Lets make the factions a bit more different and interesting… and have different ways of electing their parliaments.
Neo Charter and Castillo-Ito Mercantile are both corporate based with Neo being more monarchy affiliated and Castillo being more mergerish. Insitor Cooperative is South American, and Antares is colonial. Unaffiliated worlds are… well, unaffiliated.
Neo Charter could be based on a corporation setting up set up its slate of candidates. By default, this is sorted by shareholders for the corporation, and then by seniority. This list can be adjusted by the chairman of the corporation (if one is set, if not this defaults to the largest shareholder and seniority as the tiebreaker). A player running in a planet’s election would enter the corporation into the election. Players voting would be for a corporation rather than a player - that is a vote would be cast for GTU rather than an individual. Parliament would be proportional with an asterisk - only players from the corporation that have established a base on the planet can be members of the parliament.
For example, if GTU got 60% of the vote on a 5 seat planet, they would entitled to 3 seats. If only 1 player from that corporation had a base on that planet, they’d only get 1 seat.
Single non-transferable vote for a party (corporation). Proportional representation of the corporations (given that they have eligible candidates). Side bit with this is that it would be a way to introduce players with less name recognition into politics. NewPlayer wants to get into politics? Put them as the #2 or #3 slot for the slate for some planet in a trusted corporation.
Castillo-Ito Mercantile would use cumulative voting. (Cumulative Voting | Investor.gov)
This system is the one used in shareholder systems where you have a fixed number of votes but can split them. Everyone gets 6 votes. You can cast 6 votes for one person. You can cast 1 vote for each of 6 people. You could cast 3 votes for one and 3 votes for another. For fun, make this 2 votes for each permit the player has allocated to the base on the planet. Shareholders. This could be an interface similar to assigning experts.
Insitor Cooperative uses ranked choice single transferable vote. This is the one that people point to as one of the better ones. It’s a bit more complicated and I have seen some “wait, why did that person get elected” (STV with moderators on a stack exchange site - Gaming Stack Exchange 2014 mod election and Arqade Moderator Election 2014 | OpaVote where Ashely won despite having fewer overall votes because there were more 1st choice votes for her).
I think this is the “best” system, but its also the most confusing one. To that end and the nature of Promitor’s election it might be one to swap with Antares to have one that is simpler to understand.
Antares uses block approval voting. Vote for as many candidates as you think will do the job well. This is a “simple rugged system” that matches the martian colonists “I don’t have time for that complicated vote counting.”
Unaligned worlds… lets go for some light chaos. Random ballot. Pick a random ballot from all the ones cast. That’s the governor. Fill out the rest of the parliament by picking random ballots (toss if its a repeat). Players cast a single vote for their preferred candidate.
Why? Mostly because these are worlds where one could have some churn. Mostly it won’t matter. Several of them have had no governor for several cycles and then its “oh, we need someone to do something” and then again no governor for several cycles… or the same person with one vote for several cycles. To that end, I don’t think it would matter much other than that people would go read about it. In the realm of social choice and political philosophy its a system used as a thought experiment or as a comparison with other systems.
The cumulative voting approach is one with the least hold on the favorite betrayal criteria (where it is adventagous to vote for someone who isn’t your preferred choice)… but its also kind of the best one thematically. The other systems are ones where it is always best to cast a sincere ballot.