(a la https://slatestarcodex.com/2020/06/17/slightly-skew-systems-of-government/)
The Island of Saba very much admired the old UK approach of having one republican camera and one Are You Really Sure You Want To Do This camera. As a result, their Senate was composed of professors, bishops, guild leaders, ethicists, and hereditary aristocrats. Unfortunately, the Senate ended up being too left-wing for the comfort of the House of Representatives, and so the House of Representatives abolished first the Senate and then undesirables.
Upon declaring independence, the Grand Duchy of Baden needed to draw up a codified Constitution – but who should do the drawing up, and who should approve the final result? There had been much time and ink over how to select the membership an organization designed to counter tyranny of the majority. Ideally it would be people whose political opinions were all influenced by factors which were not shared between them. The final lineup was: the Archbishop, a randomly chosen citizen from each square unit of land determined by a per-capita maths formula, a randomly chosen citizen under the age of twenty, a randomly chosen citizen over the age of eighty, the judge who scored the highest on a test of standard dogma the composition of which a supermajority of judges had to approve, and a representative of the High Queen. All votes of the committee had to be unanimous, but any such votes would be fully binding as to the Constitution.
After eleven years of fierce debate, the membership selection process was finally approved and entrenched by a majority of virtually every conceivable political grouping in Baden. Two years of implementation later, the committee was selected. The resulting Constitutional Committee deliberated for one week, and then bindingly adopted a constitution which was virtually identical to the Constitution of Germany.
The Cracovian Metropolis was tempted to have a direct democracy, but they worried that the general populace would be unreliable, capricious, and uneducated – even more so than politicians. But they also believed strongly that handing power to elites not accountable to the general populace would be asking for disaster. So they decided that they would have a republic. The general populace would elect persons whom they believed were wise, intelligent people who agreed with core values and some major issues; and then the elected officials would make decisions in accordance with their own best judgment as to what would be best for the general populace.
Of course, that doesn't mean that the officials don't pay attention to what the people wanted. That would be crazy. Before every single decision, there is a general referendum of the whole populace on the issue. The referendums aren't binding – that would be a direct democracy – but the officials are honour-bound to weigh the result heavily, for after all, people are usually pretty good at knowing what they need. The Cracovians think that it would be laughably dishonest to have a "democracy" where the populace isn't even consulted on a given decision.
The people of New Sydney thought that the American Founding Fathers were on the right track by adapting the bicameral Westminster System to their own needs as a federal democracy. They did, however, think that the poor Founding Fathers had been constrained by the limits of eighteenth-century communications infrastructure in having to limit the House of Commons to a roomful of representatives just because popular referenda would have been totally impractical. (This wasn't necessarily what happened with the American Founding Fathers, but it's what the Sydneysiders thought had.) Still, even with modern communications, it was nevertheless too impractical to have the chamber of initiative be the entire voting population. So the Sydneysiders drew up a simple system. The National Assembly was elected by party-list proportional representation; each representative in the chamber had the right to propose any bill; and if at least two representatives voted in favour, the bill would go to the general populace, who could veto it or vote it through. Thus, the representatives were less like MPs and more like diplomats or barristers, acting as the agents of their constituents for solely bureaucratic purposes.
But the Sydneysiders weren't satisfied with that; they didn't necessarily want to get as close as possible to a direct democracy, but rather to get as close as possible to the idea of the bicameral legislature. They had the house of commons, and now they needed the senate, a body of wise and cautious individuals who could check the worst populist impulses of the lower house. So the Sydneysiders drew up the composition of their House of Magnates: bishops, professors, judges, generals, Cabinet ministers, hereditary lords, civil servants, knights, guild heads, corporation heads, the heads of various activist groups and legal funds, mayors, one spot that is auctioned off each term, representatives elected by the student bodies of various universities, representatives elected by the faculties of various universities, and the representative of the High Queen.
Foreigners complain that this is undemocratic. The Sydneysiders respond that they're missing the point.
When the State of Saarland announced that it would select cabinet ministers via competitive examination, cynics protested that the exams would be rigged; the constitutional convention replied that the exams would cover only standard dogma, the questions would be multiple choice for the sake of neutral grading, and the Supreme Court would be permitted and compelled to overturn blatantly rigged questions upon appeal by concerned citizens. This worked out quite well, with two downsides. The first downside was that the ministers tended to be people who had spent their entire lives studying only standard dogma, and thus usually had no creativity or original thinking whatsoever; however, this was no different from any parliamentary democracy. The second downside was that the Minister of Commerce formed a secret pact with the economics departments of the country and slowly turned the study of economics into a secret conspiracy in which only supporters of the Shadow-Ministry of Commerce were permitted to learn enough to pass the examination; this worked for about fifty years until an immigrant with an econ degree from Fenwick University took the test, became Minister, and reopened the universities.
The new Kingdom of Lombardy was of the opinion that rhetoric was so powerful that whoever hired the best lawyer won the debate. They didn't want to be a plutocracy, so they immediately eliminated public debate on anything related to politics. While most citizens patriotically obeyed this ban, many dissidents immediately launched underground newspapers. The official government commanded citizens to stamp out samizdat (saying that if the citizens didn't want to then they could elect different governors in three years provided that they didn't tell their fellow citizens that that was what they were doing). Unfortunately, it turned out that the Capitol had some security blind spots, unnoticed and unsecured because the government had outlawed itself from hiring white-hat hackers; as a result, the rebels overran the senate chamber. The Kingdom of Lombardy is now a rhetoritocracy; all governors are elected via voting at the end of public debates, and old-fashioned citizens complain that 99% of legislators are former barristers or newspaper columnists.
Seeing the Lombardian Civil War, a spooked Catalonia decided that the problem wasn't earnest debate but rather rhetoric, and inculcated in its population a deep-seated belief that if someone was using rhetoric it was because their actual data couldn't stand up on its own. After a brief race to the bottom in which people tried to get elected by arguing more and more stridently that they would make the worst legislator in world history, the nation soon settled on a culture in which arguments were presented in the barest way possible, ideally in randomised order. There is much controversy over whether spoken campaign speeches should be allowed, or if oral intonation does too much to sway voters.
The Federation of New Prussia adopted a unicameral legislature much like the American Senate, but thought that the Westminster System had an interesting idea in having the head of state being separate from the head of government. This seemed like a good idea, for efficiency reasons in addition to separation of powers, but their revolutionary leader General George Bolivar had a 99% approval rating and everyone wanted him to be running the show. So they were pondering whether and how to implement headship in their own constitution, when one of the convention's members happened to consider the example of the Duke of Boston. The Dukeship of Boston was almost entirely a powerless honorific, but the individual who happened to be the Duke also happened to be the elected Doge of the Free City. The Duke had said that they had considered writing the Bostonian constitution to provide that Their Grace's descendants would be powerless figureheads of state, "But it would have just been too silly for the supposed head of the country to have no power whatsoever. I mean, come on – how can you take anyone like that seriously? No offense to William, who's nice and intelligent enough, but he's just some random rich guy." (As this statement illustrates, the Bostonians often contested with the English that the more western head of state "Tells it like it is".) The constitutional convention hit upon a brilliant solution. Prussia would have one President and one Lord Chancellor. The President would be elected by direct popular vote, and serve a four-year term; the Lord Chancellor would be appointed by a House of Lords existing specifically for that purpose, and serve a twelve-year term. The President would be a direct representative of the populace, and have great power; whereas the Lord Chancellor would be a meritocratic magistrate, and have great influence. The President would have veto power over bills passed by the legislature, command of the military, and veto power over cabinet ministers; the Lord Chancellor would have a non-majority seat on the Supreme Court, the legislature, and the House of Lords, and the ability to nominate cabinet ministers. Thus, the common populace could have control over the powers of government, while the wise elders could have sway. In accordance with his popularity, General George Bolivar was simultaneously elected President and appointed Lord Chancellor; after three terms of the former and one of the latter, he retired, and the two offices were never again held by the same person.
The City-State of Dresden agreed with Clement Atlee that a powerless figurehead chief of state is tremendously useful to democracy, in that all of the patriotic hero-worship that would otherwise go to someone who could abuse that loyalty to start wars is instead directed to someone who can't do anything at all and whose authority is so blatantly illegitimate that any attempt to seize power would be met with pitchfork, torch, and possible guillotine. The difficult part was striking a balance in having a chief respectable enough to receive such patriotism yet disreputable enough to never be given any actual power. Hereditary was considered, but this was deemed insufficiently fair, for everyone should have the chance to be king. Since everyone should have the chance to be king, it was naturally decided that the monarch should be selected via absolute sortition. However, it transpired that the populace somehow considered absolute sortition to be more random than accident of birth, and thus a lottery winner would not be sufficiently respectable. So the City-State eventually settled on a complicated scheme in which one hundred randomly chosen youths had to pass a Test of Courage and the winner became King. The downside was that to be authentic, the Test of Courage had to have a high fatality rate; but on the plus side, this drastically reduced the risk of the second-place winner pretending to the throne.
(You know, since he'd be dead. Geddit?)
Greater Gargellen elects the entire legislature via sortition. They reasoned that while it's true that sortition has a chance of putting a bunch of completely evil nutcases into the office, so do ordinary elections, for there are no certainties in this universe, only probabilities; election by lot simply makes the uncertainties more visible than election by popular vote does. The neighbouring country of Samnaun sees the logic of this, but has a House of Elites that can slow or restrain any particularly crazy acts by the House of Randos. Greater Gargellen objects to this, saying that Samnaun is concerned only about bad laws being passed, rather than the House of Elites killing thousands by preventing the passage of desperately needed government programs.
Saxon Hanover didn't want the Supreme Court to be a monolithic block of identical ideologues. So they wrote in the Hanoverian Constitution that two of the five Supreme Court justices had to be the two circuit judges who were overruled the most by the previous Supreme Court. This immediately created the perverse incentive of circuit judges writing the most outrageous rulings possible, in order to win most-overturned status. Saxon Hanover hastily amended the constitution to say that only overruled cases in circuit splits would be counted for this. The Supreme Court had to do a little pushing-around when the circuit judges then tried desperately to make their own hearing of a given issue somehow qualify as a different issue which another circuit had ruled differently on, but it worked out alright in the end.
The shining city of Nouvelle-Paris is the neutral capital of the European Union, and as such is directly under the rule of European Parliament and the High Queen. However, the European government has almost entirely devolved home rule to the local municipal government, although it reserves the right to reassert their control for any reason. This latter action happened once, when the city police finally cracked down on illegal parking by MPs; Parliament and Council responded by withdrawing all funding for the arts until the Queen stepped in.