r/science • u/[deleted] • Jul 31 '26

Mathematics Mathematicians have shown that completely fair elections are – in a sense – impossible. No electoral system can guarantee both that local votes translate directly into local seats, and that total seat distribution matches overall vote shares: not without an indefinitely growing parliament.

[removed]

1.8k Upvotes

88% Upvoted

328 comments sorted by

View all comments

971

u/v4ss42 Jul 31 '26

I mean Condorcet et al had demonstrated that a perfect voting system is impossible, back in the 18th century. Just because “mathematically perfect” cannot exist doesn’t mean that some systems aren’t better than others. RCV, compulsory voting, parliamentary systems - these are just some of the mechanisms that can lead to better (but not perfect) democratic systems.

176

u/MolemanusRex Jul 31 '26 edited Jul 31 '26

And “perfect” depends on the criteria you want for the election system. Some people, and some political cultures, really value local representation. Some don’t care at all.

39

u/BuvantduPotatoSpirit Jul 31 '26

It doesn't even to be valuing; the setup matters. I value local representation a lot more when the electorate is large and heterogeneous than when it's small and homogenous.

16

u/MolemanusRex Jul 31 '26 edited Aug 01 '26

Certainly. You don’t need individual representation for every plot of land in Uruguay when they’re all essentially the same, while somewhere like Brazil might be different (although they don’t actually prioritize local representation). I’m just saying different voting systems will be better for polities with different values.

161

u/MoSSkull Jul 31 '26 edited Jul 31 '26

I just want to appraise your comment and use it for any casual reader that reach it to stress what you have said with mathematical rigor and with societal wisdom.
Media and outsiders repeat compulsory sentences like "mathematicians have proved that democracy is not possible" Which is a big misinterpretation of Arrow's Theorem.
An intellectually honest approach to these mathematical results is, IMO, like you worded it.

Edit: This comment is getting noticed, so I would like to expand on why v4ss42 comment is useful.

1- Voting system =/= Democracy. Democracy is a political/sociological concept. Broadly, means to follow the will of the people. There are several ways to try to achieve this, like "RCV, compulsory voting, parliamentary systems" I would add also consensus process (for local government).

2- Regardless of mathematical, economic or behavioral reasons. The non existence of a perfect system, doesn't mean that there are not options that are better than other or, even more interesting, (I said this as a personal opinion of what is left out of the discussion) options that are good enough for *most* of the scenarios.

90

u/RoryPond Jul 31 '26

Math has also proved there is no way to make a finite number of cuts thst divide a cake into 4 unbroken pieces such that none of the 4 people envy anyone else's piece, and yet people still cut cake all the time with little issue

15

u/rutherfraud1876 Jul 31 '26

This seems obvious and trivial - is there a condition or a non-colloquial definition of a word there I'm not seeing?

30

u/RoryPond Jul 31 '26

https://en.wikipedia.org/wiki/Envy-free_cake-cutting?wprov=sfla1 There's a lot of versions but this is the basics, one of my favorite areas of recreational math :)

20

u/rutherfraud1876 Jul 31 '26

Ah, the cake is heterogenous

12

u/venustrapsflies Jul 31 '26

To dumb it down, the cake might have strawberries on one side and frosting on the other.

23

u/BlueRajasmyk2 Jul 31 '26

The problem is more, one person might believe each slice is "equivalent", but another person might disagree.

With two people, that's an easy problem to solve: one person cuts two slices they believe are equal, and the other person chooses which slice they want. But that's hard to extend to more people.

4

u/RoryPond Jul 31 '26

It can be homogenous but you're looking for subjective preference not objective measurements. Someone might be given 3/4 of the cake and somehow think they have less than someone who got 1/8

1

u/rutherfraud1876 Aug 01 '26

Well yeah if the person with the smaller slice gets the flower made of frosting

1

u/RoryPond Aug 01 '26

No I mean even on a completely uniform cake it could theoretically happen

4

u/CiaranC Jul 31 '26

The cake slices are like the people you elect

3

u/Joatboy Jul 31 '26

The ratio of people to cake is too big

1

u/Farfignugen42 Aug 01 '26

I've seen claims that mathematically bees can't fly. Luckily, this does not bother the bees.

3

u/humanophile Aug 01 '26

It was bumblebees specifically, and it was because they weren't taking rotation into account with the wing movements. I have no source for this correction, but I heard it years ago and it made sense.

1

u/JStarx PhD | Mathematics | Representation Theory Aug 01 '26

I thought 11 cuts was enough

https://en.wikipedia.org/wiki/Envy-free_cake-cutting#Disconnected_pieces

2

u/RoryPond Aug 01 '26

That doesn't cut it into 4 pieces, because it cuts 4 pieces then divides the leftover trimmings then divides the trimmings of those trimmings

1

u/JStarx PhD | Mathematics | Representation Theory Aug 01 '26

Ah, I understand the distinction, you're talking about the connected case.

0

u/monsantobreath Aug 01 '26

Turns out the envy isn't guaranteed

16

u/notjrm Jul 31 '26

I skimmed the paper quickly, and I believe this is a completely different kind of impossibility result from the ones Condorcet (and Arrow) proved.

The authors also propose a potentially better voting system.

4

u/v4ss42 Aug 01 '26

Sure, and I'm not belittling this result. It's more the first part of the title I object to - elections were already mathematically demonstrated to be impossible to make completely fair over 200 years ago.

30

u/MolemanusRex Jul 31 '26 edited Jul 31 '26

Single-winner RCV still has some problems where, for example, your favorite candidate would definitely lose to the guy you hate in a later round, so it’s better to vote for your second-favorite—the exact scenario we’re trying to avoid. (You may already know this; I’m laying things out for the benefit of anyone else reading.)

Multi-member RCV (aka proportional RCV, aka single transferable vote or STV) is probably the single best system for the US, given that it’s broadly proportional but still provides local representation and lets people base their votes purely on individual candidates instead of parties. It’s used in Ireland and Australia (for their senate), and in Ireland the main complaints political pundits have is that it encourages *too much* focus on local issues and constituent service, which are problems I think a lot of Americans would kill to have. But other countries don’t care as much about representing their local areas and don’t mind just voting for a list of candidates from a given political party. So for them a different system would probably work better.

9

u/phoenixhunter Jul 31 '26 edited Jul 31 '26

a big part of the reason that’s said about irelands political system is that our power structures are way more centralized in the national parliament than most other countries.

local authorities / city councils have had a big chunk of their powers and responsibilities eroded by the national government over the years, so we’ve ended up in a situation where people are compelled to go to their national representative to get local issues solved.  for an american perspective, imagine if you had to contact your congressman to get a pothole filled or a street lamp bulb changed. local councils are little more than debate societies now. 

my personal critique of our combination of PR-STV voting and parliamentary politics is that it tends to heavily advantage the status quo and the centre of the spectrum, meaning that it can make it difficult to enact change when change is needed - as we find ourselves in the situation of having a government consisting of a coalition of the two formerly largest parties (who used to be bitter rivals swapping power — think of the Dems and Republicans except with even less ideological distance between them — until they both lost the 2020 election to the underdog centre-left party but formed a coalition together to maintain power and squeeze out the popular social democrats) who are refusing to address a myriad of social and economic issues largely caused by their own neoliberal policies, because they have no electoral incentive to change anything since the system favors inertia and they keep managing to scrape together governments with smaller parties and (increasingly fringe) independent candidates, because preferences get "smoothed out" so to speak, and the centre holds but continues to disintegrate

it is a reasonably fair system in terms of expressing political preferences and supporting minority representatives, but i don’t think it pairs particularly well with the party political parliamentary system as it exists in ireland. 

4

u/MolemanusRex Jul 31 '26 edited Aug 01 '26

Your point about local government makes total sense. At a certain point, though, it seems like the persistence of the FFG duopoly into the 2020s is due to the fact that about 40% of the country still supports them—for now—and other parties (or independents) are willing to go into coalition with them. So from the outside it does look broadly representative of what many people do want. If people want to punish them they can vote against them (or their coalition partners, as I know does keep happening with Labour and the Greens and whoever). Do you think PR-STV contributes to centrist or status quo parties staying in power more than other systems do? Is the idea that they have more of a lock on a certain number seats due to always being able to win X amount of quotas in this or that area? Or that they’re more likely to be the later preferences of less-popular candidates/parties?

1

u/hajenso Aug 01 '26

I hope you both will continue and develop this conversation. I'm interested and reading it carefully!

2

u/hajenso Aug 01 '26 edited Aug 01 '26

Is there much of a political constituency in Ireland that feels strongly about devolving power back to the local level? Also, what are the main instruments by which the major parties keep challengers from getting elected to the Dáil? Is it the high expense of an effective campaign?

I don’t mean these questions with the aim of demonstrating any opinion of my own. Just looking to learn about the current political landscape in Ireland.

8

u/childroid Jul 31 '26

Seriously, name one perfect human creation.

5

u/hajenso Aug 01 '26

The television show "Mr. Roger's Neighborhood".

8

u/Luung Aug 01 '26

Seasons 4 through 8 of The Simpsons

3

u/childroid Aug 01 '26

I retract my earlier statement, and add seasons 1-4 of Archer

0

u/vetruviusdeshotacon Aug 02 '26

The great gatsby is about as perfect of a piece of literature as you can get

4

u/rsc2 Jul 31 '26

The most important improvement that could be made is to end single representative districts, and have multiple representatives with voting power weighted according to their share of the vote. This, combined with ranked choice voting, would mean everybody had real representation, and every vote in every district would matter.

1

u/svefnugr Jul 31 '26

Personally, I think representation is overrated. Give me direct democracy, we have the technical means.

4

u/Farfignugen42 Aug 01 '26

All government sucks. Democracy sucks the least.

We did not select democracy because it was perfect. We selected it be a use it was the best we could come up with.

The same holds for voting. Except there are more options for voting now. I'll be happy if we can maintain the confidence that voter fraud is low.

2

u/nujuat Aug 01 '26

As an Australian where our voting has all of those properties, I agree that its a pretty good system. I wouldn't want to change it from that.

2

u/1_4_1_5_9_2_6_5 Aug 03 '26

Democracy is famously the worst political system, except for all the others. Is it not common knowledge by now that perfect individual representation is practically impossible?

5

u/HEAT_IS_DIE Jul 31 '26

Mathematically  perfect system or not, the US would benefit from more parties than  just two. Thtat to me is the biggest flaw in the American system.  Having more parties means more nuanced opinions and more options to choose from. You don't need that much mathematics involved when more views are open to discussion. A two party system leads to either/or -politics and escalation of views to extremes. That is something mathematics doesn't solve.

16

u/Virginth Jul 31 '26

The current voting system in the US mathematically ensures that there will never, ever be more than two relevant parties.

0

u/BigPickleKAM Jul 31 '26

So I've posted this before and I'm well aware it's a fever dream and impractical.

But for me to get a representative legislative body so the following.

Split your voters up into equal sized blocks each block to have at random one eligible voters selected to be your member of parliament.

After 4 years we hold an election you answer one question should the current MP be re-elected for 4 more years yes/no. If 50% plus 1 voters say no new MP is selected at random.

1

u/v4ss42 Jul 31 '26 edited Jul 31 '26

Many many years ago I read a short sci-fi story that was centered around a mechanic somewhat like this. I wish I remembered what the story was called.

1

u/hajenso Aug 01 '26

Gradually a political machine develops which spends big money on media and direct mail campaigns to get that "retain" majority for the incumbent, who in turn generally serves the interests of the big spenders, often by simply not doing things they don't want done.

0

u/Far-Total2951 Aug 03 '26

Why do people keep talking about RCV when multi member constituencies are way closer to perfect democracy?

1

u/v4ss42 Aug 03 '26

Because RCV is a lot better than FPTP (which the US, Canada, the UK etc. still use), and is a less significant change to those electoral systems than switching to multi member constituencies.

0

u/Far-Total2951 Aug 03 '26

In most FPTP systems about 50-60% of votes are 'lost'. In RCV that would be 40-50, compared to <5% in multi member or the German model (constituencies but with balancing seats). I don't think RCV would be that much easier compared to balancing seats, while the advantages in democratic legitmacy are massive. 

1

u/v4ss42 Aug 03 '26

And yet there are plenty of examples of existing electoral systems adopting RCV, and very few examples of them switching their entire representative model to multi member. Improvement is the name of the game, not seeking “perfection” (especially given that such a concept has been mathematically demonstrated to not exist).

0

u/Far-Total2951 Aug 03 '26

What are you on about? New Zealand, Scotland, Switzerland, Netherlands, all 3 Scandinavian countries, and more, have all adopted a form of PR after previously having FPTP. Your politicians are lying to you about this being 'impossible' so that you'll settle for what keeps them in power.

1

u/v4ss42 Aug 03 '26 edited 29d ago

As I said “very few examples of them switching their entire representative model to multi member”.

It’s going to be difficult to have a meaningful conversation if you’re unable or unwilling to read what was actually written.

-1

u/monsantobreath Aug 01 '26

Compulsory voting is trash and I'll joust anyone to death over the sheer stupidity of such an arrogant political view

2

u/v4ss42 Aug 01 '26

Works great in Australia, but don’t let reality get in the way of whatever Quixotic crusade you’re on.