In honor of the forthcoming visit of Glen Weyl to the University of Rochester, I thought I’d offer a post explaining the idea behind one of Glen’s signature policy reforms: quadratic voting.
Suppose we’re going to hold a referendum on, say, whether to build a street light in our neighborhood.
The problem with giving everybody one vote is that (on both sides of the issue) some people care a lot more about that street light than others do. We’d like those who care more to get more votes.
In fact, we’d like to allocate votes proportional to each voter’s willingness to pay to influence the outcome. There are excellent reasons to think that willingness-to-pay is the right measure of “caring”. Those reasons will be evident to readers with some knowledge of welfare economics and opaque to others, but it would take us to far afield for me to get into them here. (For the record, if you’re encountering this measure for the first time, you’re almost surely raising “obvious” objections to which there are non-obvious but excellent rejoinders.) For this discussion, I’m going to take it as given that this is the right way to allocate votes.
Here’s the problem: If I allocate votes based on willingness to pay, people will simply lie. If you’re willing to pay up to $1 to prevent the street light, but know that you can get more votes by exaggerating your passion, that’s what you’ll probably do.
Okay, then. If we want to allocate votes based on willingness to pay, then we have to make people actually put some money on the table and buy their votes, thereby proving that they care. We could, for example, sell votes for $1 each. That way, people who care more will buy more votes and have more influence, as they should.
Unfortunately, that’s not good enough. If you care more about the issue than I do, you might buy more votes than I do — but there’s no reason to think you’ll buy more votes in direct proportion to your willingness to pay. Let’s suppose, for example, that the ability to cast a vote is worth $2 to you and $4 to me. Then I should get twice as many votes as you. But if votes sell for $3, I might buy quite a few, whereas you’ll buy none at all. That’s a lot more than twice as many.
So let’s try again: Instead of selling votes for a fixed dollar amount, we sell them on an increasing scale. You can buy one vote for a dollar, or two votes for four dollars, or three votes for nine dollars — and we’ll even let you buy in tiny fractions, like 1/10 of a vote for a penny. The price you pay is the square of the number of votes you buy. That’s the definition of quadratic voting.
Why the square, as opposed to the cube or the square root or the exponential? There really is something special about the square. To appreciate it, try an example: If a vote is worth, say, $8 to you, you’ll keep buying additional votes as long as you can get them for less than $8 each, and then stop. With quadratic voting, one vote costs you a dollar. You’ll take it! A second vote costs you an extra $3 (bringing the total to $4). You’ll take that too! A third vote costs you an extra $5, a fourth costs you an extra $7, and a fifth costs you an extra $9. So you’ll buy 4 votes and then stop. You can similarly check that if a vote is worth $24 to your cousin Jeter, Jeter will buy twelve votes and then stop. Jeter cares three times as much as you do, and he buys three times as many votes. And with a little calculus, you can check that if Aunt Murgatroyd’s vote is worth four or five or nine or twenty times more to her than your vote is to you, she’ll buy exactly four or five or nine or twenty times as many votes as you do. That’s exactly what we wanted. In that sense, this voting scheme works — and, except for minor variations, it’s the only scheme that works.
For the math geeks: If a vote is worth $N to you, and if C(V) is the cost of V votes, you’ll want to maximize NV-C(V). (This assumes that you’re a small enough player that your votes don’t materially affect the likely outcomes, so that if your first vote is worth N, then so are all your subsequent votes. It also assumes that the payments you’re making are small enough so that you don’t feel substantially poorer, and therefore don’t change your mind about what your votes are worth.) When C(V)=V^2, the solution to this problem is V=N/2, which is (of course) proportional to N. Essentially no other function can make this claim.
(We could replace C(V) with some function like 10V^2 instead of just V^2, but then all that would happen is that everyone would buy 1/100 as many votes as before, leading to no change in either the amount they pay or the election outcome.)
Now, in Glen Weyl’s preferred version of quadratic voting, you buy your votes not with dollars but with “voting credits” that are issued every year, in the amount of, say, 100 credits per voter. You can use your credits to buy votes in any of the various referenda that come up from time to time. But the main insight is quite independent of whether you’re paying with dollars, voting credits, or anything else — what matters is not the currency, but the pricing scheme, which has to be quadratic to get the right results. Out of all the possible voting schemes, this is the only one that works.
Edited to add: Here is some thoughtful criticism. Thanks to the ever-thoughtful John Goodman for the pointer.
Would voting credits (like dollars) continue to accrue so long as you don’t use them?
In a dollars-based system, clearly politicians would have a strong incentive to appeal to the rich. But in a voting credit system, politicians would have a strong incentive to appeal to non-voters. They might promote lower tax rates for people who had spent years in hospitals or prisons, in an effort to appeal to people who hadn’t been to the polls in a long time.
Some thoughts here.
1. “Minor variations” I assume means k votes costs P(k) units for some quadratic P? The triangle numbers might be “best” because then the nth vote costs n dollars/credits?
2. How does this get administered exactly? I’m not sure it can’t be done but it doesn’t feel particularly obvious that no one would be able to abuse this. Say by showing up n times and casting a single (or fractional) ballot each time (or paying someone indifferent to cast a single borrowed vote for them etc).
3. I might be willing to pay an awful lot for the first vote. But after a certain amount of bought votes I’ll be pretty sure I’ve won and will pay nothing. Heck if I think victory (or defeat) is certain I’m pretty likely to not vote at all! Why waste voting credits on a sure (or lost) thing. Is there any work on how people will change voting behaviour under this kind of pressure to conserve votes?
@Jonathan regarding #3, this assumption is made explicit: “This assumes that you’re a small enough player that your votes don’t materially affect the likely outcomes, so that if your first vote is worth N, then so are all your subsequent votes.”
@Bennett Haselton: Ah, I missed that. I could see how the calculus would go and skimmed the actual calculation. Thanks.
@Jonathan however you’re right that sometimes people will feel (correctly or incorrectly) that the election is a foregone conclusion and so they won’t buy votes to reflect their preferences.
A voter would only bother if they think the election might be close. But if they think the election will be close, then they might assign different values to their different votes, and then another of the main assumptions is no longer valid.
It seems to me like there’s a simpler way to do this which avoids the free-rider problem (where people withhold votes if they think the election won’t be close) and the problem where people’s votes have different values if they think the election *will* be close.
Why can’t you do it like a giant multi-player second-price auction, as follows:
Everybody writes down their vote and a dollar amount. The side with the higher sum wins. We then compute the difference $x between the winning sum and the losing sum, and then anybody on the winning side who write an amount greater than or equal to $x, has to pay $x. (Possibly nobody has to pay, if the winning margin is greater than the individual amounts that anybody wrote.)
I submit that under these rules, everybody will write down the exact amount that their preference is worth to them.
Consider the election from a point of view of an individual voter, taking everyone else’s votes as a given. If, even without your vote, your side’s total is already greater than the other side’s, then it doesn’t matter what you write down — your side will win and you won’t have to pay anything.
But it’s possible that, without your vote, your side’s total is less than the other side’s total by some unknown amount. In this case, if victory to you is worth $8, then you’ll write down exactly $8. You wouldn’t write down $7, because maybe the margin will be $7.50 and then you will lose the election when it would have been worth paying $8 to win it. And you wouldn’t write down $9, because then maybe the margin will be $8.50 and your side will “win” but you’ll have to pay $8.50 even though victory to you was only worth $8. So you write down exactly $8. (This is the same argument why people in a second-price auction bid exactly what they think the item is worth.)
It seems like this logic is sound no matter what you expect anybody else to do, so you don’t have to worry about free-riders holding out because they think their side will win anyway, and you don’t get into complicated problems about whether each additional vote is worth “the same” if you think the election will be close.
Isn’t this solution optimal? Am I missing anything?
Many legislatures–including the legislatures of 41 US states–are forbidden to adopt legislation addressing more than one subject. Among other things, this policy is designed to prohibit unpopular policy from getting passed by bundling it in with popular policy.
Courts have been reluctant to strike down legislation merely for violating the single subject clause. Admittedly, the boundaries between related “subjects” is vague. But also, judges–many of whom are former legislators–may feel reluctant to undo compromises hammered out by the legislature. How else could anything get passed?
I wonder if quadratic voting by a legislature would persuade courts that legislative compromises are no longer necessary to pass bills, and that each provision really should rise or fall on its own merits?
Conversely, would the party process manipulate quadratic voting? Clearly a party would marshal its members to vote for the bills it supports. But would quadratic voting provide some new, strategic advantage (or disadvantage) to a coordinated group?
Annoying typo: “but it would take us to far afield for me to get into them here.” Should be too.
The quadratic voting is fascinating. The obvious concern is the wealthy buying all the votes. Bill gates is worth,say a million times the average American, However, he only gets 1000 times as many votes for the same proportion of his worth, which is not enough on its own to buy the election, but is probably enough for people to have some concerns.
Issuing 100 credits to be used as desired would work best if there were lots of referenda. If they carry over from year to year, lots of people in, say Scotland, would not spend any until the independence referendum came along. This may not be a bad thing, allowing those who do care more about street lighting to carry the day when it comes up.
Very thought provoking. I like the second price auction idea too, but not thought it through yet.
I believe Weyl used same sex marriage as an example of the type of legislation where those that care deeply could pay quadratic prices to get it passed. However, there are many on both sides that seem to care deeply, so they could cancel out. I am sure Weyl did a somewhat fuller analysis than that.
Still pondering quadratic voting by legislatures. And cards.
1. I have an initial adverse reaction to the idea that we’d be able to look at each party’s balance of credits and know that a minority had the resources to sway any vote. This seems like a poker table in which one player clearly has sufficient resources to outbid all the others, regardless of the merits of this cards–thereby eliminating anyone’s incentive to play. Many poker tables deal with this problem by imposing a maximum bid. Would such a modification unduly mar the elegant simplicity of Weyl’s proposal?
2. Imagine that Paul Ryan, Republican Speaker of the House, proposes a bill to double the tax rates applicable to Democrats. Democrats hate this, and so outbid the Republicans in order to kill the proposal. Ryan then changes a semicolon in the bill and offers the revised version as a new bill. Democrats would again have to out-bid the Republicans. And so it would go, with Democrats burning credits at a faster pace then Republicans.
In the game of Contract Bridge, we call this “flushing out the trump”–but in Bridge, the stronger team generally will expend at least as many trump cards as the weaker one in implementing the strategy.
Maybe we’d want to impose some kind of limit on the number of times an issue could be raised in a given session? (Policing that rule would not be a lot of fun.)
Professor Landsberg, are there an infinite number of votes available? While the degree to which one cares about an issue may best be expressed economically, the ability to quantify one’s passions will be a function of net worth: what does the value of a vote mean to Jeff Bezos as compared with Jeff Nobody? To avoid the street light, Bezos may care $1 million worth; he might not care about cost as much as he cares about comfort. If the winner always is the one with the highest net worth, why not simply auction the decision? And where does the winners’ cash go after the vote? Respectfully submitted, Chas Phillips
Charles G Phillips:
The response to your concern about Jeff Bezos versus Jeff Nobody is the same as the response that comes up in any discussion of economic efficiency. Namely: Bezos is much richer than Nobody, so he’s going to get more of *something* than Nobody gets. If you think that’s a problem and want to remedy it, then you should be talking about income redistribution, not voting schemes. But, taking Bezos’s wealth as given, if Bezos values both an election victory and an Olympic swimming pool at the same $1 million, and if Jeff Nobody would rather lose the election than lose access to the resources that build Bezos’s swimming pool, then it is indeed better for Bezos to win the election than to build the swimming pool.
The money collected of course can be distributed to the Jeff Nobodys of the world. If you do things write, the Jeff Nobodys will prefer this reform to the status quo, precisely because they lose a lot of elections (which makes them sad) but get a lot of payments (which more than makes up for the sadness).
Professor Landsburg, Quadtartic voting is new to me but if you are saying that the ultimate extension of the concept is that wealth has the ability to control elections, I am all for it. I was not clear in my post: if we followed economic efficently in all matters, life would improve for all Americans. Thank you for your response. Regards, Chas Phillips
You all may find it helpful to preview other talks Weyl has given. For example, at the link below he discusses the robustness of QV to relaxing its assumptions:
It’s an interesting concept and I have enjoyed reading about it. I don’t understand what problem it is trying to solve, though. More precisely, it does not strike me that the cardinality of voting is the biggest problem with the process, or even a mildly concerning one.
The document you link to at the end points to the problem of collusion. So let’s say I want to buy 10 votes, and you aren’t going to vote. I’m better off paying you 25 dollars to cast 5 votes for my side, slipping you a fiver for your trouble, and casting 5 votes myself for a total cost of 55 dollars.
While this is definitely possible, I want to point out that quadratic voting is almost certainly an improvement on the status quo in this respect. More generally, consider a p-voting system, where it costs x^p dollars to cast x votes. Then QV is a p-voting system with p=2, and ordinary voting is a p-voting system with p=infinity. It’s easy to convince yourself that buying votes outside the system is more attractive the greater p is.
I’m all for trying to make voting less irrational…although in that sense this struck me as like buying a bunch (but say still a fairly small %) of raffle tickets (say for an event where more tickets are sold than the item is worth)…does doing something irrational several times make it rational?
More to the point here, I found myself pausing at the “obvious” suggestion that “we’d like those who care more to get more votes.”
1) This seems to play into special interest politics. If personal gain is a stronger motivator than perception of public good, a virtue of one-vote is you have to persuade others of the benefits to others.
2) It also seems to play into more “busybody” politics. Governments like to do a lot of goofy / trivial things that have idiosyncratic appeal to some, which fortunately is sometimes offset by the “meh” of the broader public (e.g. who may be taxed ratably for these things). It seems this proposal raises the required “meh” factor quite a bit.
I fully expect there are non-obvious but excellent rejoinders to these concerns.
How does this fit with Arrow’s theorems? As I see it it is introducing another factor – peoples intensity of desire – which Arrow did not consider, at least in his Impossibility Theorem. If we introduce this factor (assuming we could measure it) would this affect Arrow’s outcomes?
I had a look at their paper:
https://chicagounbound.uchicago.edu/cgi/viewcontent.cgi?article=1649&context=law_and_economics
it is well presented and argued with lots of interesting historical context and discussion of tyranny of the majority.
One minor niggle
” And quadratic pricing minimizes the impact of large wealth disparities because the cost of buying votes increases exponentially.”
I think this is not technically exponential as the power is 2 not X. Is this right? However, this makes the point I made above that a rich person needs to spend $1 million to outvote 1000 poor people spending $1 each.
That aside I recommend anyone interested to read it.
A key component is the redistribution of the vote revenue on a pro rata basis which compensates the losing side. In the example above the 1000 poor people would receive $1000 each from the rich person, compensating them to a large extent for the loss of the vote.
They point out that QV should never allow direct transfer of money from poor to rich since the rich would have to spend more money buying the votes than they gain from the poor. This assumes everyone can work out how much to spend on the votes and behaves rationally.
” Just like today, rich people will have an advantage under QV. But while rich people under the current system can improve their electoral prospects by using personal funds to buy advertising, rich people under QV would use their funds to buy votes,”
They point out that Berlusconi would still be better off spending his money in advertising than buying votes, since he could only get 0.2% of the votes with 2.5 billion Euros, about half his wealth.
“In fact, under QV it would still be necessary to outlaw extra-system vote buying that could be used to undermine the quadratic nature of costs by allowing one individual to buy votes as a proxy for another.”
This could be a big problem.
“Suppose, for example, that people generally overestimate the probability of being pivotal, or vote for reasons unrelated to the specific gains from an election such as a sense of civic duty.”
This could be another major flaw. People generally vote for reasons such as civic duty and breaking the one-person-one-vote criterion might suddenly remove peoples’ sense of duty. The reasons for voting are difficult to elucidate, but risking undermining the reasons should not be undertaken lightly. it is true that rich people can buy influence in elections through advertising, but by making the link direct and part of the system may weaken the general emotional boost people get from voting.
On the Arrow point, they say “Lurking in the background is Arrow’s
theorem, which proves that under relatively broad conditions, no voting system can produce outcomes that are both Pareto-efficient and non-dictatorial. Arrow’s theorem assumes ordinal preferences; the quadratic voting system we discuss below does not.”
I think they are saying that QV should in principle be both Pareto Efficient and non-dictatorial, thus fulfilling a new set of reasonable conditions including Arrow’s original ones plus a new one of intensity of desire. I am not sure if this is the case and would welcome any opinions.
Clarification on Arrow.
They say “While it might be better if they were fully compensated, full compensation (which would guarantee Pareto outcomes) is not practical.”
I think this refers to the pro rata redistribution. It would in theory be possible to design a redistribution system that fully compensated people, but this is not practical rather than impossible. Thus the QV as described would not be Pareto efficient, but would be much better than current systems, but a Pareto efficient, not dictatorial QV system is possible in theory.
I like Bennett Haselton’s idea above of combining it with a 2nd price auction. Theoretically, it seems right, but I think it’s too complicated a scheme to ever become reality (a fact that will also prevent quadratic voting from taking off, too).
The second price auction sounded appealing to me also but on reading the paper the function is taken by the redistribution of revenue from the vote sales. I also don’t see how you combine the two ideas.
I am not quite sure how the second price thing works.
A park is proposed. Votes are $1.
There are 5 people nearby who would benefit a lot. They each buy 10 votes for $10. Votes for = 50
There are 10 people a bit further out who would benefit a bit. They each buy 5 votes for $5. Votes for = 50, plus the 50 from above = 100
There are 80 people further out who don’t mind much, but would slightly prefer to spend the money on a ring road. They each buy 1 vote for $1. Votes against = 80.
The proposal is carried by 100 to 80.
X (the difference between yay and nay votes) = 20
Total votes cast = 180
Total number of voters = 95
Average difference per vote = $0.11
Average difference per voter = $0.21
What value do we use for X to decide who pays? In the scenario above, if we use an average, then the winners are getting a massive bargain.
If we use the total figure it is overwhelmingly likely that almost nobody will have to pay.
As an example, say the no voters cared a bit more and paid $1.10 each. They would have 110 votes, so NO vote wins. X = $10. Nobody pays. (It seems a shame to keep charging nobody, but that’s Homer for you.)
Harold,
Arrow’s theorem applies only to plurality voting where each person has 1 vote. I didn’t read their paper like you did, but it sounds like they are just saying, since Arrow’s proof did not cover a system such as QV, we don’t know whether the theorem holds for QV or not.
Henri, thanks, that was roughly what I thought. I was wondering if an Arrow-esqe series of conditions could be applied to this type of voting and if it could then be proved to be impossible to meet all them simultaneously.
looking further it seems Gibbard’s Theorem may be more applicable, this states (wikipedia version)
“for any deterministic process of collective decision, at least one of the following three properties must hold:
1. The process is dictatorial, i.e. there exists a distinguished agent who can impose the outcome;
2. The process limits the possible outcomes to two options only;
3. The process encourages agents to think strategically: once an agent has identified her preferences, she has no action at her disposal that would best defend her opinions in any situation.
I presume this would comply with Gibbard’s Theorem and thus it can be said that this is not a mathematically perfect system (not that I suggest anyone made that claim).
Harold —
In the specific scenario that you proposed, in my system, nobody would have to pay. The margin of victory ($20) is greater than the amount that anybody individually voted ($10 or less). My $X is just the margin of victory, not an average of anything.
My reasoning for this second-price structure was that it motivates everybody to vote by an amount that is exactly what they think their preferred outcome is worth. (The argument for this is given in my comment.)
With any other system, you would have people trying to game the system depending on what they think other people would do. For example, if the rule is “Everybody on the winning side has to pay their bid amount,” then if someone thinks their side already has enough votes to win anyway, they’ll bid less than what victory is really worth to them, so that they won’t have to pay as much.
But in the second-price system, everybody votes exactly what they think victory is worth to them, and the winner is the side with the greatest total, which is what we want, right?
You wrote, “If we use the total figure it is overwhelmingly likely that almost nobody will have to pay.” Well yes, but in any system with a lot of voters, it is overwhelmingly likely that your vote will not influence the outcome anyway. What the second-price system does is force the voter to think: If your vote *does* end up making a difference, what would it be worth to you to win? And then add up all those values on both sides.
Perhaps I’m not understanding, do you think my original argument and conclusion is correct, or incorrect?
“Well yes, but in any system with a lot of voters, it is overwhelmingly likely that your vote will not influence the outcome anyway” OK, I think I am seeing it now. You explained it correctly and the lack of understanding was mine.
In my example, if instead 5 people spending $10 each it was one person spending $50, or 2 people spending $25 each, then those people would end up paying $20.
If it was one person and they valued it at $30, the total votes for and against would be 80 and the difference zero. So if this one person valued it at $31 he would win and pay $1. If he valued it at $29 he would lose and pay nothing. The more votes he buys, the bigger is X and the more he ends up paying, so in principle he will not buy more than he is willing to pay for. If he wants to be really sure he could buy 1000 votes. X is then $970 and he pays $970, almost what he paid. So he would not buy that many votes unless it was worth it to him.
I think I have spotted a problem with this arrangement, which ios the wealthy buying the election. In the above case if the person is wealthy he might not miss $1000 and would be happy to pay this for the park. In fact, he could outbid the poor for a direct transfer from the poor to himself.
If the proposal had been that everyone else pays him $1. A rational person would spend up to $1 to avoid paying the dollar, so the other 85 people all buy one vote for $1 since this is the maximum they are prepared to pay to win. He buys 86 votes, pays X which is $1 and receives $85 from the rest.
Is this a genuine problem? QV avoids this because the wealthy need to spend more and more to buy each extra vote. They would need to spend $7396 to buy 86 votes for a receipt of $85.
Let me know if I have made an error.
@Harold,
Whether “the wealthy buying the election” is a problem or not is probably a philosophical question; I wouldn’t want this system used for every election or every referendum (why should the super-rich be able to pay to make gay marriage illegal again?), but it could be useful for voting on something that affects everyone, such as the original example of whether a streetlight should be built. Remember, if the super-rich buy the election, the winners pay money in to the general fund, which should in theory benefit everyone in the form of lower taxes or increased benefits (or could just be given to everyone as a stimulus check), so the losers do get compensated.
However, you could also use Weyl’s idea of 100 voting credits that are issued annually, rather than money. I think all of the logic still holds if you’re using voting credits rather than dollars.
@Harold (actually everyone),
Sorry, I made a major error in my description of how the second-price voting system would work (although with the error fixed, I still think the conclusion is correct and it’s the optimal solution). I wrote:
“We then compute the difference $x between the winning sum and the losing sum, and then anybody on the winning side who write an amount greater than or equal to $x, has to pay $x. (Possibly nobody has to pay, if the winning margin is greater than the individual amounts that anybody wrote.)”
What I should have said was:
“We then compute the difference $x between the winning sum and the losing sum, and then anybody on the winning side who write an amount greater than or equal to $x, has to pay the amount that they wrote down, minus $x. In other words, they have to pay the amount that *would* have been the other side’s margin of victory, had it not been for their own vote. (Possibly nobody has to pay, if the winning margin is greater than the individual amounts that anybody wrote.)”
This is how a true second-price auction works. If you imagine a system with two voters voting on who should get a painting, then if your bid is more than the other voter’s bid by $x, then you pay your own bid minus $x — in other words, you pay the other bidder’s bid.
But the rest of the logic was correct, although I wrote the wrong numbers in some places. (For example, I wrote, ‘And you wouldn’t write down $9, because then maybe the margin will be $8.50 and your side will “win” but you’ll have to pay $8.50 even though victory to you was only worth $8.’ The first ‘$8.50’ should instead say ‘$0.50’: ‘And you wouldn’t write down $9, because then maybe the margin will be $0.50 and your side will “win” but you’ll have to pay $8.50 even though victory to you was only worth $8.’)
Here’s a much simpler way of looking at it:
Imagine you as one bidder, and the collective rest of the electorate (the voters on your side and the voters on the other side) as the other bidder, and you’re “bidding” on how some resource should be used. If the rest of the electorate (without you) is already voting in your favor, then your vote is moot; this is the equivalent of an auction where the other bidder might just bid $0 and give you the thing for free. However, if the rest of the votes might be against your favor until you cast your vote, now the system I’ve described works like a second-price auction. And now you can just use the well-established result that both parties in a second-price auction will bid exactly what the item is worth to them.
(What I was trying to spell out explicitly in my first post, was the logic why everyone in a second-price auction bids exactly what the result is worth to them, but I bungled the numbers.)
@Harold,
I was thinking about your objection: “In fact, he could outbid the poor for a direct transfer from the poor to himself.”
I crunched the numbers and my conclusion is: (1) you’re right; (2) however, this is a problem with *any* voting system that just weighs people’s preferences against each other, and (3) the real issue is not who is “rich” but who has the ability to get an initiative like that on the ballot in the first place.
(1) Suppose there are two voters, Alice and Bob, voting on the measure “Bob has to give $100 to Alice.” In my second-price system, Bob will bid $100 against and Alice will bid $100 for. If things are “off by a little” and Alice bids $101, she pays $101 to take Bob’s $100, so she’s worse off by $1 but Bob is worse off by $100. If Alice bids only $99, Bob “wins” but pays $99 and is almost as badly off as if he’d lost. (Equivalently, if you flip a coin in the case of a tie, then you get one of these two scenarios and Bob is about equally badly off in both of them.)
It’s even simpler if Bob has something that Alice values more. Suppose Bob values his car at $1,000 but Alice is willing to pay $1,200 to get it. In a market system, he’d just sell it to her, but if “Bob gives Alice his car” is up for a vote, she will win the vote $1,200 to his $1,000 and she’ll pay a $1,200 voting fee for the car and Bob gets nothing. This is “economically efficient” since the resource goes to the person who values it more, but it’s not very fair to Bob.
(2) This is, as I said, a problem with any system that just weighs people’s preferences. Alice’s preference for Bob’s $100 is exactly equal to Bob’s preference for keeping his $100, so any system that weighs preferences is going to result in a tie, and if Alice wants it slightly more, she’ll get it. (Presumably this would happen with quadratic voting too.) We might protest, “But it’s Bob’s money/car!”, but a preference-weighing system doesn’t take that sense of fairness into account.
(3) Note that none of this has anything to do with Alice being rich or her ability to “outbid” Bob. If Bob has something and Alice wants it more, in a pure preference-weighing voting system, she’ll be able to get it.
The problem, I think, is that an initiative like “Bob has to give Alice his car” could end up on the ballot in the first place. Once that measure is on the ballot, Bob is kind of screwed, because he could lose if Alice wants it more, but even if Bob wins, he’ll have to “spend” something of value (money, or voting credits) to defeat the initiative. This is true for almost any type of voting system.
I think the only way to avoid this problem is to have some a priori rules about what kinds of things can end up on the ballot. Because we want to use total-preference-weighing whenever possible, however if we use that system to vote on ownership of Bob’s car, it will end up being taken away from him if someone else want it more.
Bennett. This gets confusing, even though it is really quite simple.
1) In your Alice case, wouldn’t she pay $1200 – X? That is a voting fee of $1000, which makes sense in a second price auction as it was the second best “offer”.
2) The point is that you will probably pay less than the maximum you are willing to pay, but you may have to pay up to that maximum. The elegant factor pertaining to elections is that the amount you pay is a direct result of the effect you had on that election. You only pay close to your maximum bid IF you had a big effect.
3) In QV they say it is impossible to win by outbidding the poor for direct transfer. I don’t doubt they crunched the numbers correctly. It works because 1000 poor people get 1000 votes for a dollar each whereas a rich person has to spend $1 million to get 1000 votes. It apparently works out that the rich person will always be worse off, if everyone votes according to their actual gains/losses.
4) QV only maximises social welfare if the voting fees are re-distributed pro rata. If this were applied also to your Alice and Bob example, only re-distributing to those that didn’t pay, then Bob gets $1000, which is exactly what he valued the car at.
@Harold:
1) Oops yes I should have said Alice pays a $1,000 voting fee for the car.
2) That sounds correct.
3) In that example, the reason the rich person can’t outbid the poor for direct transfer is that there is 1 rich person and 1,000 poor people. But in my simplified example where there are just two people, you can still get a tie vote when voting for one person to take the assets of the other person. If 10 people want to divide up the assets of 1 person, QV makes it easy (the 11th person will have to “spend” a lot to defend themselves).
4) I was not assuming that the voting fees get paid *to* the losing side.
Actually, I tried to find an algorithm that would work if the voting fee is paid to the losers, to counteract the effects of “Alice and Bob voting on whether Bob gives his car to Alice.” At first it sounds like you could solve this problem by having the winners pay the losers, but the problem is that if you generalize this to an election with many voters, it changes people’s voting incentives. If you think your preferred side will win anyway, now you have an incentive to vote for the other side, so that you “lose” and you get a cash payout.
Now people will try to game the system by writing down votes and amounts that don’t reflect their true preferences, and so you can no longer sum up everybody’s preferences to decide what to do.
I’m not sure if it’s possible to design a system that achieves both:
– ensures that everyone will reveal their preference accurately, so that you can sum up their preferences to decide what to do; and
– guards against the scenario of “Alice and Bob voting on who gets Bob’s car” by ensuring that Alice has to compensate Bob.
The challenge is to either find a solution or prove that it can’t be done.
Bennett 27 – “the only way to avoid this problem is to have some a priori rules about what kinds of things can end up on the ballot.”
I agree this is a problem with any voting system *at all*; some call such rules “individual rights”. But Weyl proposes (at least in theory) getting rid of private property precisely because everyone can be better off if we auction off Bob’s car (and everything else) to the highest bidders…or compensate the couple who can’t get married with the “honeymoon” vacation of their dreams?
Again I find myself stuck on the initial premise e.g. from the Amazon synopsis: “Many blame today’s economic inequality, stagnation, and political instability on the free market.” Who and why? (Not politicians I mean real people.) Compared to what? Is all inequality equal or does the source matter (e.g. voluntary exchange vs kleptocracy)? To echo Henri Hein, “what problem exactly are we trying to solve again?”
@iceman
It seems like private property helps to *avoid* this problem, doesn’t it? If Bob owns his car, he *can* auction it off to the highest bidder — which might be himself, if nobody else values the car more than Bob does.
#29 Bennett.
3. Weyl et al do make the point that QV works best with many voters.
“QV works best with a large number of voters: the more voters there are,the more accurately the system works. QV’s efficiency relies on all voters perceiving the chance of their changing the outcome with an additional vote as the same.”
On the poor ganging up to rob the rich, they have a section in redistribution in general. The result of a Rawlsian “veil of ignorance” vote they claim would be optimum social welfare.
If investments had all been made, but individuals were uncertain about their income, then there would be temptation to impose 100% taxes. “even though, having known this beforehand, no one would have made any investment. This is a well-known time-consistency problem that can easily lead to excessive taxation of accumulated capital…”
I think this is close to the case under discussion. They do acknowledge that QV does not block time -consistency problems like this, but neither do majority or supermajority systems. Countries tend to avoid this situation because they have an eye in the future.
A key claim is “But QV, unlike those other rules, does block time-inconsistent policies that are purely redistributive to the majority (or supermajority).”
I don’t see why this is the case. The poor majority is not only likely to outvote the rich minority, but even if they lose the redistribution will pay them anyway.
Their example of why the rich cannot rob the poor:
”
. Consider a scheme that took a dollar from every individual in the bottom 50% of the income distribution and gave $50 to every individual in the top 1%. A member of the top 1% would not pay more than $49 to enact such a proposal, which would give them 7 votes each. Such a proposal could be defeated by the bottom 50%, each buying 0.5 of a vote, which would only cost him or her a quarter each. Robbing the poor to pay the rich never prevails under QV.”
If we swap this round, the proposal is to take $50 from each in the top 1% and give $1 to each person in the bottom 50%. The rich people get 7 votes each. The poor outvote them for a quarter each – well worth it.
I don’t understand the claim that:
“QV blocks purely redistributive projects because it permits only efficient projects and redistributive projects are not efficient because they do not generate wealth.”
#39. Iceman. ““Many blame today’s economic inequality, stagnation, and political instability on the free market.”
In my (limited) experience people are more likely to blame it on Capitalism than the “free market.”
However, if “free market” means a market left entirely to its own devices, complete with all well known market failures such as externalities, monopolies, time inconsistencies, information asymmetries, principal-agent problems, public goods etc, then we know we will end up far from an economically efficient outcome.
Bennett 31 – yeah I would certainly include private property under “liberty rights”. If by auction to highest bidder or himself you mean Bob simply has a reservation price for selling. Not sure how Weyl’s public auction for public benefit makes everyone better off including Bob. Maybe he has in mind a repeated game. I was wondering if he’s fully accounted for the incentive effects of eliminating private property in a repeated game context. I suppose I could read the book.
Harold 32 — “blame today’s…” could only mean free markets as / to the extent practiced in the real world, so again the question is, compared to what? Where do we see political instability and economic stagnation? More monopolies? What system doesn’t have to deal with agency issues, information asymmetries, externalities (where do we even see more pollution)? My point was simply that there may be tweaks to improve on “pure capitalism” but the sales pitch for the book seemed to set up a much bigger straw man.
Yes, I am one of those people raising “obvious” objections. The most obvious – willingness to pay would be a great measure of “caring” in a world where everyone was equally able to pay; but clearly some people are richer than others, and such a vote would be distorted in favour of the richest.
My second objection is that these “votes” (and perhaps “shares” might be a better word, since this model is closer to the management of a public company owned by shareholders, rather than the one-member-one-vote purity of a representative democracy) are a positional good, their value is highly dependent on how many are being bought by other people. In particular, the more people buy shares, the less valuable they are because each individual share is less likely to swing the result. This creates an unappealing first-mover advantage, where the first few people buy a bunch of votes, and this deters other from participating.
My final objection is specifically the quadratic nature of the voting. It’s mathematically elegant but vulnerable to abuse. Say I am willing to pay $10 per share. Then you predict I’ll buy five shares for $25, and no more. But I could get other people to vote on my behalf. For the same $25, I could pay twenty other people (who otherwise don’t care at all about the street light) a dollar to vote the way I tell them to, and they could make a commission of 25 cents each to make it worth their time. I’ve now achieved four times the influence for the exact same financial outlay.
I think this is one of those economic ideas that is beautiful in theory but absurd in practice.
Polltroll: Your first objection is indeed obvious, but is also wrong.
If Bill Gates is richer than me, then he’s going to have more of some things than I do. The goal is to induce him to make choices that account for my preferences.
In particular: Suppose Bill and I live next door to each other and we’re arguing about whether the town should install a street light. It’s worth $20 to me to have that light there, and to me that’s a lot of money. It’s worth $10,000 to Bill to keep the street dark, and to Bill that’s pocket change.
If Bill spends $10,000 to buy votes to get his way, and if that $10,000 is then redistributed to the people who live on the block, I’m going to end up with more than enough money to compensate me for having to live without a street light. That’s a good outcome. The alternative is that Bill spends his $10,000 in a way that hurts me a lot more (say by buying up so many Thanksgiving turkeys that I can no longer afford one).
What you seem to be thinking is that you don’t like the fact that Bill is so much richer than me, and you’re annoyed that quadratic voting fails to solve that problem. You might as well be annoyed with the polio vaccine, which also doesn’t solve that problem. But it does an excellent job of solving the problem it’s designed to solve.
If you don’t like the income distribution, you can think about ways to change that. But if you choose to address the question: What is the best way to make social decisions *given* the existing income distribution — and given that rich people who don’t spend their wealth on one thing will spend it on another — then quadratic voting speaks to the question directly. There are all kinds of legitimate questions one might raise, but yours isn’t one of them.
I want to push back a little on Steve’s reply to polltroll. Like the top-notch economist he is, Steve simply assumes that the money Bill spends on buying votes to get the outcome he wants on the street lamp issue will somehow magically be redistributed to the losers of the street lamp vote.
But given transaction costs (and given Coase’s critique of welfare economics generally), that possibility is ludicrous! The administrative costs of making this transfer will be large, especially if we are voting on micro issues like street lamps. (Just imagine how large the ballot would be if it encompassed which potholes to fill and which lamps to replace.)
My larger point, however, is that perhaps there are some things in which we don’t want efficient (or welfare enhancing) outcomes and that elections might be one of those things. After all, once elected into office, the candidates are the ones who get write or enforce or interpret the rules, depending on the type of office the candidate is running for. Do we want the rule-making process (or rule enforcement or rule interpretation, as the case may be) to go to the highest bidder? Earlier in my career, I would have said yes. See: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1568905.) But now, I’m not so sure.