Category Archives: voting

Some small single transferable vote elections

Attention conservation note: This is very much a stamp collecting post. Even I’m not sure it’s that interesting, but I thought I might as well write it up.

Single Transferable Vote is often referred to as a voting system, but it’s not really: It’s instead a very large family of voting systems, with a near infinite number of dials to turn to get different behaviours.

I’ve never really had much intuition for what the different dials do, so I thought I’d have a bit of play and construct some small elections that get different answers.

The two things I wanted to compare are the Droop quota vs the Hare quota, and the effect of the restart rule in the Wright system (in this system whenever a candidate is disqualified you start the whole process again from scratch with that candidate removed, resetting any candidates who have already been elected).

Droop vs Hare

Suppose we have three candidates, labelled 0, 1 and 2, and are trying to elect two of them. We have the following votes:

  • 6 votes for 0, 1, 2
  • 2 votes for 2, 0, 1
  • 1 vote for 0, 2, 1

Then the Hare quota elects candidates 0 and 2 while the Droop quota elects candidates 0 and 1.

The election plays out like this: In the first round, we elect candidate 0 as the clear winner, as it exceeds both the Hare and Droop quota. Then in the second round, neither remaining candidate has enough  votes to clear the quota so one must drop out. With the Droop quota, candidate 2 drops out and candidate 1 is subsequently elected, with the Hare quota the reverse happens.

The reason this happens is that the Droop quota is slightly lower than the Hare quota (4 vs 4.5), and as a result the voters who voted for 0 retain slightly more of their score for the next round. Because of the strong block voting 0, 1, 2 this means that under the Droop quota 1 beats out 2 in the next round, whileas under the Hare quota it’s reversed.

This seems broadly consistent with the descriptions I’ve seen about the Hare quota favouring smaller parties. I don’t know whether this shifts my opinion of it or not.

Aside note: One thing I hadn’t noticed before is that even with complete votes, the Droop quota need not actually produce a complete result. If you’re trying to elect two out of three candidates and you have three electors, then the droop quota is 2. This means that you “spend” 2 votes electing the first candidate and no longer have enough to elect the second. I ended up excluding this case from the elections I considered.

Wright Restarts

Running with the Droop quota, the following election changes results if you restart it every time someone is disqualified (retaining the set of previously disqualified candidates):

We’re now electing two candidates out of four, once again with nine voters. The votes go as follows:

  • 6 votes for 0, 1, 2, 3
  • 2 votes for 2, 0, 1, 3
  • 1 vote for 3, 0, 2, 1

Without the Wright restarting this elects 0 and 2. With it it elects 0 and 1.

Without restarts the election plays out as follows: We elect 0, then 3 drops out, then 1 drops out, then we elect 2 and are finished.

With restarts what happens is that after 3 drops out, we rerun from scratch and then 2 drops out instead of 1.

think what’s happening here is that the vote “3, 0, 2, 1” has a higher weight at that point without the Wright restart: With the restart it got counted as a vote for 0 in the initial round, and so got down-weighted along with the other votes for 0. This means that when it came time to design between 1 and 2 dropping out, the decision goes the other way.

I feel like this makes sense as a tactical voting mitigation step: Without the restart, I can just put a no-hoper candidate as my first vote, wait out the rounds it takes for them to drop out and then have a stronger vote.

Methodology

I wrote some code implementing STV with a couple of flags and asked Hypothesis to compare them. You can see the implementation here, but it’s very hacky.

A couple things to note about it:

  • Despite using Hypothesis, it’s not very well tested, so it might be wrong.
  • There are still a lot of variations and elided details about what sort of STV this is. I implemented what I think of as “vanilla” STV but I’ve no idea if that’s an accurate depiction of the status quo for it.
  • One design choice I made was to throw out all elections that caused any ambiguous behaviour, for either choice of the flag. The reasoning for this is that these small elections are really proxies for large elections where each voter is thousands of real voters, so the ties would end up being broken by small random variation in almost all practical cases.
  • I was actually surprised how good a job Hypothesis did at generating and minimizing these. I thought I might have to write a custom shrinker but I didn’t.

Conclusions

I don’t know. This was less enlightening than I hoped it would be.

I feel like I’m slightly more in favour of Wright restarts than I was, but I was already modestly in favour of them, and I don’t really feel like I’ve shifted my opinion about Hare vs Droop one way or the other.

It might be interesting to expand this to other STV variations (e.g. Meek’s method) but they differ more in implementation than these simple flags did, so I didn’t feel like implementing that right now.

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Why I’m in favour of proportional representation

Suppose I offer a group two options I like and most of the group hates. They vote on them, and you get the result that the majority hates the least and we go with that one.

This process is “democratic” – people voted, and got the result that they voted for – but hopefully it’s clear that I’m the one with the power here.

This is, of course, the trick to getting what you want out of democracy: Give people a heads I win, tails you lose choice and it doesn’t matter how they vote. All power rests with the people presenting the choices.

And, at least for now, we don’t have any good way of coming up with those choices except a bunch of people in a room talking to each other. There is no algorithm or voting system for taking a population with a diverse set of opinions, preferences and knowledge and automatically turning that into a coherent and concrete set of policies.

And I’m not sure there ever could be:  Many of these choices are complex and require days or weeks of study to understand the ramifications, which is hard to do if you also have to hold down a full time job, and your opinion without taking that time might be very different from your opinion if you had taken it. Which should we respect?

So we’re back to talking. This is quite hard to scale up to millions of people, so this is what we have parliaments and other houses of representatives for: It reduces the problem of millions of people without enough time trying to have a conversation and come up with the choices to one which is small enough that… well at least there’s the possibility of a productive conversation.

But in order, as an individual, to have any sort of say in what policies parliament creates, you need to have someone with a voice close to yours. It doesn’t matter if you elected someone who will vote more or less the same way you want (it doesn’t even matter if you live in a direct democracy and parliament will put the bill to a referendum!): If a voice close to yours is not part of the political debate, your interests are not represented. You can vote as much as you like, but it won’t help much – you’ll just be getting the less bad option of the two someone has handed to you. You don’t just need votes, you need a voice in parliament.

And this is true at the group level too: Even if a strong voting bloc manages to consistently push for the less bad option for then, if that bloc does not have actual representation of their views in parliament, they’re never going to do better than that.

And existing non-proportional systems have consistently failed to produce that representation. Instead they take us right back to the beginning: Which of these two parties that you don’t like would you prefer to have in power?

A move to a proportional system gives you that, by giving significant groups actual representation in parliament. Instead of just getting to voting between a few choices, you get to be part of the process for creating the choices instead, and you get a voice in government as well as a vote.

 

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Trading influence: A half-baked parliamentary design

Advance warning: I’m pretty sure this is a bad idea. I’m sharing it because I think it might be an interestingly bad idea. Or maybe it makes a better game mechanic than democratic system, I don’t know.

The starting question from this was: How would you design a voting system for a parliamentary government (that is, the system that the parliament itself uses, not how they are elected) so that it encourages compromise and finding acceptable middle grounds rather than just doing straight up majority rule?

The key idea is to make in an explicit system of trading influence (to replace or augment existing implicit systems of trading influence). On questions you are mostly indifferent to, you can trade your vote away for influence on later votes.

In order to do this, we introduce a system of virtual currency representing influence. The currency is given value through an annual tithe, designed such that the amount of influence in circulation is only sufficient for, say, 80% of parliament to pay the tithe. Anyone who cannot pay the annual tithe drops out and comes up for re-election (thus creating a system of rolling elections). Some sort of additional term limit is probably required too.

Votes for or against a bill are then cast as follows:

Each representative casts one of two types of votes:

  • firm vote, which is a vote for or against plus an amount of influence the representative is willing to spend to ensure that vote.
  • tentative vote, which is an abstention plus an amount of influence the representative would require to cast their vote for or against.

Once the vote is cast, the auction begins. I’m a little unsure on the optimal design of the auction here. I believe the following is reasonable, but I’m pretty sure it’s not optimal (for any sense of optimality). The design is basically that of a Vickrey Auction, but with votes instead of influence.

For each side, we determine the number of purchasable votes: That is, counting from the smallest offer upwards we buy votes until we would require more influence than we have allocated left to spend. However many we bought is our set of reachable votes.

For each side we add up the number of purchasable votes with the number of firm votes cast for that side. This gives the number of accessible votes.

If both sides have the same number of accessible votes, that’s a tie. Move to tie breaking procedure (e.g. default against the bill) and nobody pays anything.

Otherwise, whichever side has the larger number of accessible votes wins the vote but must purchase votes to the tune of one more than the other side’s accessible votes (this may be zero votes) using their influence. This works by buying up votes starting from the cheapest until you’ve purchased enough votes. The cost of these votes is then distributed amongst the people paying for them using a system I described previously that basically splits the bill evenly amongst everyone who can afford it, and everyone who can’t afford the even split pays their whole budget.

I don’t think this encourages entirely honest play unfortunately – although the vickrey mechanism forces you to be honest about the number of votes you need, it’s a bit too disconnected from the actual bids (and I don’t think people are encouraged to be honest in the prices they set their vote at at all?) but it’s probably not too bad. Insert better auction design here.

Problems

There are a number of obvious problems with this and probably some more non-obvious problems with this.

The fundamental problem with using any sort of currency mechanism for voting is what you do when someone submits the same bill twice. I think this is fixable here by having a “parliamentary behaviour” notion that when you’re bought you stay bought. In order to enforce this it may be useful for MPs to declare blacklists where they refuse to buy votes from certain candidates.  This complicates the calculation of how many votes are accessible significantly (I think it may turn it NP-hard. It’s certainly solvable through a mixed-integer linear program, but that starts to sound like something you don’t want in your voting procedure).

This is vulnerable to bad actors who just always put their vote up for sale, taking currency out of circulation and thus increasing turnover in the parliament. By doing this preferentially only when the makeup of the parliament is not to your liking it might be possible to stack the parliament that way.

If you have either term limits or are obviously not going to be re-elected because it’s coming up to tithe time and you don’t have even close to enough influence it creates an incentive to just go wild on your spending. Of course term limits already do that.

It’s unclear how this interacts with party systems and party whips. I’d like to avoid that by having more fluid parties and no party whips, but I think reality may work against me on that one…

Advantages

Well I think it’s interesting, and interesting is obviously good, right?

But the concrete advantage of it is that by creating a system of trading influence fairly explicitly it creates a much firmer platform for small parties who are really mostly interested in a core set of issues to get influence over those issues while achieving a sort of “compromise liquidity” for the larger more influential parties.

Moreover, in systems with small parties you already have this sort of influence trading happening, only it’s behind the scenes and thus illegible to the public. By making it official and part of the system you get greater public oversight into what is going on.

Over all, I think the disadvantages beat the advantages by some margin, but maybe that will give the undecided middle some interesting influence that they can spend on a better system later…

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First Past The Post is not the problem, districts are

It will come as no surprise to anyone that I am thoroughly against the systems used for electing representatives in both the UK and the US.

What may come as a surprise is that I’m pretty much indifferent to the fact that they both use first past the post, or simple plurality voting.

Don’t get me wrong. First past the post (FPTP from now on) is a rubbish voting system. It has essentially no redeeming values that are not also shared by approval voting, which is in all ways a superior system to it (I’m not massively in favour of approval voting, but it’s unambiguously better than FPTP). The point is not that FPTP is good, it’s that in this context it is irrelevant.

The reason it is irrelevant is that almost any choice of voting system in its place would produce just as bad or worse results, because the most major failure is in how we are applying voting, not the voting we’re doing.

To see this, consider the following scenario:

  1. We’ve got some geographical districts and each one gets exactly one representative.
  2. The representatives are divided into two parties. Call them the Purple and Green parties.
  3. The purple party has 50.1% of the populace who think that Purple is amazing and Green is literally the worst. The other 49.9% think the opposite.
  4. This division is uniformly spread over the entire country with little to no local variation

What happens? Well, you get an entirely purple house of representatives! Because in each district you have a strict majority of people who prefer Purple. Green gets exactly zero representatives despite the fact that 49.9% of the populace are strongly in favour of Green.

This is manifestly ridiculous, and the resulting government cannot be said to have any significant democratic mandate over an equally ridiculous 100% Green party. I’ll leave it up to you to decide how many more seats they should have than Green, but I’ll take it as read that getting all the seats is not an acceptable answer.

And getting this scenario required almost nothing about first past the post. If you replaced it with approval voting, alternative vote, range voting, whatever, the result would still be the same, because it boiled down to a simple majority vote and the majority in each district genuinely preferred Purple.

The choice of voting system influences what sort of distortions you can get. As your voting systems get better the above tends to be the only sort of distortion you get, but it will always remain possible. With districts, you can always take whatever single result you’d get if you ran the vote over the whole populace and just give them 100% of the seats by just spreading the population out this way (note: There are some technical conditions required for this to be strictly true, but it is in practice true with basically everything).

In practice the distortions are usually not this extreme because most of the time the distribution is not uniform. But that doesn’t make the result more representative of the populace, it just makes it a consequence of the vagaries of where people happen to live. You then get Gerrymandering happening, which is essentially the art of deliberately creating these distortions in a way that furthers your political aims.

The only way to avoid this problem altogether and also have districts is to have a system where some districts can genuinely have a majority preference for a candidate but that candidate still doesn’t get elected. This isn’t as unreasonable as it sounds. Two examples of reasonable ways to do this are random ballot and biproportional apportionment. However both of these are niche and you’re unlikely to get them.

Amongst systems with mainstream support, you really need to do away with districting in part or in whole. You need to move to a full proportional representation system across the whole country, or across larger multi-member districts.

But one thing you can’t do is just tinker with the system that you use in each district if you want to make a difference. Changing the voting system you use within districts is just rearranging deck chairs on the titanic, and continuing to focus on first past the post as a problem is just going to get you another vote on the optimal deck chair rearrangement at best rather than getting people off the sinking ship.

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Three thought experiments on majority voting

I present to you three examples in which we employ majority voting between two options (that is, we ask a population “Would you like A or B?” and we choose the option that the largest number of people preferred).

You can and should infer the obvious context for this, but I am not going to comment further on it in this post.

Tea and Cake or Death

Suppose 51% of the population vote for a bill that will result in the death of the remaining 49% of the population.

Questions
  • Should the country go along with this?
  • If they decide to, do the remaining 49% have a democratic obligation to accept that?
  • Does the answer change if it’s 90% and 10%? 99% and 1%?
  • What if the remaining 49% are merely financially ruined? Moderately inconvenienced?

Pizza or Barbecue

A group of nine friends regularly meet for dinner. Of these, five of them really like pizza and four of them really like barbecue. As good citizens of a democracy, they put this to a vote. Unsurprisingly, this results in them always having pizza.

Questions
  • Is this fair?
  • Suppose only four of them really like pizza, and the ninth person changes their mind regularly and thus always gets to decide where they go for dinner. Is that fair?
  • Suppose the friends in question are tired of putting it to the vote each time and some of them push for a vote to go to a regular meeting place. They put the question “Should we have Pizza or Barbecue for all future group dinners?” to a vote. Pizza wins. Is that fair? Does that answer change if we have the previous 4/4/1 split?

Which president?

Fair warning: This one is by far the most complicated of the three thought experiments.

Our student mathematical society decides to elect a president. There are three candidates, Alex, Kim and Pat. We’ve read all this confusing stuff about voting theory and we can’t really decide what we like except that majority rule is clearly the best for two candidates, so we decide to reduce this to the solved problem. I pick two candidates, we vote between them and the majority winner stays in. We then vote again between them and the third remaining candidate, and the winner of that becomes president.

Note that the student body is roughly equally split between the following three preferences:

  • Alex, Kim, Pat
  • Kim, Pat, Alex
  • Pat, Alex, Kim

As a result, the majority of people prefer Alex to Kim, the majority of people prefer Kim to Pat, and the majority of people prefer Pat to Alex.

Note that:

  • If I put Alex and Kim together in the first round, then Pat is president because Alex beats Kim then Pat beats Alex.
  • If I put Alex and Pat together in the first round, then Kim is president because Pat beats Alex, then faces Kim and the majority prefer Kim to Pat
  • If I put Kim and Pat together in the first round, then Alex is president because Kim beats Pat in the first round and then Alex beats Kim
Questions
  • How strong is the resulting president’s mandate?
  • Does the answer to the previous question depend on which round they were in?
  • Does the answer to the first question depend on how we chose the rounds?
  • Does the answer change if instead of explicitly voting for three candidates as part of a single batch we instead have a system where a new candidate always runs against the incumbent?

Concluding Statement

Majority vote between two options is often held up as some pinacle of uncontroversial democracy where at least in this case we know what the right answer is, even though voting is complicated in general.

I hope I have convinced you that is not the case.

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