Quadratic voting is a proposal for a voting system that ensures participants cast a number of votes proportional to the degree they care about the issue by making the marginal cost of each additional vote linearly increasing—see this post by Vitalik for an excellent introduction.

One major issue with QV is collusion—since the marginal cost of buying one vote is different for different people, if you could spread a number of votes out across multiple people, you could buy more votes for the same amount of money. For instance, suppose you and a friend have $100 each, and you care only about Cause A and they care only about Cause B, and neither of you care about any of the other causes up for vote. You could spend all of your$100 on A and they could spend all of theirs on B, or you could both agree to each spend $50 on A and$50 on B, which would net times the votes for both A and B as opposed to the default.

The solution generally proposed in response to this issue is to ensure that the vote is truly secret, to the extent that you cannot even prove to anyone else who you voted for. The thinking is that this creates a prisoner’s dilemma where by defecting, you manage to obtain both the $50 from your friend and also the full$100 from yourself for your own cause, and that because there is no way to confirm how you voted, there are no possible externalities to create incentives for not defecting.

Unfortunately, I have two objections to this solution, one theoretical and one practical. The theoretical objection is that if the two agents are able to accurately predict each others’ actions and reason using FDT, then it is possible for the two agents to cooperate à la FairBot—this circumvents the inability to prove what you voted for after the fact by proving ahead of time what you will vote. The practical objection is that people tend to cooperate in prisoner’s dilemmas a significant amount of the time anyways, and in general a lot of people tend to uphold promises they make even if the other party has no way to verify. I think there’s even an argument to be made that the latter is at least partially a real world instance of the former.

I’m still optimistic that this might not be a showstopping problem in practice, since there is a limited pool of people who you would trust in practice, which puts a ceiling on how many times your vote can count. However, I think this is still a major unavoidable flaw with QV for both practical and theoretical applications.

• Isn’t “collusion” here just another way of describing political organizing? Like, different interest groups which each care only about their own issues get together and decide to support each other’s issues so that they can win power and each accomplish their agendas.

This seems....at the very least not bad, and probably actively good.

A good historical example of this would be the Teamsters joining in coalition with the MLK-era civil rights movement. Are you saying this is bad collusion? Or is this good solidarity and organizing? How would one tell the difference?

The “problem” you are describing here is people making political choices to join coalitions, and exists in normal democracy just as much as it exists in QV.

• Within the context of this post, collusion means any way of getting around the quadratic cost of voting by spreading the votes across multiple people. This is undesirable because it defeats the purpose of QV, in much the same way that allowing some people to cast multiple votes in regular voting would. The purpose of QV is to get an accurate picture of how much people care about things. You could add a separate layer to allow coalition forming, but the QV layer is the wrong place to allow this.

Further, while the examples I present in the post are symmetrical, there are also less symmetrical examples of collusion. For example, even with a secret ballot, if threatened with bodily harm, I expect a significant fraction of people to be intimidated into voting the way they are instructed (and this argument applies doubly for agents whose actions can be proven).

• But it seems like a lot of this political organizing (or “collusion”) occurs through the channel of convincing people to start caring about your issue. This is a good thing! The Teamsters in the 1960s really put themselves on the line for Civil Rights, I don’t think it’s correct to say that they did it for purely Machiavellian reasons. At least now, the Teamsters seem really proud of their history on this and have a whole section of their website devoted to it.

Actually paying people to vote your way would be illegal of course. But it’s illegal now, and not widespread in the United States AFAIK, because the risk/​benefit on it is so bad. To expand on this a bit, to the extent this is what you’re referring to as “collusion”, in a polity with any significant number of voters, you’d have to buy a lot of votes to influence the outcome (bare minimum thousands, most of the time). But every single person that you approach with an offer to buy their vote is a risk to turn you in, and you face serious felony charges if caught. Totally not worth it.

If what you really mean is using policy to “buy” the votes, then we’re back to this seeming good. The outcome is that a lot of people get a policy that they like, and they use their votes in a way that seems likely to them to get that. Again, seems good.

To the extent that you are using actual dollars to “vote on” (really buy) policy outcomes, I guess you have a lot of other problems with the system. To the extent you are using “voting tokens” or something as described in Vitalik’s post, then the proper, virtuous, and pro-democratic strategy is to convince more people to spend some voting tokens on your issue. Then you end up with an outcome broadly acceptable to many, which is what you are supposed to get in a democracy. Of course, it’s not that clear to me that this in practice works out all that differently than regular democracy, since convincing the mass public to support you is much more effective than spending a lot of tokens yourself.

• I don’t see how that’s a useful hypothetical scenario. If you have enough secret agents that you can coerce a large percentage of voters to do what you want, what voting system is able to stop you?

“I will help you with your thing if you do the same for me” is the core ethos of non-dictatorial civilization. IMO, QV encouraging cooperation (not “collusion”) is a point in its favor, not against.

• A good historical example of this would be the Teamsters joining in coalition with the MLK-era civil rights movement. Are you saying this is bad collusion?

In general, collusion is “secret or illegal cooperation or conspiracy, especially in order to cheat or deceive others” (Google dictionary). When the Teamsters joined in coalition with the MLK-era civil rights movement, this was neither secret, nor illegal, nor intended to cheat or deceive others. So it was not collusion.

In the opening post, the term “collusion” presumably comes from Vitalik’s article: Quadratic Payments: A Primer.

Collusion is also tricky. If we can’t prevent people from selling their votes, the mechanisms once again collapse into one-dollar-one-vote. We don’t just need votes to be anonymous and private (while still making the final result provable and public); we need votes to be so private that even the person who made the vote can’t prove to anyone else what they voted for.

If two people are trading votes in a Vitalik’s Quadratic Voting, they are bypassing the mechanisms to make votes anonymous, private, and unprovable, which are the same mechanisms that are intended to prevent the collusion of selling votes. I don’t know if Vitalik intended for vote trading to be illegal under QV, but that’s my interpretation.

• The “collusion” issue leads to a state of affairs that two political groups can gain more political power if they can organize and get along well enough to actively coordinate. Why should two groups have more power just because they can cooperate?

• The impression I got, was that collusion between likeminded people created an “indirect democracy” where causes supported by the most people could most efficiently advocate their position.

If that is the case, then this system does punish parties that are less willing (or able) to cooperate, which could feasibly be a bad thing, if it means that unpopular results occur because one side is less nuanced on it’s position (e.g. a 40% group beats three 20% groups who cannot cooperate).

One way around this, maybe, is a Negative Vote (allowing a united method opposition), but that has foreseeable issues, especially if Negative Voting is as efficient or more efficient than Positive.

• I don’t understand. A 40% group will (and IMO should) beat 3 non-cooperating 20% groups in pretty much any voting system.

A system that encourages groups to work together for their collective benefit seems like a solution for that situation, not a problem.

• A 40% group will (and IMO should) beat 3 non-cooperating 20% groups in pretty much any voting system.

tl;dr: I disagree. Other than first-past-the-post, which is terrible, 3 non-cooperating 20% groups with similar preferences will and should beat a co-operating 40% group. This is also true for quadratic voting.

Here is a detailed scenario matching your 40/​20/​20/​20 example. Suppose we have the following voters:

• Alice prefers apples to other fruit, and strongly prefers fruit to vegetables.

• Bob prefers bananas to other fruit, and strongly prefers fruit to vegetables.

• Charlie prefers cherries to other fruit, and strongly prefers fruit to vegetables.

• Yasmine prefers yams to other vegetables, and strongly prefers vegetables to fruit.

• Zebedee prefers zucchini to other vegetables, and strongly prefers vegetables to fruit.

Y and Z are able to coordinate. A, B, and C are not. This is not because Y and Z are more virtuous, nor because vegetables are better than fruit. It’s for the prosaic reason that A, B, and C do not share a common language. All voters have similar utility at stake, for example Charlie is not allergic to yams.

In a first-past-the-post voting system, with apple, bananas, cherries, yams, and zucchini on the ballot, Y and Z can coordinate to gets yams and zucchini on alternating days. This is good for Y and Z, but does not maximize utility.

However, in a (good) ranked voting system, we instead get a tie between apples, bananas, and cherries, which we break randomly. This is good for A, B, and C. Proportional representation would get a similar result, assuming that the representatives, unlike the voters, can coordinate. Approval voting would get a similar result in this example.

Quadratic voting calculations are a bit harder for me, and I had to experiment to get a near-optimal voting strategy.

Let’s suppose that A votes as follows:

• $30 for Apples (+5.48) •$15 for Bananas (+3.87)

• $15 for Cherries (+3.87) •$20 against Yams (-4.47)

• $20 against Zucchini (-4.47) B and C vote similarly but according to their own preferences. Naively this maps to A preferring apples to other fruits (by$15) but strongly preferring fruits to vegetables (by $35). I don’t have good intuitions for whether A would vote this way in practice. Meanwhile Y and Z coordinate and vote as follows: •$10 against Apples (-3.16)

• $10 against Bananas (-3.16) •$10 against cherries (-3.16)

• \$70 for yams (+8.36)

On alternate days Y and Z coordinate to vote for zucchini, as in the first-past-the-post coordination example. Again, I don’t have good intuitions for how they should vote, but I experimented with a dozen strategies and this one was best I found.

The results are:

• Apples, Bananas, and Cherries: 6.90

• Yams: 3.32

• Zucchini: −13.4

So although coordination/​collusion allowed Y and Z to boost their effective voting power, they are not able to enforce rule-by-minority in this example.

• OK, by “cannot cooperate”, you meant “unable to coordinate communication about their already-shared values” rather than “unable to agree to support each others’ unrelated interests”. Got it.

• Okay, I accept your point that a cooperating 40% group will beat three non-cooperating 20% groups with unrelated interests in pretty much any voting system. That doesn’t change whether A+B+C are physically incapable of communicating, or they lack sufficient trust to make an agreement stick, or there is a law against voting agreements that they are following (and Y+Z are not), or something else.

So it’s not the case that QV is vulnerable to collusion/​cooperation when other voting systems are not. I think the remaining debate is whether QV is more vulnerable, or vulnerable in a worse way. I’m not sure what the answer is to that.

(I’m not brgind or EdgyCam, I can’t speak to what they meant)

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• Thanks for writing this. You clearly explained the collusion problem.

Is there much experimental/​real-world data out there around how QV performs in practice?

We have laws against insider trading. Similarly, we can create laws against collusion (we already have anti-collusion laws for 1P1V), at least to discourage it. Eliminating collusion and insider trading is impossible, but laws nevertheless do a good job discouraging such actions.

I read Radical Markets recently and liked it. It’s written by Eric Posner and Glen Weyl, who are experts on mechanism design. The works of Vickrey inspire their ideas. They talk at length about QV. They highlight that with QV, people are expected to feel a lesser need to ‘scream’. Under 1P1V, you’ve got a single vote, so you better make it count, so you’re incentivised to ‘scream’. This potentially incentivises polarisation within democratic systems. This is one of the things that stood out for me from the book.

• since there is a limited pool of people who you would trust in practice

It might be possible for large parties to collude too. (To the detriment of everyone outside of the party)

Each person goes and records their vote with a party authority. The party authorities then tally votes and tell everyone the result. (Tallying can maintain privacy using ZK proofs or a trusted third party) Then the authorities tell you to vote proportional to this result in the official vote because it is the “right thing to do”. They could use lie detectors and generally psychologically pressure you before you go to vote—such that it is difficult for you to lie about what you will go and vote. As long as you know a majority of people aren’t lying about what they voted, it’s in your interest to keep this tradition alive.

• Wait hold on, I thought this was a feature of QV that made it well suited to funding public goods 😄? (The more individuals each find the same thing beneficial, the more it must be a “public good” and thus underfunded)

• Doesn’t this, by extension, seem to more directly lead to a cost-benefit problem of coalitions?

At some point the marginal cost of additional votes leads will be greater than the marginal cost of influencing other voters, either via direct collusion, or via altering their opinions through alternative incentives, such as by subsidizing voters who agree but care less, or by offering payments or commitments that mitigate the reasons opponents care about the issue.

I’m not sure that’s necessarily a bad thing, but there are lots more ways to influence other voters with resources than just colluding to vote to eachother’s advantage.

• I agree collusion is not a showstopper, because individual people very rarely bother to try anything dishonest, and even when they do it isn’t effective. Also political parties will simply disseminate recommended spending plans. To prevent this would require something like absolute power over all communication, wielded by an entity over which no political party has any influence.

The truly secret voting suggestion is possibly the most awful idea I have ever heard with respect to voting, because while individual voters rarely commit fraud or do anything else inappropriate with their votes a very common and highly successful method of cheating an election is for the people who tally the votes to simply declare victory for one candidate or the other. If we cannot prove who anyone actually voted for, we can’t prove who actually won at all.

• If we cannot prove who anyone actually voted for, we can’t prove who actually won at all.

Using zero-knowledge proofs it is possible to prove that votes were counted correctly, without revealing who anyone voted for. See MACI [1], which additionally provides inability to prove your own vote to a third party.

• From the MACI link, my objection is a generalized version of this:

### Problems this does not solve

• A key-selling attack where the recipient is inside trusted hardware or a trustworthy multisig

• An attack where the original key is inside trusted hardware that prevents key changes except to keys known by an attacker

This is the level where trust is a problem in most real elections, not the voter level. I also note this detail:

It’s assumed that is a smart contract that has some procedure for admitting keys into this registry, with the social norm that participants in the mechanism should only act to support admitting keys if they verify two things

Emphasis mine. In total this looks like it roughly says “Assuming we trust everyone involved, we can eliminate some of the incentive to breach that trust by eliminating certain information.”

That is a cool result on the technical merits, but doesn’t seem to advance the pragmatic goal of finding a better voting system.

• Vitalik’s Optimism retro-funding post mentions a few instances where secret ballots are used today, and which could arguably be improved by these cryptographic primitives:

• The Israeli Knesset uses secret votes to elect the president and a few other officials

• The Italian parliament has used secret votes in a variety of contexts. In the 19th century, it was considered an important way to protect parliament votes from interference by a monarchy.

• Discussions in US parliaments were less transparent before 1970, and some researchers argue that the switch to more transparency led to more corruption.

• Voting in juries is often secret. Sometimes, even the identities of jurors are secret.

In general, the conclusion seems to be that secret votes in government bodies have complicated consequences; it’s not clear that they should be used everywhere, but it’s also not clear that transparency is an absolute good either.

• I have not read this one, thank you for the link!

• Just as a note, this also applies the QV version that does not use money. Each human has 100 “points” that they must allocate, the only choice being how much they allocate to which candidates.

• if the two agents are able to accurately predict each others’ actions and reason using FDT, then it is possible for the two agents to cooperate

Couldn’t you equally require QV participants pre-commit to non-collusion?

• I think this could work in small groups. However pre-commitment to non-collusion requires trust between the entire electorate, whereas collusion requires trust only between the two people who are colluding.

• I think the main difficulty here is that defining non-collusion might be a bit tricky, but it could probably work with some assumptions.