting is ideal in many circumstances, the standard 1p1v and majority rule system tends towards a winner-take-all dynamic of left vs. right; if you aren't convinced, ask Ralph Nader.

Accounting for Nuanced Preferences

There is an old expression that if your only tool is a hammer, every problem begins to look like a nail. The logic is simple – you solve problems with the tools available. If you lack good tools, you lack reasonable solutions. If your only tool is one vote, then every political issue begins to look left or right. While there are many ways to frame the flaws of 1p1v, the easiest way is to realize it doesn't account for nuanced preferences. To illustrate consider the following question:

What's the best pet: dogs, cats, fish, or birds?

If you answered dogs, you’re certainly not alone; they are man’s best friend, always there to give you unconditional love.  Yet cats are wise, cuddly, and low maintenance, seemingly the perfect companion for some. Yet, many love the elegance and tranquility of fish. And who can deny the beauty of birds, with vibrant colors and melodic songs? It is difficult to choose a favorite, which is why many own more than one and some have all four.

In other words, most of us like dogs and cats and fish and birds. It is difficult to treat these options as binaries; asking someone to choose one animal forces her to filter her preference through an artificial constraint. In the process, information is lost. Such a constraint actively interferes with the decision-making process, thus skewing the arrived at consensus to something which does not accurately mirror the truth.

Imagine a vote was conducted using a 1p1v system to decide which pet everyone should have. Does anyone imagine the result would be remotely meaningful? The tool used to arrive at the consensus isn’t reflective of nuanced preferences regarding the question posed. We are taking a hammer to a problem that requires a better tool.

Think of this in political terms. Think of the dogs, cats, fish, and birds as climate change, inequality, health care, and military spending. In a similar light, it isn't illuminating to ask – which issue do you care about? For most, the answer is all of them.

They are complex issues where preferences require weighing each option relative to the others. Perhaps your true feelings are something to the effect of – I feel we should spend much less on our military, and though I care about inequality and climate change, my biggest concern is health care, given I have a child with serious medical issues. 1p1v doesn't capture complicated preferences over such diverse and non-binary topics. Instead, it takes the multi-dimensional complex set of issues and smashes them into a box labeled either left or right.

There have been many proposed solutions to the problem. For example, Ranked-Choice Voting is a mechanism that allows the ranking of preferences thus avoiding some of the pitfalls of 1p1v. Indeed, there is a long list of such proposed solutions. With that, let's turn our focus to a potentially powerful tool capable of resolving these glitches.

Quadratic Voting

Quadratic Voting, also known as “QV” or “Plural Voting,” was first described by Glen Weyl and Eric Posner in their 2019 book Radical Markets in a chapter entitled Radical Democracy: A Market for Compromise in Our Shared Lives. QV works by providing each voter with a fixed number of “voice credits” which voters are able to assign to options. This naturally allows a more nuanced picture of democratic consensus.

QV weights your ability to affect the outcome in a way that isn't just a one-for-one mapping between person and vote. Instead, your influence is weighted by how a voter assigns their voice credits to each issue. In practice the idea is fairly simple, your support of a given issue is worth the square root of voice credits assigned, lets's call voice credits Z.

If we think of Z as the credits required for a given number of votes, we can square both sides to solve; this is why the term quadratic is used (a quadratic function has an exponent of 2, it's squared, while the square root function has an exponent of ½).

This method has the following property,  the more Z you assign to a given issue, the greater the influence, but with diminishing returns. If you assign 1 Z, you get a weight of 1. If you assign 100 Z, you get a weight of 10 – 100 times the Z only gets you 10 times the influence. Thinking of it in terms of costs implies the cost of additional votes depends on how many credits you have assigned to a given issue; additional votes cost more and more Z; see table below for illustration (also see example at the end of the article).

QV at the Margin

It can be challenging to understand how such a system solves so many problems related to 1p1v. The critical insight is to understand how QV affects marginal decisions, marginal meaning additional. There is a common phrase "rational people think at the margin" as "thinking at the margin means to let the past go and to think forward to the next hour, day, year, or dollar that you expend in time or money"; see link.

Under 1p1v, you only have one margin. It is almost trivial in that sense. You vote once for your top decision. Thus, you have one next step, one marginal decision. With QV, you have many margins. Depending on the number of voice credits issued, to be precise. A good starting point to understanding this is to consider an example. Suppose we want to choose between climate change, inequality, health care, and military spending. The vote might be for which candidate you support, given a choice between four candidates that each prioritizes one of the issues.

Let’s use the hypothetical person described above; let's call them person X. Recall the person cared about each issue but cared the most about health care. In the 1p1v model, X casts their single vote towards health care. Contrast this with the options and trade-offs under QV. Suppose each voter is given 10 voice credits. Given X cares the most about health care the voter will assign the most credits to health care, but that doesn't mean X will assign all their credits to health care!

To illustrate this suppose X assigned 9 credits to health care and is thinking about how to use their 10th. If X puts all 10 towards health care the total weight on health care would be the square root of 10, approximately 3.16. Now, consider the incentive to put one vote towards one of the other issues. If X assigns 9 votes towards health care and 1 towards climate change. The weight towards health care drops to 3 (square root of 9), while climate change gets 1 full vote (square root of 1 is 1).

This is subtle, but it is crucial to emphasize. Switching the final credit to climate change results in 6.25 times more weight when considering the final credit. Climate change gets 1 vote at the cost of .16 votes towards health care (√10 - √9 = .16 and 1/.16 = 6.25). It follows that if X doesn’t care about health care 6.25 times more than climate change, they should assign their final credit to climate change. The optimal choice is to find where the additional benefit of moving your vote equals the additional cost of assigning another vote to the same issue.

The same reasoning applies to every issue and the allocation of every credit. It can be subtle, but it is a transformative change. The question goes from which do you like best to how do you rank your preference. For each credit, you weigh a diminishing effect of allocating your credit to the same issue vs. your desire to support a different issue. It mimics the real world and how we make day-to-day decisions. Not in terms of absolutes but nuanced preferences. Not in terms of a binary yes or no choice, but on a spectrum of relative likes and dislikes. Mathematically QV allows more marginal decisions to be made, allowing for a more optimal solution. In non-technical terms, it gives people more choices to make a decision reflective of their true preferences.

It might be the case that after deliberation X still casts all their credits towards health care just like in the 1p1v system. However, they now have an incentive to think pluralistically and not focus exclusively on one issue; the opposite of the incentive structure imposed by 1p1v. In other words, it takes the issues out of an artificial two-dimensional box and gives the issues more dimensions. It forces us away from polarized binary thinking toward plurality and nuanced thinking.

In summary, quadratic voting is a method that weights your influence on an outcome by taking the square root of an assigned voice credit. It allows voters to express how much they care about an option relative to other options and thus provides a more nuanced mechanism for reaching collective consensus and establishing the will of the people.

Better Tools, Better Solutions

It isn't easy to describe all the beneficial aspects of QV. All the implications taken together would result in a truly profound change to politics, voting, and the world as we know it. It would change the options available to voters and change the incentives faced by political parties when forming their agendas and platforms. One tool, one solution. Many tools, many solutions. The dynamics that propel us towards two parties and polarized debate would be gone. Voters can demand solutions to specific problems. If we allow n votes, where n is much greater than 1, we allow n choices and thus generate a supply of n solutions. Taken together, we permit the emergence of better and more focused resolutions to critical issues.

While historically certain checks and balances have limited some of the downfalls of 1p1v and majority rule, technology hasn't traditionally been available to implement something akin to QV. In other words, we have been stuck with a hammer to solve all our problems. We now have a better tool; we now have the technology to implement QV. Blockchains and Web3 enable such a world! One-person-one-vote served the world well in the absence of a more effective tool, but it's time to do better.

Written by Scott Auriat (scotta.eth) – This article was made possible with funding by the CityDAO Research and Education Guild

Example:

Trying to decide which token to use to pay contributors in a DAO. Three people – Alice, Bob, and Zach – are asked which token they prefer: ETH, BTC, or USDC?  Each person is given 10 votes which he/she can allocate amongst the options, casting as many votes for or against an option as he/she wishes. Alice is a bitcoin maximalist. Bob loves ETH but also likes the idea of paying in USD. Zach hates volatility, thus prefers USDC, and apposes either ETH or BTC.

First, The votes are cast

Additional Resources:

Arweave TX
_ktfjW4Yg35j-c8LIxbPdwgnQey7kX2jNR_eQyAd7TY
Ethereum Address
0x5b807f0274FA7F4B56DCd0d8F24294A92b1716a1
Content Digest
oOOqyUpr7VdgDxLbywwk-DS3mUxdQ18o6OvGrWsvKeY