I have some ideas and questions that I know some friends of mine would be interested in, but I don’t know where to post them, so I’ll just post them here in case that turns out to be not too bad an idea.

I’m going to talk about proportional representation.

Someone recently suggested that the best number of elected people in any jurisdiction’s overall government might be the cube root of its population. For the current global population that’s about some number between 1920 and 2400.

So imagine a global parliament with 1920 or 2400 members.

480 or 600 of them are elected from 480 or 600 single-member districts using ordinal voting and instant-runoff; something better than first-past-the-post. For this purpose the globe is divided into 480 or 600 districts of equal population.

Another 480 or 600 are elected from 160 or 200 three-member districts by proportional representation. Again ordinal voting will be used. I don’t know whether single-transferable-vote or some other kind of PR will be used, but I don’t like the “party lists” idea, because that means the parties (rather than the voters) choose the representatives. Let each three-member district be composed of three of the single-member districts. As a general rule each district will choose two members of its majority party and one member of its second party. (Sometimes one might choose a member of its third party.). It will be VERY difficult for any party to elect more than two-thirds of these representatives. And the “second party” will seldom have many fewer than one-third of these seats.

Another 480 or 600 representatives will be elected similarly from 80 to 100 six-member districts. I guess each six-member district could consist of two of the three-member districts from above. Most voters can rank their top four choices with no trouble; and their “bottom” four “can’t stand the thought of them” disapprovals with no trouble; so there needn’t be any “donkey voting”, is my guess. Most of these districts will usually return representatives in one of the ratios: 3:3:0:0 or 3:2:1:0 or 3:1:1:1 or 2:2:2:0 or 2:2:1:1. So third-party and 4th-party representatives have good chances of winning seats in this segment of the parliament. I’m guessing the largest party has about a 60% chance of winning about half the seats, and little chance of winning more than half or fewer than a tihrd of them; the 2nd party has around an 80% chance of winning at least a third but also an 80% chance of winning fewer than a half; the 3rd party has about an 80% chance of winning at least one seat plus about a 20% chance of winning a second seat; and the 4th party has about a 40% chance of winning at least one seat. But for all I know my math is all wrong? I doubt any single party will often get significantly more than half of these seats.

The last 480 or 600 members will be elected from 12 or 15 40-member districts. I don’t know everything there is to know about PR but I hope there is some way to do this that doesn’t require “party lists” and also doesn’t require “donkey voting”. Let each of these 40-member districts consist of 40 of the single-member districts. They’ll probably return an average of 12 or 13 each of first-party and 2nd-party members; 6 or 7 each of 3rd-party and 4th-party members; and 1 or 2 each of 5th-party and 6th-party members. They’re not too unlikely to return one fewer than the lower estimate or one more than the higher estimate of each of those. They’re pretty unlikely to be further off the averages than that. In this section of the parliament, then, just about any party that has more than 2.5% of the voters in one of those 12 or 15 districts is likely to seat a representative.

So a quarter of the parliament will come from single-member districts; a quarter will come from 3-member districts, in which the second-largest party is unlikely to win many fewer than about one-third of the seats; another quarter will come from 6-member districts, where the third-largest party is likelyish (80% chance?) to win at least one seat (+ a 20% chance of winning a second seat?) from each such district, and the fourth-largest party has (I guess) at least a 40% chance of winning at least one seat; and finally the fourth quarter will come from the 40-member districts, where the six largest parties will probably-ish be proportionately-ish represented.

.....

I am leaving aside any problems of currently-existing small-population national-states not wanting to join such a government for fear of having their “voices” ignored by the highly populous national-states. And other problems of getting there from here. Let’s just suppose we can get there from here; would the above be a good “there” to get to?

I am also leaving aside any problems about picking the executive and judicial branches. ... Who has a comment?

I think any super-majority of 5/8 or more of such a parliament are going to have to be multi-partisan.
That is I think it unlikely many more than 62,5% of them will often be all from one party.
Again, my math could be way off; my statistics here is being done in my head, using rather naive methods.

Hard to follow this. Could you make it simpler? It might also help if you begin at the base level and work your way up to the parliament seats.

Some other comments:

I think there would have to be leadership assignments within the parliament, like party leaders and parliament leaders.

Also, in any hypothetical world government, supranational organization (like the EU), or a country, there's more than just a parliament. There would also be, I presume, an executive branch of the government and a judicial branch. The effectiveness of the world government may very well have more to do with the powers vested in the parliament versus the other branches of government, than with the proportionality of the parliament representation.

Cube root does not make much sense... At lower numbers it takes a while to get reasonable representation, and higher numbers get cut off very quickly. What we're seeing today is the influence of lousy candidates and corruption, so quantity doesn't matter as much as quality. A benevolent dictator would always be preferred, the problem is any system that creates a good dictator would inevitably produce a bad one.

Cultures vary, as well as issues across the globe. I don't think a couple thousand is enough. We have 536 (435 reps, 100 senators, 1 president), and I don't think that's enough.

The UN makes more sense than a literal global government. Some international projects can go through the UN, like the 17 Goals and HVDC powerline, but a global government can't literally run every country. For starters, some of these countries oppose freedom of speech.

Way simplified, without any reasoning or motivation included beforehand:
Just one chamber of the parliament:
One-quarter of MPs elected from 1-member districts
One-quarter “ “ “ “ 3-member “
One-quarter “ “ “ “ 6-member “
One-quarter “ “ “ “ 40-member “

Emphasis on proportional representation and ordinal voting.

The number of members in each quarter would have to be some multiple of 120 for everything to come out even.

So in this scenario, every single person on Earth is part of an equal number group, each with an elected representative. I understand that proportionally it seems fair, but I think most people would end up as part of a pretty arbitrary grouping. That's way ideally voting districts are drawn in such a way that accounts for real communities. So I don't think it could be represented well by any one leader drawn from that group.

Also it would seem that the parliament representatives would be, what, half Chinese and Indian? Unless we're somehow able to supress national loyalties in favor of loyalty to the group (which as I've pointed out seems like an arbitrary construct), I don't see how that would be acceptable to countries that are smaller, culturally different, or richer.

@Agis:
On re-reading there seem to be several places in your remarks that make me unsure you understand.

Let’s suppose there are 1,200,000 voter-citizens.
120 single-member districts of 10,000 voters each; every voter is in exactly one of them.
40 3-member districts of 30,000 voters each; every voter is in exactly one of them.
20 6-member districts of 60,000 voters each; every voter is in exactly one of them.
3 40-member districts of 400,000 voters each; every voter is in exactly one of them.

Then everyone would be represented by 1+3+6+40 = 50 MPs. Not just one.

The smallest groups would have very good odds of electing about 1 to 4 representatives from a 40-member district. As long as they constitute about 2.5% or more of the population of at least one such district they’ll probably elect at least one. If they are half that numerous they might still be most of a coalition to elect one.

Groups who constitute about 16.67% or more of a 6-member district are pretty much bound to be able to elect someone from a 6-member district. If they’re 8.33% or more of such a district they still have about a 50% chance, I think, or they will probably be able to dominate a coalition that will elect such a member.

Groups who constitute 33.33% of a three-member district will almost surely be able to elect one of their members from such a district. If they are only 16.67% of such a district they still have a better than 50% chance of doing so, I think; and probably could dominate a coalition with a near certain chance to elect such an MP.

Groups who comprise 50.01% or more of some single-member district will probably elect one of their number from that district.
Groups who amount to 25.01% or more of such a district could probably form the center or nucleus of some coalition that could elect such a member. Compromise and log-rolling would be required; the more numerous the group the less compromising and deal-making they’d need to do. A 49.99% group wouldn’t have do as much as a 49.9% group, who wouldn’t have to do as much as a 49% group, who wouldn’t have to do as much as a 40% group.
And in single-member districts they’ll do instant-runoff ordinal voting or single-transferable-vote or something, so that the voters whose preferred candidate has the patheticest minorityest support can switch to supporting someone else; lather, rinse, and repeat until the favored candidate has over 50% of the vote given to candidates who still stand a chance.

(Something similar will be done in multi-member districts, not sure what.)

Each of the multi-member districts’ delegation will proportionally represent their constituency.

Yes, it’s true that the more numerous a group is the more MPs like them there’ll be in parliament.

But a group whose membership is so scattered (by, for instance, a diaspora) that, for instance, they’re less than 8.33% of any 6-member district, might still be more than 2.5% of some 40-member district, and seat an MP.

....

For the most part history has shown that partisan, ideological bonds, will quickly cross geographic boundaries between districts, and in most questions these bonds will be more governing of how MPs vote than just being neighbors will.
There’ll certainly be some questions that come up wherein most of the MPs will vote to support those from their own geographic area rather than those from further away. This may be less strictly true of those elected by some diaspora.
...
The idea of deliberately gerrymandering the districts to be of unequal population in order to have each “natural community” represented by the same number of delegates, even if the least populous “natural communities” are much smaller and the most populous “communities” are much larger, is not compatible with the kind of representative democracy I’m looking for with proportional representation. If you just don’t like PR then, we could talk about some other system; I’d rather do it on another thread, though.
....
The fact that current smaller nation-states would rather have at least one kind of representation in which they get a bigger vote than they’d get compared to a larger nation-state, is a “how do we get there from here” problem, and I’m not ready yet to discuss it about this thread-topic. Maybe I will be soon.