Here’s an intuition for why it’s important that it’s quadratic (based on standard microeconomic reasoning).
By spending your votes, you pay some cost, and get some benefit.
The cost consists in: voting credits, time, attention, energy, reputation costs if you have odd views...
The benefit consists in: higher probability your post-of-choice ends up in the book, which comes with a host of externalities like further influence of your values and epistemics on readers of the book.
As long as cost < benefit, you want to keep voting (otherwise you’d be leaving benefits on the table). You’ll do this until the cost of the last vote = benefit of the last vote.
If your total cost of voting is f(V)=V22 then the cost of each marginal vote is f′(V)=2V2=V. By the above reasoning, you’ll stop voting when your marginal benefit = marginal cost = V.
Hence, your distribution of votes V1,V2,... across options o1,o2,...will measure how much you value each option being included in the book.
(After I’d written this I found this blog post by Vitalik which explains it even better.)
An issue with the proposal is the failure of the assumption of utility of votes being proportional to number of votes.
It seems plausible to me that there’s some threshold of votes above which a post will very likely end up in the book. If this is true, then I think my utility in buying more votes for that post would be ~linear up until that threshold, and then flat (ignoring potential down-voters).
If this is the case, I’d significantly understate my preference for this post, instead spending my points elsewhere.
So, for example, Embedded Agency might be undervalued by this scheme.
This point isn’t relevant when thinking about quadratic voting for elections with millions of voters, since then it makes more sense to assume that the probability of passing a proposal is linear in the number of votes I can influence.
Moreover, this would lead to weird equilibrium dynamics…
If I’ve voted for a post that gets above the threshold, then I want to remove my votes and place them elsewhere. If I don’t do this, but other users do, then I am effectively subsidising their preferences.
I don’t know how this would pan out, and can see it messing things up as everyone tries to model everyone else and be clever.
It seems like an open numerical question whether this issue would be relevant for the current round (i.e. whether the utility of most users would be linear in the region of influence they could expect with their votes):
Here are some numbers that I wrote down but now don’t really know how to take this further. Under Ben’s initial scheme, each user can buy at most ~32 votes per post, by spending all their money. There are ~500 voting users. Which gives an upper bound of ~16000 votes for a post. There are ~75 nominated posts, out of which ~25 will end up in the book. If all users distribute their votes uniformly, we’d have about ~3.65 votes per user per post, and ~1800 votes per post. Let’s handwave and say that with 7000+ votes a post is as good as guaranteed for the book.
I think a solution to this might be that if instead of voting on what should be in the book, you decide on some subsidy pile of karma+money, and you use quadratic funding to decide how to allocate that pile to each post (and then giving it to the author/eventual co-authors).
You might just include the top posts in this scheme in the book, also making sure to make their scores prominent (and perhaps using scores in other ways to allocate attention inside the book, e.g. how many comments you include).
It seems more plausible to me that under this scheme users utility would be linear in the amount of votes/amount of funding they allocate to posts.
This makes a lot of sense, thanks for writing it down.
Aside: I think it’s way harder for readers to read stuff bulletted like this, and would personally find it easier if this was identical but without the bulletting.
Don’t think that would help—instead of knowing the actual votes Vi for post i, I would have some distribution P(Vi) over the votes cast for i, and my intuition is that as long as I have sufficient probability mass above the threshold, it would skew my incentives.
But doesn’t the knowledge that everyone else is also doing this converge to just stating my true preferences… Or something? I don’t really have a feel for the game theory behind this, but it feels like knowing that everyone is trying to vote strategically makes it hard for me to count on other people voting for things that I don’t vote on.
I don’t expect everyone to vote strategically. In fact, I expect most users to act in good-faith and do their best. I still think these things can be a problem.
That’s interesting, because I expect most people to vote strategically when using QV. The structure of QV heavily encourages thinking about the value of each marginal vote.
Trying to allocate your budget truthfully in accordance with your preferences about posts != trying to game the rules as an unbounded EV-maximiser would.
Strategic voting for me = trying to think how much value your vote has relative to the outcome you’re trying to achieve. I don’t see for instance looking at how many people have already voted on something as “gaming the rules”, it just changes the value of a marginal vote of my own. I expect most people to think like that because QV is already making you think about the marginal value of another vote.
I agree that you will stop voting on a post when the marginal cost = marginal benefit. This means if I take my votes from above:
Post A: 5 votes
Post B: 8 votes
Post C: 1 vote
This means I valued a vote on post A at price 5, post B at price 8, and post C at price 1, where the prices are all relative to each other.
So my noob question is: Don’t I know this relative pricing having to say what the marginal cost function is? If each marginal vote had costed 2n, wouldn’t I still have stopped at the same relative prices? Like, if the above votes were done with the marginal vote costing n, here’s what it would look like if the marginal vote cost 2n.
Post A: 2 votes
Post B: 4 votes
Post C: 0 votes
Which are all basically the same ratios (with a little bit less resolution), right?
If you scale it by a constant k that will happen (as the constant will just stick around in the derivative, and so you’ll buy votes until marginal cost = marginal benefit / k).
If you were to use like f(V)=V33 then each marginal vote would cost f′(V)=V2 , and so you’d buy a number of votes V such that V=√B (where B is your marginal benefit).
Some of the QV papers have uniqueness proofs that quadratic voting is the only voting scheme that satisfies some of their desiderata for optimality. I haven’t read it and don’t know exactly what it shows.
Yeah, I guess I’m hearing that all versions of it still cause the voter to work out prices, and that the information is findable by transforming their votes, but that quadratic doesn’t require doing any transformation and makes things simple.
I think that it’s not just about having an easier time reverse engineering people’s values from their votes. It might be deeper. Different rules might cause different equilibria/different proposals to win, etc. However I’m not sure and should probably just read the paper to find out the details.
Here’s an intuition for why it’s important that it’s quadratic (based on standard microeconomic reasoning).
By spending your votes, you pay some cost, and get some benefit.
The cost consists in: voting credits, time, attention, energy, reputation costs if you have odd views...
The benefit consists in: higher probability your post-of-choice ends up in the book, which comes with a host of externalities like further influence of your values and epistemics on readers of the book.
As long as cost < benefit, you want to keep voting (otherwise you’d be leaving benefits on the table). You’ll do this until the cost of the last vote = benefit of the last vote.
If your total cost of voting is f(V)=V22 then the cost of each marginal vote is f′(V)=2V2=V. By the above reasoning, you’ll stop voting when your marginal benefit = marginal cost = V.
Hence, your distribution of votes V1,V2,... across options o1,o2,...will measure how much you value each option being included in the book.
(After I’d written this I found this blog post by Vitalik which explains it even better.)
An issue with the proposal is the failure of the assumption of utility of votes being proportional to number of votes.
It seems plausible to me that there’s some threshold of votes above which a post will very likely end up in the book. If this is true, then I think my utility in buying more votes for that post would be ~linear up until that threshold, and then flat (ignoring potential down-voters).
If this is the case, I’d significantly understate my preference for this post, instead spending my points elsewhere.
So, for example, Embedded Agency might be undervalued by this scheme.
This point isn’t relevant when thinking about quadratic voting for elections with millions of voters, since then it makes more sense to assume that the probability of passing a proposal is linear in the number of votes I can influence.
Moreover, this would lead to weird equilibrium dynamics…
If I’ve voted for a post that gets above the threshold, then I want to remove my votes and place them elsewhere. If I don’t do this, but other users do, then I am effectively subsidising their preferences.
I don’t know how this would pan out, and can see it messing things up as everyone tries to model everyone else and be clever.
It seems like an open numerical question whether this issue would be relevant for the current round (i.e. whether the utility of most users would be linear in the region of influence they could expect with their votes):
Here are some numbers that I wrote down but now don’t really know how to take this further. Under Ben’s initial scheme, each user can buy at most ~32 votes per post, by spending all their money. There are ~500 voting users. Which gives an upper bound of ~16000 votes for a post. There are ~75 nominated posts, out of which ~25 will end up in the book. If all users distribute their votes uniformly, we’d have about ~3.65 votes per user per post, and ~1800 votes per post. Let’s handwave and say that with 7000+ votes a post is as good as guaranteed for the book.
I think a solution to this might be that if instead of voting on what should be in the book, you decide on some subsidy pile of karma+money, and you use quadratic funding to decide how to allocate that pile to each post (and then giving it to the author/eventual co-authors).
You might just include the top posts in this scheme in the book, also making sure to make their scores prominent (and perhaps using scores in other ways to allocate attention inside the book, e.g. how many comments you include).
It seems more plausible to me that under this scheme users utility would be linear in the amount of votes/amount of funding they allocate to posts.
This makes a lot of sense, thanks for writing it down.
Aside: I think it’s way harder for readers to read stuff bulletted like this, and would personally find it easier if this was identical but without the bulletting.
I came here to say ‘nah bro bullets are great’ and then I looked at the parent and the was like ‘oh geez thats too many bullets’
One interesting way to get around this problem would be to have votes be private.
This would probably create other weird dynamics.
Don’t think that would help—instead of knowing the actual votes Vi for post i, I would have some distribution P(Vi) over the votes cast for i, and my intuition is that as long as I have sufficient probability mass above the threshold, it would skew my incentives.
But doesn’t the knowledge that everyone else is also doing this converge to just stating my true preferences… Or something? I don’t really have a feel for the game theory behind this, but it feels like knowing that everyone is trying to vote strategically makes it hard for me to count on other people voting for things that I don’t vote on.
I don’t expect everyone to vote strategically. In fact, I expect most users to act in good-faith and do their best. I still think these things can be a problem.
That’s interesting, because I expect most people to vote strategically when using QV. The structure of QV heavily encourages thinking about the value of each marginal vote.
Trying to allocate your budget truthfully in accordance with your preferences about posts != trying to game the rules as an unbounded EV-maximiser would.
Strategic voting for me = trying to think how much value your vote has relative to the outcome you’re trying to achieve. I don’t see for instance looking at how many people have already voted on something as “gaming the rules”, it just changes the value of a marginal vote of my own. I expect most people to think like that because QV is already making you think about the marginal value of another vote.
I agree that you will stop voting on a post when the marginal cost = marginal benefit. This means if I take my votes from above:
Post A: 5 votes
Post B: 8 votes
Post C: 1 vote
This means I valued a vote on post A at price 5, post B at price 8, and post C at price 1, where the prices are all relative to each other.
So my noob question is: Don’t I know this relative pricing having to say what the marginal cost function is? If each marginal vote had costed 2n, wouldn’t I still have stopped at the same relative prices? Like, if the above votes were done with the marginal vote costing n, here’s what it would look like if the marginal vote cost 2n.
Post A: 2 votes
Post B: 4 votes
Post C: 0 votes
Which are all basically the same ratios (with a little bit less resolution), right?
If you scale it by a constant k that will happen (as the constant will just stick around in the derivative, and so you’ll buy votes until marginal cost = marginal benefit / k).
If you were to use like f(V)=V33 then each marginal vote would cost f′(V)=V2 , and so you’d buy a number of votes V such that V=√B (where B is your marginal benefit).
Some of the QV papers have uniqueness proofs that quadratic voting is the only voting scheme that satisfies some of their desiderata for optimality. I haven’t read it and don’t know exactly what it shows.
Yeah, I guess I’m hearing that all versions of it still cause the voter to work out prices, and that the information is findable by transforming their votes, but that quadratic doesn’t require doing any transformation and makes things simple.
I think that it’s not just about having an easier time reverse engineering people’s values from their votes. It might be deeper. Different rules might cause different equilibria/different proposals to win, etc. However I’m not sure and should probably just read the paper to find out the details.