Quadratic voting is a potential voting method that has gotten a fair amount of discussion in various places, one of the most notable presentations on this is in Radical Markets. While the game theoretic justification for this voting method is sound under optimal conditions, with low information/transactional costs, and perfectly rational actors, I believe that there are flaws in this idea that make it unusable in most real world circumstances where it is being proposed. It is a system that is perfect on paper, but unsuited to the real world.. While the game theoretic justification for this voting method is sound under optimal conditions, with low information/transactional costs, and perfectly rational actors, I believe that there are flaws in this idea that make it unusable in most real world circumstances where it is being proposed. It is a system that is perfect on paper, but unsuited to the real world.

A flaw of many real world voting systems is that there is not a good way to allow voters to provide information about the relative importance of issues. This means that people who only have a weak preference on an issue will be in effect over represented in political outcomes on that issue. Quadratic voting is a proposal to fix this issue.

In a QV ballot a voter has a number of points that they can allocate across issues. Allocating more points to an issue makes the vote on that issue weighted more. The value of each point declines as you add more points to an issue. Accordingly, there is an incentive to split your points across multiple issues.

I won't go into too many details about the justification and game theory here because its been covered by other sources quite a bit. I would assume that if you are reading this blog post, then you probably have some knowledge here. However, I have provided a bit of a summary at the end of this blog post — focusing on the quadratic funding variant because it is somewhat easier to build an intuition around.

What do I see as the problems with this proposal?

In summary, it runs into issues with very large elections, and breaks down when people don't act as Homo economicus purely rational self-interested actors.

For now, I will leave you with two examples of what I am talking about. – How to optimally allocate voting points

The optimal strategy in a QV system would be to allocate votes in proportion to the value of a vote, not the subjective importance of the outcome of that particular issue.

By value of a vote, I mean the probability that the voter will be the pivotal voter that decides the outcome of the election * the importance of the outcome.

This can result in the game theoretic optimal allocation of voting points being fairly counterintutaive in cases where both large and small elections are on the same ballot.

Lets imagine that there are two elections on the ballot 1) the election for the U.S President, where you have a one in a million chance of becoming the pivotal voter, and 2) the county dogcatcher where you have a one in 10 thousand chance of becoming the pivotal voter. Lets say you care about the outcome of the presidential election 10,000 more than the dogcatcher.

Because of the probability of being the pivotal voter in each case the value of a vote in the dogcatcher election is 100 times more. Therefore, the strategically optimal way to fill out your ballot is to allocate approximately 10 times as many points to the dogcatcher.

In a real world election using this method, most people may not be that extreme in how strategically they vote — and there will probably be wide variance in how people allocate their votes taking into account the pivotal voter probability. This mix of strategic and non strategic voting will eat at the efficiency benefits of the system. – High minimum threshold for issue importance

Consider the funding-matching version of quadratic voting (picking this variant because it is particularly easy to follow the math here). (picking this variant because it is particularly easy to follow the math here).

Lets imagine that there is a public good that 1 million people donate one cent to. In this case, if quadratic funding was used to allocate matching funds, $10 billion would be allocated — $10,000 per capita.

This scales upward with population, if there were five million donors donating one cent each — 250 billion would be allocated to the project — $50,000 per-capita.

And if someone for various reasons donates a larger sum, that will be magnified to an absurd degree — making any voting behavior other than being perfectly rational and self-interested a system breaker.

As you can see, quadratic funding/voting can only really be used for very huge issues and in fairly small communities given the practical limits of how people think — defeating one of its main features, allowing more day to day political participation in every-day political decisions without the limitations of direct democracy under more standard voting systems.

Additional notes / Overview of Game Theory Justification for QV

Here is a brief intuition for how QV works / where the theoretical justification comes from.

Imagine you are in a home owner's association that is considering building a swimming pool. You and the other 100 members each get $100 of value out of the pool. This means that you would be willing to pay up to $100 to have the pool built.

If contributions were completely voluntary, the marginal value of each dollar you contribute would be equally distributed across everyone in the association. Ie. if you contribute enough to completely build the pool, you would only get $100 of value out of that.

In effect, for each dollar of value that you create with your personal contribution, you only get one cent of personal value.

Therefore, if you were a perfectly rational and self interested actor you would only want to pay this if the pool's total cost was $100 or less. This applies even if the total value for everyone of the pool was much more than that.

But in a world where everyone is a perfectly rational actor, individual willingness to pay can be a very useful source of information on what people's preferences are.

Lets imagine that your HOA comes up with an idea — using fees to create a matching fund for projects submitted by members.

Is there in effect a way to translate the individual willingnes to contribute money into a estimate of the total value of the project?

Lets also say that everyone is behaving like a perfectly self-interested perfectly rational actor. Furthermore, lets say that theb good/service being funded is non-excludable — its not walled off to only the people who contribute.

For that swimming pool example, what ratio of matching donation would it take to make it rationally self interested for yourself to donate enough such that your donation + the matching funds = *the social value of the project).

For the example of a project where the benefits are split across 100 people, the correct matching is $99 dollars for each donated dollar.

The optimal maximum funding for a project where each person donates a dollar is actually, 100 * 100 or $10,000

You can by this logic derive a general formula here for the case where everyone's preferences are identical.

Optimal project funding is equal to the individual donation amount times the square of the population size.

Which is the core idea and why quadratic voting is quadratic.

For the matching funding game, you can derive a general rule for cases where indvidual preferences (and therefore donation ammounts) vary. The total funding = the square of the sum of the square roots of the donations.

Similar logic can be used for voting on funding projects with abstract points instead of matched monetary contributions (the aforementioned formula can be used to calculate the relative importance projects, and the funding avaliable can be allocated proportanate to this).

And finally, similar logic can be applied to voting on issues/canidates. A quadratic voting ballot would allow voters to allocate points to each issue — with the weight given to that issue being equal to the square root of the number of points allocated.