Collusion In Quadratic Voting

Two weeks ago I posted on the idea of quadratic voting, where voters pay a cost to buy votes, a cost that goes as the square of the number of votes they buy. Under certain reasonable assumptions, this voting system should produce economically efficient outcomes! Since so many get so obsessed with the objection that the rich might buy more votes, I focused on a “voting quarks” variation, wherein everyone gets the same number of points to spend across many elections.

I mentioned that this system could make agenda-setting more important. And if we did not ensure anonymous voting, there could also be a problem with some paying others to vote certain ways. On reflection, however, what I most worry about is that collusive voting becomes a bigger issue under quadratic voting, relative to ordinary voting.

How strongly you care about an election outcome depends on how much of a difference it makes to outcomes you care about, and on your chance of being pivotal, so that the election turns on your vote. Under ordinary voting, how much you care influences 1) if you bother to vote at all, and 2) how much effort you put into getting relevant info. But under quadratic voting, we must also add 3) how many votes you buy.

Without collusion, i.e., with each voter choosing independently, then under ordinary voting everyone has the same chance to be pivotal. So then if voting were mainly done to influence election outcomes, the election outcome would become a weighted average, among those who care enough to bother to vote, of how well informed each voter is, times the sign of their preference on the election. Note that the weights satiate, however; once you care enough to bother and are well informed, it doesn’t matter if you care a lot more or get much better informed.

When a group of voters colludes to vote as a block, then their chance of being pivotal is roughly proportional to the number of votes that their block controls. This proportionally increases their collective interest in bothering to vote, and in getting info. So they have a stronger interest in getting themselves to vote, and in getting info about which way to vote. But, this effect satiates at the point where they will pretty surely vote and are pretty sure which way to vote. So the possibility of block voting does end up adding an additional weight favoring groups who can coordinate, but all groups who can coordinate above some level count the same.

Under quadratic voting, colluding groups acquire an additional advantage, because they also have a stronger group interest in buying more votes. And importantly, this advantage does not satiate, but continues to grow with the size of the group. So the election outcome much more strongly weighs the ability of people to form larger groups that coordinate to vote as a block. I’m not at all sure this would be a good thing.

Added 5a: I was wrong to say that collusion gains satiate once a group is sure to vote and sure how they want to vote. A group also gains from internal vote trading, and this gain continues to larger groups. This gain happens in both ordinary and quadratic voting.

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    Robin, what exactly do you mean by groups buying votes in a system with anonymous voting and a ban on paying people to vote a certain way?

    • RobinHanson

      You and I are a group of two. We agree that I buy ten votes, you buy ten votes, and we both vote for candidate A. We can’t directly verify this, but we might still coordinate to achieve it.

      • IMASBA

        How would we buy votes if quarks were not for sale and bribing people to vote a certain way was illegal (or how would attempts to circumvent those restrictions be more effective in a system with quark voting than in a system with single, discrefe votes)?

      • sflicht

        Well circumvention would be more effective partly because quarks are intrinsically transferable. They would have to be represented either by physical tokens or entries in a digital database that at least one person (the voter in question) can manipulate to move them from one eligible election to another. Whatever the mechanism, moving quarks around would be easier than moving votes around in our world today, which involves bring busloads of actual human beings from one place to another, or at least finding a bunch of dead people and adding their names to the voting registry.

      • IMASBA

        If quarks are only transferable to your future self and if referendums work without quarks then what exactly would be the problem?

        Is it that the Americans on this blog are so used to referendums being slapped onto regular elections that they think this is the only way things could work and therefore quark voting would necessarily have to be applied to referendums as well?

      • sflicht

        Well, as an American, I’m comfortable stating that using different voting systems for referenda and other elections would confuse the hell out of most of my compatriots… 😛

        But I’m not sure that using QV for referenda would be a bad idea. The common criticism of direct democracy is that it allows for excessive majoritarianism. QV would actually alleviate this concern, right?

      • IMASBA

        If you’re so worried about “majoritanism” in referedums that you’d entertain the idea of using the quark voting bazooka to do something about it then it’s probably better to oppose referendums altogether. In other countries referendums are more rare and use separate ballots, often even separate dates from the regular elections, so using a different voting system would be feasible, that is if people can intuitively understand quark voting in the first place (then again, is it really so much more complicated than electoral colleges and having different rules regarding districts, etc… per state?)

      • Tom Stroop

        You and I both have 100 quarks saved up. You really care about issue A and I am indifferent. I really care about issue B and you are indifferent. Ordinarily we might both buy 10 votes each on our respective issues. But if we can trust each other, we can agree to buy 7 votes each on both issues.

      • IMASBA

        I’m just not concerned about that (why would you give up 4 years of representation because you are indifferent about a single issue in the current election?), but here I’m apparently in the minority on that.

  • charlie

    This seems like a legitimate concern, especially if ‘collusion’ is understood broadly.

    Isn’t the basic theory of interest group politics that groups which can effectively coordinate can have a disproportionate effect on election outcomes, even in a one-person-one-vote system?

    • IMASBA

      If no one is forced or paid to participate in an interest group it’s still democratic. The Rhineland Model and its derivatives, as used in Germany, Sweden and the Netherlands, among others, offer ways to reign in unrealistic demands of the largest interest groups (essentially they force those groups to negotiate with each other).

      • charlie

        You are using “democratic” as a code word for “good” or “acceptable.” There are a wide variety of democratic voting procedures that are undesirable and make people worse off. Having lots of referenda on single policies, for example.

      • IMASBA

        I was using it as a code word for “respecting the wishes of an electoral majority in a way very similar to that of today”.

        P.S. The Swiss don’t share your negative few of having lots of referendums (there are a lot of different voting procedures that “work”, or at least work in some cultures).

  • Tom

    The collusion issue gets interesting when you factor in something like gerrymandering. If you can win a lot of “safe” elections without needing many votes, you can pool the excess to use in contested elections.

  • Cambias

    I’ve noticed that some people — mostly economists and engineers — put way more emphasis on voting mechanisms as a solution for the problems of democracy than is useful. Would changes to the voting system alter the media and party-driven hyper-polarization in contemporary America? I doubt it; the quasi-religious/tribal nature of party identity (at least on the liberal side) has little to do with how elections are structured.

    Constitutions, laws, and formal structures are only as good as the people who use and enforce them. Some of the world’s nastiest despotisms had (on paper) wonderfully humane and liberal constitutions. One of the world’s longest-lasting and robust democracies — the UK — has a horrible ad-hoc kludge of a constitution.

    It is impossible to devise a system which will always produce good results. You have to pick good people.

    • RobinHanson

      Even if the voting rules matter a lot less than the people who vote, it can be a lot harder to change the latter than the former.

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