Note from the future (2019): There are much more mainstream proportional representation systems than this one. I do think this one is a great idea, but it’s a political nonstarter, so don’t get too hung up on it.

A long time ago I wrote A “perfect” voting system. It’s my most popular post ever. Unfortunately it’s really badly written. This is an attempt to write a replacement version of it. I’ve written a lot about the idea since, and this will incorporate some of that, but it’s very much intended to be a stand-alone piece.

Modern British democracy is broken. It’s not a problem unique to us by any means, but it’s a problem we feel acutely. We have an entrenched two party system, with the Liberal Democrats being the only real contender for a third candidate. Small parties find it incredibly hard to get a foot in the door because the system works in such a way that the amount of representation you get is massively out of line with the amount of support you have. Regardless of whether you believe full proportional representation is a desirable goal, what we have is so far from it that it’s hard to even call it democracy: Most peoples’ votes simply don’t matter because they live in a constituency which is “safe”: In the 2010 general election, the electoral reform society predicted the outcome of 382 safe seats. They got two wrong.

Much of the blame for this has been leveled at the first past the post voting system. This is entirely fair – it really is an awful system. However equally a lot of the blame can be leveled at any of the commonly used systems when applied to geographic constituencies: It causes the geographic distribution of your voters to matter intensely.

One way this problem manifests is through gerrymandering: By manipulating the boundaries you can significantly change the number of seats a party gets. Here’s a good explanation of how it works.

It’s important to note that this is true with any of the normal class of voting systems applied to each constituency: If you have a system where in a two party fight the person with the most support always wins (which is true of basically every system commonly proposed. Indeed you might currently be thinking that it’s a property every system should have. Hold on to that thought) you can gerrymander to manipulate the result.

The obvious solution to this is to switch away from a regional constituency system to a proportional representation based one. This is a perfectly valid solution and one that works for a number of countries.

I’d like to propose a… well not an alternative per se, in that the system I am proposing is also in many ways a proportional system. A different way of achieving proportional representation which preserves the regional constituencies.

Why might we want that?

Well the main reason from my point of view (there are many other reasons, and different people will find different ones important) is to ensure every region of the country has its interests represented: It’s all very well having proportional representation by party, but surely you also want proportional representation by region? Otherwise you can end up with your political interests represented but your local interests neglected.

There are ways to achieve this under PR: Essentially you form larger constituencies, have each be multi-member and elect those members under a PR system. This is an entirely respectable hybrid solution.

It does however end up inheriting the problems of both worlds – the smaller your regions are the more vulnerable you are to gerrymandering, the larger your regions are the less individual areas are represented. There is surely a decent middle ground that trades off optimally between the two, but what that is depends on your priorities, and both will remain at least partial issues.

Fortunately it turns out there is a way we can do better. There is a system with small constituencies in which everyone votes locally but we get global proportional representation.

The system in question has many major advantages. In particular:

It has true proportional representation across every axis, not just party. If 40% of votes go to people who support a specific issue, about 40% of candidates elected will support that issue. The same is true of things like race, class, gender, etc. This system might actually successfully reduce the prevalence of old rich white men in parliament (they’d probably remain a majority for some time due to being who people vote for, but it should even out)

Despite being a regional method it is completely immune to gerrymandering. The only way to distort the results by manipulating regions is to make some regions larger or smaller than others, which is intrinsically a problem with regional systems and comparatively easy to avoid.

There is no incentive for tactical voting. The optimal vote for you to cast is a tick next to the name of the person you most want elected.

There are no safe seats. Politicians remain extremely accountable to their electorate, regardless of whether they have a majority. They will always want and need more people to vote for them unless they literally have 100% of the vote (though they’ll probably be feeling pretty comfortable at upwards of 99.9% of the vote I imagine).

This sounds like a pipe dream. There are mathematical theorems (Arrow’s impossibility theorem and the Gibbard–Satterthwaite theorem) which constrain the possibilities for voting systems. How does this supposedly perfect voting system defeat the mathematics?

Well, it does that simply by not being part of the class of voting systems to which they apply. Arrow is very restrictive indeed – it only applies to preferential voting systems. Gibbard-Satterthwaite is less restrictive – it says that any non-dictatorial deterministic system which can elect at least three candidates is subject to tactical voting.

It’s time to come clean and stop teasing: We avoid this by not using a deterministic voting system. The system we use instead is one called Random Ballot. Conceptually it works as follows:

Every voter casts a single vote for their preferred candidate, as under our current system of first past the post When all the votes are cast, we select a random voter and elect the person they voted for

This is not an implementation strategy you would actually want to use (it’s quite fragile), but fortunately there is a robust equivalent strategy that I will mention later. So if you have practical implementation based objections to this idea, hold on. They’ll be addressed. For now, keep this in mind as your mental model of how the system works.

Lets look at the consequences of this systems.

Functionally what we are doing is saying that each candidate has a chance of winning equal to the fraction of the votes they hold. Say for the sake of simplicity we have three parties and 100 votes in a constituency. The Red party have 60 votes, the Blue party have 35 and the Teal party have 5. You can think of each of these votes as lottery tickets with a guarantee that someone will win – the more you’ve bought, the more likely you are to be that winner. So Red are a bit less than twice as likely to win as Blue and are twelve times as likely to win as Teal. But even poor little Teal there is in with a shot at winning – the fact that they’re a minority party means they’re very unlikely to but it’s not completely infeasible.

So what I’m saying is that a party with a tiny minority can sometimes get into power in a seat. Is that not massively unfair?

Well, no, not really.

The thing about random systems is that if you run them enough times they start to look very predictable – sure you can’t predict what any given one will do, but you can make pretty good bets as to the number that will do a specific thing.

If we were electing a president, random ballot would be a colossally bad idea. We don’t elect many presidents (in this country none at all, but that’s not important right now), so the randomness factor will play up massively. On the other hand when you’re electing 650 constituencies at once as we are in a general election, you’re running the result so many times that the results start to look downright deterministic.

Supposing you capture 1% of the popular vote. Under this system we expect you to have about 1% of the seats – i.e. about 6 or 7. You might have more, you might have less. But you’re not going to have a lot more or a lot less. It would be surprising if you had more than 10 or fewer than 2 or 3. It would be extremely surprising if you had no seats or more than about 15. It’s pretty much inconceivable that you’d get more than 20 – you’re better off banking on winning the national lottery and using that to fund your next election campaign to improve your chances.

This averaging effect also works over multiple general elections, though there are fewer of those than seats so its effect is necessarily a bit more random. But assuming its numbers stay roughly the same our hypothetical constituency would be Red about 60% of the time, Blue about 35% of the time and Teal about 5% of the time.

One thing that often concerns people about this system is the possibility of it putting a party into power over all who doesn’t deserve it. Allow me to state categorically: This doesn’t happen. If you’re going to get a majority of seats, you need to get incredibly close to a majority of votes. At around 48-49% of the popular vote you’re in with a chance at a majority. At 50% it’s a coin flip. Much above 50% you’re probably going to get that majority. Compare this with the current system: Literally the only general election under the current system where the party in power has had the majority popular vote happened in 1931. There have been some extremely high 40s (in 1955 and 1959 for example the conservative party had 49.7% and 49.4% of the popular vote), but we have never since that point broken 50%. If anything, election by random ballot is an excellent defence against parties who gain power without real support.

And remember: All of these things are true not just for parties, but for every other characteristic you choose to imagine. Are our parties not actually along our desired political lines? No problem! Not only can we vote for independents who better support us, parties become free to run multiple candidates in a given constituency which reflect different aspects of their ideology: If you’re a progressive party you can run two candidates, once who is more economically progressive than the other to see if you’re getting the social or the economic vote. Yes this splits the vote but that’s OK because it doesn’t reduce the total vote for your party (and may even increase it): If the votes simply split between your two candidates then each of them is less likely to be elected, but the chances that one of them will get elected remain the same.

So that’s why it’s fair and balanced at the country level, but what about our constituencies? Are we sacrificing their good for the common good by making them take candidates that they don’t want?

It turns out not.

You see, although our candidates may not be from the party that the constituency wanted, they have a really strong incentive to make their constituency as happy as possible.

Right now, and indeed this would remain true with any deterministic regional single-candidate method, suppose a party captures 60% of the vote in a constituency. What do they do about the remaining 40%? Well, they ignore it mostly.

If you manage to capture a strong majority under a more traditional system then you can pretty much rest on your laurels. You have a safe seat, it’s going to be hard to kick you out unless you manage to really alienate your constituents.

Under random ballot every vote counts. Those 40% who don’t vote for you? That represents a 40% chance that you will not be re-elected. If you want to keep your cushy job in parliament you’re surely going to do your best to appeal to every voter in that 40% and convince them you’re a great MP – anything to get your numbers up and boost your re-election chance.

It’s very much the AV “Make your MPs work for their seat” position, only more so. The MPs must care profoundly an deeply about their numbers of votes, and it’s in their best interests to keep their local populace happy. It’s representative accountability on an unprecedented scale.

So we have a system which is pretty close to ideal and a local and national level. What’s not to like?

There are some common objections. I will try to address them.

The implementation you describe is fragile and easy to game

This is true. Actually selecting a random ballot is a bad way to implement this – there’s possibility for sleight of hand, it’s very hard to verify and the whole election can come down to a debate about whether a specific ballot is spoiled. Here is how to fix that:

Rather than selecting a random ballot, we count up the votes as we currently do. Calls for recounts are permitted, but must be performed before the next stage. Once a candidate has been elected it is too late to call for a recount.

What we then do is use these counts to simulate drawing a random ballot: We create one virtual ballot for each counted one, lay them out in order and use a computer program to pick a random one.

This program must be open source so that it can be verified and run by third parties. It uses a pseudo-random number generator seeded by a commitment scheme from each of the candidates to be elected and an impartial administrator. These commitments and their corresponding secrets are published at the time of the election.

If you didn’t understand that, don’t worry. The highlights of it are:

The results are entirely reproducible. Although they are effectively random they are generated from a process with known inputs that are kept secret up until the time of the election. Given those (published) secrets the full result can be reproduced

Any third party with a computer can verify the results are correct

It requires a collusion between every single candidate plus the election administrator to fix the results

People just won’t understand it

I’m getting a little tired of this objection, truth be told. It’s the most common objection I get to the idea, and there’s definitely some validity to it.

But the framing makes me uncomfortable. You see every single person I have explained this scheme to in person has got it fairly rapidly, as have most people who have read about it online – there have been objections, but they’ve generally been objections that showed that they understood the core idea but thought it had other crucial problems with it. Granted there’s selection bias going on here – the audience for my blog and the people I’ve described it to are definitely more mathematically inclined than most, but it definitely suggests that it’s really not all that hard to understand.

This makes this objection much more along the lines of “Well I understand it obviously, but those stupid proles won’t, will they?”

I’m pretty uncomfortable with this sort of paternalism.

Yes, understanding the fairness of this system is tricky and involves maths and probability, which people aren’t that good at. Yes, it would require education for people to truly understand it. But participation in the system is incredibly easy, for people who truly care enough to want to understand the details it’s decidedly within their capabilities, and for those who aren’t that interested they will still be much better off participating in a fair system they half understand than an unfair one they fully understand.

Why not just use a sortition?

Excellent question! I’m glad I asked me.

For those who don’t know, a sortition is when you simply draw your MP at random from the eligible population.

Using a sortition instead of my proposed system may in fact be an entirely good idea. One of the nice things about this system is that it transitions seamlessly into a sortition if people decide they want one – Anyone can run, and if everyone runs then what you get is precisely a sortition. Some of the practical details will need changing and scaling, and you’d need to lower or remove the barrier to entry of becoming an MP (there’s a deposit you pay at the moment which you get back if you get enough votes), but fundamentally there’s nothing stopping this behaving exactly like a sortition.

Here are some of the possible advantages of this system over a sortition:

It keeps more experienced candidates in the house, much like traditional voting systems.

It has much higher accountability of candidates to those whom they represent – under a sortition you have no chance of re-election, so no incentive to endear yourself to your constituency. Under this system you are strongly encouraged to do so.

Not every member of a group is well suited to representing that group. Disadvantaged groups and those who experience prejudice pay a much greater cost for being in the public eye, and as a result many of them will be strongly disinclined to speak out for group issues. By allowing them to delegate to someone they feel will represent those issues well and fairly you are in fact giving them more power rather than less.

What if a party with barely any votes gets a seat?

There’s no doubt: This can happen. A party with 1/1000th of the vote has about a 50% chance of getting a seat (this is not unreasonable – they’re “owed” about 0.65 seats). They’ve even got a small but appreciable (about 1 in 2000) chance of getting as many as 5 seats. If the party is some extremist group that you don’t want to get power this can be perceived as bad. How dare parliament represent this group?

First off, the important thing to note is that this balances out. You’re not going to get a parliament which is dominated by many tiny parties that together are not capturing an appreciable proportion of the vote. The bottom 1% of the vote occupies no more than about 20-30 seats in parliament – anything above that is less probable than a meteor strike. It’s an appreciable voting block to be sure, but it’s not a unified one – all the different minority party voices are likely to be at odds with each other rather than a single reinforcing block.

But back to the single party case, what do we do if an unpleasant group like the BNP get themselves a couple seats in parliament?

My inclination is to sit back and watch the show.

British news has a bit of an obsession with our fascist minority parties, far out of proportion to their actual voice. I think the only reason they can get away with this is that there’s no visible signal for how much support they get. It’s much easier if you can go “Yes, but you only have 1 seat in parliament. Clearly no one cares that much about you, do they?”. Also, honestly, I expect them to do a pretty bad job if they gain power and be laughed out at the next election.

It’s certainly possible that this won’t happen, they’ll actually do a “good” job as MPs and this will build their support. That would be… unfortunate. But if that happens we will have a level playing field to fight them on which we can and should do. In the meantime, we shouldn’t let a fear of fascism be used as an excuse to throw away democracy.

This will limit the experience of MPs in parliament – too many people will lose their second election

This turns out not to be the case. Although any individual may have a high chance of losing their second election, over all the strong proportionality feature means that the total number of MPs who have made it through a previous election is a direct sign of how much people like the current government: The fraction of people who vote for their incumbent MP will be approximately the same as the fraction of MPs in parliament who retained their seat from the last election.

This is a nice feature. It essentially means that people get exactly as much change as they want – if people are generally satisfied with their current representation in the house they will generally vote for the incumbent, if they are dissatisfied they will vote for change. And unlike in the current system they will get it.

It is worth noting that this does tend to limit term lengths. If an MP has only 50% of the vote they’ve a decent chance of re-election once, but they’re unlikely to make it two or three times. On the other hand, if they have 75% of the vote they’ve a decent chance of a few more terms than that. This too is a nice feature: It means the MPs with the most experience are the ones who do the best job at keeping their constituents happy.

It just feels wrong

A friend used a great analogy the other day. When your objections come down to “it just feels wrong” you are no longer making a reasoned argument, you are that guy who is saying “But baby, I don’t want to wear a condom because it just doesn’t feel the same” and you are sacrificing other peoples’ health and wellbeing for your own aesthetic preferences. Don’t do that.

This system is unfamiliar certainly. There is some historic precedent (indeed, Athenian Democracy had a high reliance on lotteries. More recently the Doges of Venice were elected through an outrageously complicated system involving iterating a sequence of votes and lotteries), but it’s been a long time since we’ve done anything like this.

But unfamiliar is not the same as bad, and in the face of such overwhelming advantages, the fact that it’s different from the current extremely broken class of systems should not be taken as a downside. We are used to living in a democracy that simply doesn’t work as one. Is it any surprise that when given the opportunity to live in a more perfect system we find it unfamiliar and strange? Let us have the courage to believe we can achieve perfection, and not let its unfamiliarity deter us.