Tag Archives: Institutions

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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Trade Quarks, Not Votes

If you don’t care about some election today all you can do is abstain, but what if you could instead save your vote to have extra votes in a future election? Or what if you could transfer your vote from a topic where you care less, say mayor, to a topic you where care more, say president? Or what if you could trade votes with other people, like your next two cycles of mayor votes for one of their president votes? Or what if you could buy and sell votes for cash on an open market?

All of these options have intrigued people over the years. But they all have the same problem: they tend toward having each election decided by the few people who care the most about it. True, ordinary elections don’t reflect people’s strength of preference; people who care a lot have the same influence as people who care a little. But these alternate ways to collect, transfer, and trade votes all have the opposite problem; most everyone’s preferences may be ignored except for the few extremists who care the very most.

However, a simple yet amazing variation can allow collection, transfer, trading, and selling in the voting process, while having elections tend to be decided by a weighted average of how much each voter cares. This amazing variation is: voting quarks.

Hadrons are basic particles in physics that can fly around and smash into things. They are made out of quarks, but (in our world today) those quarks are always stuck inside hadrons, and can’t fly around by themselves. Quarks influence the world via the hadrons of which they are part.

By analogy, a voting quark is a part of a vote that can’t influence an election by itself; it must be part of a vote particle. And voting quarks must be formed into square arrays in order to make votes. So you can use one quark to make a vote of size one, or four quarks to make a vote of size two, or nine quarks to make a vote of size three, and so on.

The key idea here is that elections are won by which ever side has the most votes, with bigger votes counting for proportionally more; but what voters are given are quarks, not votes. For example, each election each voter might be given four new quarks. If no collection, transfer, or trading were allowed, this would be pretty boring, as the only useful option would be to convert those four quarks into one vote of size two to use in this election. (After which that vote, and those quarks, would be gone.)

But if quark collection was allowed, a voter could choose to instead save all these new quarks for future elections. Or they might use one quark this election to make a vote of size one, and save the other three for future elections. Or if they had collected at least five previous quarks, they might add them to these new four quarks to create a vote of size three to use in this election.

Abilities to transfer or trade quarks would work similarly; you’d move the quarks around as allowed by the rules, and then form votes from the quarks as desired to use in each election. The system might even not directly give voters quarks at all, but only sell them quarks for cash.

The main point is that in a system like this people have an incentive to vote in each election roughly in proportion to their strength of preference on that topic. Which allows elections to produce more economically efficient outcomes. And the wider the scope over which quarks can be moved, the wider the scope over which choices could be more efficient.

This point has been plausibly argued in a paper called “Quadratic Voting” by Steven Lalley and Glen Weyl. (Weyl has a related paper with Eric Posner.) They talk about this in terms of buying votes directly with cash, paying proportional to the square of the votes bought. This is an extreme version that I suspect most people will find hard to swallow, at least as the first change to accept. So I designed the above quark language to show how we might move there gradually, such as perhaps by first allowing collection, then transfer, then trading. And we might slowly increase the number of quarks given per election, to approach a more continuous voting.

Steven Levitt has commented positively on the quadratic voting idea, but Tyler Cowen criticized it for encouraging “intense preferences of minorities”. I find that a rather odd criticism, and agree with Eric Posner’s response.

I do have a concern though: this approach would require us to pay more attention to agenda setting. Once votes or quarks can be moved between elections, then every election not only decides an issue, it also creates resources that be used to decide other elections. So we’d want to try to ensure that issues in elections connected by quark moves are similarly important, or perhaps set relative quark prices between them.

Also, the act of introducing an election on some topic ends up being an implicit tax on the people who expect to win that election. They will have to use up quarks there than they can’t use elsewhere. If the status quo is already in their direction, then people who favor the status quo will regret holding an election on that topic, even if they expect to win. Factions may conspire to hold repeated elections that they expect to repeatedly lose, just to tax other factions.

This isn’t an overwhelming objection. We already must pay substantial attention to agenda setting even under ordinary voting. But this does up the ante a bit. So we should try this stuff out slowly, gradually, testing and observing as we go. And we may need to invent new ways to set agendas. But this looks pretty promising, so let’s get started!

Added 11a: OK, on reflection one only has to worry about the relative importance of elections when voters can collect or transfer quarks between them. If voters can instead only trade quarks between elections, their relative importance will be reflected automatically in the relative prices of quarks traded. Also Eric Posner suggests a general agenda mechanism:

Added 10Jan: Commenters are too hung up on money. Money is only relevant in the most extreme version I mentioned, where quarks are bought with cash. Consider instead the other options to only collect, transfer, or trade quarks.

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