Arrow’s Theorem is often invoked as a criticism of alternative voting systems (RCV, etc). And not while not wrong exactly, it seems textbook “perfect being the enemy of the good”. (It’s also one reason I prefer Approval Voting, which in addition to its benefit of simplicity, sidesteps Arrow by redefining the goal: not perfectly capturing preferences, but maximizing Consent of the Governed.)
Arrow's Theorem only applies to some voting systems and only in some situations.
Yes, the theorem doesn' apply to approval voting nor does it apply to score voting.
Arrow's theorem only applies to deterministic voting systems. So sortition (or other method based on random sampling) are not affected.
The theorem also doesn't apply to proportional representation systems. (Though they have their own problems, of course.)
Most RCV systems are very gameable with tactical voting. Though they aren't that useful, I guess.
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Arrow's theorem also doesn't guarantee that you will have problems. It just says that for some votings systems you can construct voting populations with preference that can't be captured well. It doesn't say whether these situations are likely to occur in practice.
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Arrow's theorem also doesn't apply when you allow bargaining, or people compensating each other.
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Of course, the problem with democracy in practice isn't so much that existing voting systems don't capture what voters want. Even first-past-the-post seems to be doing a reasonable job of that.
And who’s going to decide that those things you mentioned at the end are good or bad? If not (elected) leaders, that is. Take the exemple of the most famous democracy, the US, its current dominance was built on a few wars (the Civil War, to settle things domestically, and the two World Wars that allowed it to extend its dominance worldwide) and big periods of protectionism (like at the end of the 19th century).
Who's going to decide what voting system is good or bad? At some point, you have to inject some judgement calls, if you want to end up with a judgement call.
Btw, the protectionism was bad for the US economy, and did not help its dominance at all. (That's assuming you like US dominance?)
> Arrow's theorem also doesn't guarantee that you will have problems. It just says that for some votings systems you can construct voting populations with preference that can't be captured well.
no, it has nothing to do with capturing preferences. it simply says that no ordinal social welfare function can simultaneously satisfy these criteria:
There is no dictator.
If every voter prefers A to B then so does the group.
The relative positions of A and B in the group ranking depend on their relative positions in the individual rankings, but do not depend on the individual rankings of any irrelevant alternative C.
To be clear: I am saying that in practice people get the _policies_ they mostly agree with, not that the candidates they prefer over other candidates get elected.
That's (partially) because the candidates in order to attract voters pick policies that voters prefer.
Yes, you can extend Arrow's theorem a bit. But again, it doesn't apply to people who can negotiate or compromise or who play repeatedly. And it also only applies to aggregating an ordering of preferences. It doesn't apply to eg filling up a parliament for proportional representation.
(Btw, the random dictatorship doesn't sound too bad. As a slightly modified form, I think it would be a good experiment to fill up parliament with a few hundred randomly selected people amongst all who are willing.)
> But again, it doesn't apply to people who can negotiate or compromise
You want hundreds of millions of people to negotiate and compromise with each other in a way that would eventually produce representatives that reflect the population's resulting preferences somehow? How would that work?
> or who play repeatedly.
I don't see why I should expect that to make the problem easier.
> I think it would be a good experiment to fill up parliament with a few hundred randomly selected people amongst all who are willing
That sounds like it could go incredibly wrong. Everyone who is willing will sell themselves out to the highest "bidder" (maybe bidding via money, maybe promises of future laws...), and the population unwilling or unable to become a member of parliament will have no say in the matter.
> You want hundreds of millions of people to negotiate and compromise with each other in a way that would eventually produce representatives that reflect the population's resulting preferences somehow? How would that work?
Tacit negotiations can work. And in practice, it's often your representatives that do the negotiations with other people's representatives.
>> or who play repeatedly.
> I don't see why I should expect that to make the problem easier.
Check out the repeated Prisoner's Dilemma for some inspiration for how repeated play can breed cooperation.
> That sounds like it could go incredibly wrong. Everyone who is willing will sell themselves out to the highest "bidder" (maybe bidding via money, maybe promises of future laws...), [...]
How is that different from people selling their vote today?
Just make sure that the legal system does not enforce these contracts, and you are good. (You can also make such contracts illegal completely, just like selling your vote today is illegal in many countries.)
> [...] and the population unwilling or unable to become a member of parliament will have no say in the matter.
You can cook up slightly more complicated versions: every voter nominates a (willing) candidate on their ballot. Nationwide, you collect 600 ballots and fill up parliament with the people named on them. Pick your favourite resolution method, in case the same person gets picked multiple times in your sample.
(Eg you could give that person more weight in parliament, or you could pick the voter's second choice, or you could pick the ballot of the guy who got picked twice to pick a replacement, etc.)
> Tacit negotiations can work. And in practice, it's often your representatives that do the negotiations with other people's representatives.
And now you're back to square one? How do you choose those representatives in a way that represents their constituents' views? That was literally the original problem.
> Check out the repeated Prisoner's Dilemma for some inspiration for how repeated play can breed cooperation.
Have you seen literature on this somewhere? On its face iterated prisoner's dilemma being more cooperative does not in any way suggest that iterates voting somehow admits an easier solution for finding collective preferences than non-repeated voting. The problems are drastically different so far as I can tell. If you've seen literature suggesting otherwise I would love a link or two.
> How is that different from people selling their vote today? Just make sure that the legal system does not enforce these contracts
You seem confused? The reason you can't sell your vote today isn't that it's illegal to sell your vote, but rather the fact that there's no way to prove how you voted, so you could just lie with no incriminating evidence.
Whereas it's pretty darn easy to see how the candidate who promised you tax breaks suddenly voted to raise them when he came into office.
> slightly more complicated version
I see nothing obvious suggesting that your (homemade?) scheme is better, so I'm gonna put the onus on you to explain it wouldn't suffer from similar problems...
> And now you're back to square one? How do you choose those representatives in a way that represents their constituents' views? That was literally the original problem.
No, why? Arrow's Theorem eg has nothing to say about proportional representation. And Arrow's Theorem only applies to aggregating orderings of a finite list of preferences. But the methods under investigation need to be 'generic', ie can't make use of any special properties of those preferences, either. (See eg https://people.mpi-sws.org/~dreyer/tor/papers/wadler.pdf for how being 'generic' limits what your methods can do.)
And to come back to iterated games: almost no matter how the representative was chosen in the first period, if she's standing for re-election, she has an incentive for keeping her represented happy.
Arrow's theorem just applies to a list of static choices; not to how the chosen might behave when trying to get re-elected.
> Have you seen literature on this somewhere?
I don't remember right now. But I think 'The Myth of the Rational Voter' might mention some research somewhere. (See https://en.wikipedia.org/wiki/The_Myth_of_the_Rational_Voter) That book mostly mentions this when it argues that the problem with democracy ain't that voters don't get their wishes, but the problem is that voters do get their wishes.
> Whereas it's pretty darn easy to see how the candidate who promised you tax breaks suddenly voted to raise them when he came into office.
Sure. But if millions of people are eligible to be drafted at random, you are going to have a hard time pre-emptively bribing them. That's equivalent to doing something nice for the entire country.
> I see nothing obvious suggesting that your (homemade?) scheme is better, so I'm gonna put the onus on you to explain it wouldn't suffer from similar problems...
Because eg people who don't want to stand for parliament still have a say? That was exactly one of the problem you brought up with naive sortition. Remember?
Well, if your theorem says A implies B; if A doesn't hold, your theorem doesn't apply, but B could still be true for other reasons. But you need a different argument or empirical data to convince people of B.
Sorry, I don't understand that. Could you explain?
Arrow's Theorem applies when you have a discrete number of choices and you try to aggregate people preferences over them (in specific ways etc).
If instead of discrete elections, Alice and Bob can negotiate that _today_ they go to the football match and _tomorrow_ they go to the opera, that opens up new spaces for coordination that Arrow's theorem doesn't touch.
Similar, if Alice is allowed to pay Bob, or if they can do political horse-trading like 'I support your foreign policy, if you support my lowering the speed limit', that's also not covered by Arrow's theorem.
The theorem really only applies to deterministically aggregating people's individual orderings of a discrete set of options into some aggregated order for the group. That's it.
So it doesn't concern side-payments, or other continuous compromises. Or repeated play.
Is there a proof or demostration somehwere about maximizing the consent of the governed?
Arrows theorem always has implied to me that the next step should be quantifuing some welfare measure for voters and then exploring which system maximized that welfare measure. "Consent of the governed" sounds like a welfare measure so I an intrigued.
I don't really think you need too much to prove it yourself.
You are being governed with consent when the person who's elected is someone you are okay being governed by. And the person who wins an approval election is the person with that has the most people fine with being governed by them. Because approval voting doesn't ask people to rank candidates voting for someone you disapprove of only hurts you and voting for any subset of people you do approve of is sincere.
It's not some deep thing because it changes the target to something much easier. Finding the best candidate is hard, finding the candidate most people find acceptable is less so.
Approval voting gets more mathematically interesting when you assume people have preferences among the candidates they approve of and whether the best candidate gets elected but IRL you don't actually care about that anymore. You're fine electing someone who isn't the best.
I'm not convinced it can actually achieve that. There is still just one winner, just as now, and I'm not sure the people who picked them under duress will really feel they were listened to. (Or they can approve of only one, and almost certainly lose if it's not one of the two most popular parties.)
Still, I'm not averse to trying. Either it will help, or tactical voting will leave us more or less where we are now. If nothing else it's an opportunity to give the current deadlock a shove.
> Or they can approve of only one, and almost certainly lose if it's not one of the two most popular parties.
"They can only approve of one" is FPTP, the existing system. Everybody knows that sucks. The whole point of approval or score voting is to avoid that.
Right now if you favor candidate C but they have 5% of the vote and candidate A and B each have 45%, your preferred candidate has no chance and your vote can only change something in determining whether the winner is A or B, so you avoid voting for your preferred candidate.
With approval voting you vote for them and one of the major parties. Then people notice that third party candidates are immediately getting 30-40% of the vote because the people afraid of wasting their vote no longer have to refrain from voting for their preferred candidate. In some districts they even win. Which dissolves the two party system because people have to take third party candidate seriously and starting a new party has a real chance at succeeding rather than being an exercise in futility.
I phrased that badly. I meant that they may choose to vote for only their favorite, as tactical voting. It says they don't approve of any other, to send a message.
But it's not clear they will feel the message is sent if their candidate loses, and they are stuck with least favorite choice because they didn't select an alternative.
tactical voting is normally what we call it when you DON'T vote for your favorite, e.g. a green party supporter votes democrat.
with approval voting they'd obviously vote for green too.
some of the people who normally vote green under the current system might _still_ only vote green with approval voting, but very few would do it unless green actually had a chance to win.
I think the idea is that if there are two popular candidates A and B, one of whom is almost certain to win, a voter feels forced to approve of whichever of A or B they prefer even if they don't really want either one, just to hedge against the other winning, exactly as in FPTP. Or they can approve of only the candidates they really want, but they will likely lose.
Approval only seems to be able to break this gridlock if there is a "hidden" commonground between the two parties which can be revealed by the extra approval votes.
The problem with FPTP is that as soon as you have more than two parties, the two most similar parties split the vote among their common constituency and give the win to the least similar party. As a result any candidate who wants a chance at winning has to run on the ticket of the major party they most agree with, or else they split the vote with them and lose. Hence two party system.
With a cardinal voting system, someone can run on a ticket which is similar to one of the major parties and should get approximately the same level of approval as that party's candidate. Which is to say, they can potentially win. Then more third party and independent candidates run, giving people more options.
It's not just about what voters do, it changes what candidates do.
That's not necessarily all that different from now. We have a two stage system. In the primary people with broadly similar platforms run against each other. The "third parties" are factions within the two major ones.
Those options exist, and it's a multi way election. Primaries receive far less attention but they are where the real work of democracy is done.
I believe people are hoping they can vote for a radical candidate and a mainstream candidate, on the off chance people will love the radical candidate if they just get on the general ballot. I'm not convinced that will ever happen, and such people will be not just disappointed, but continue to be convinced the system is rigged against them.
> In the primary people with broadly similar platforms run against each other. The "third parties" are factions within the two major ones.
No, you still have that problem of splitting the vote in the primaries.
Remember how Donald Trump used to be the most hated Republican candidates within the Republicans in around 2015 / 2016? As in the one that the most people actively disliked in polls; but he was different enough from the other candidates that he didn't suffer from the internal vote splitting that they did.
The primaries still use first-past-the-post in the US, don't they?
That's likely to reduce diverse representation vs. single-member districts. If there are e.g. 8 seats a party could run 8 identical candidates and they'd all get the highest approval ratings for the combined district if one of them would, and other parties wouldn't get any.
List voting might work as an alternative to single member districts. You vote for your favorite party, and they are allocated a proportion of the total seats.
You lose the ability to know your local candidate, but how many people really do these days? It's what we set up in Iraq, but we don't do it ourselves.
It doesn't solve the problem that there is still exactly one chief executive. You can try making that a committee but that has other downsides.
> List voting might work as an alternative to single member districts.
But now you don't have a cardinal voting system and that's even worse than single member districts.
> It doesn't solve the problem that there is still exactly one chief executive. You can try making that a committee but that has other downsides.
Committees are dealing with it the wrong way. The right way to deal with it is to take away all of the executive's power. Constrain the national government from doing hardly anything and instead pass local laws to do whatever you want to do.
Then create elected positions responsible for different portions of the government. Directly elect the Attorney General and the heads of the major government departments. Let the President be like the Queen of England -- a figurehead with minimal responsibilities.
Germany has an interesting hybrid voting system that combines proportional representation via parties with local representatives.
In Germany, you cast two different votes. The first vote elects your local representatives via a first-past-the-post system; they all go to parliament. Then you fill up parliament with more people to make the proportions match those of the second votes cast all over the country. (There's lots of special cases and rules involved. Eg to handle the case when a party gets lots of first votes, but no second votes.)
Yes, I saw that! Inspired me to look at the original paper.
The video takes a slightly different approach from the paper and uses a retraction on the möbius strip to its boundary as a contradiction.
That particular argument doesn’t generalize as well in higher dimensions (in particular, the symmetric product won't always have a boundary to retract to), so I followed the original paper’s one instead. I'll add a link to that video as well
I'm not quite sure why one would use a sphere, unless you were specifically trying to get a version of Arrow's theorem.
If anything it looks like it fails precisely because the space is not homologically trivial, but I'm a bit unsure how to make that precise. A similar set up with just [0,1]^n as preference space works perfectly fine just by averaging all the scores for each candidate.
I kind of sense that requiring a function X^k -> X to exist is somehow hard if X is not 'simple', but I'm not yet sure what the obstruction is.
I thought about averaging the scores, which gives you a point inside the circle, and then projecting onto the circle with a ray from the centre, which is continuous everywhere apart from where the average is at the centre (e.g. for two voters this is when they have exactly opposite views).
So if you have a continuous probability distribution on the domain the probability of undecidability has measure zero.
It's not the undecidability that is a problem, it's the discontinuity. Undecidable answers are manageable, random answers however are very annoying to deal with.
Yea I think one reason to restrict to spheres is because the voting function takes as input the relative preferences (like in [0,1]^n how does all 0s differ from all 1s), which implies the vectors should be normalized
As it turns out choosing a simplex instead doesn't change things much from the hypercube. I think the arithmetic mean also still works. In stark contrast to the sphere.
Am I missing something or does the article fail to explain the point of Arrow’s Theorem? Is it satisfied for the discrete case, provably impossible, or what?
> While this applies to discrete rankings and voter preferences, one might wonder if it’s a unique property of its discrete nature in how candidates are only ranked by ordering. Unfortunately, a similarly flavored result holds even in the continuous setting! It seems there’s no getting around the fact that voting is pretty hard to get right.
I agree, it could do with a little more proofreading. Arrow’s theorem states that no voting state which ranks candidates can satisfy the the given conditions.
On a glance, the Chichilinsky theorem assumption of smoothness for the mapping between voter preferences And the vote result (the relation phi) seems burdensome. For example, many people might be effectively summarized as single issue voters - the topological consequences of a typical definition of differentiation (calculus) would seem unjustified. The exercise of exploring this world may be interesting, but I’m not convinced of its utility to politics.
