US Authoritarianism

There's two main ways you can do it, which you pick depends on the things you care about. The first is to just say that a voter can cast as many votes as they like, but that the weight of their ballot is divided by the number of votes they cast. So if a voter selects N candidates, then each candidate gets 1/N votes towards their total. Top vote-getters win. This method is very simple and easy to understand, but it does encourage voters to strongly limit their support, since each vote they cast dilutes the power of the rest of their votes. In fact, going from 1 choice to 2 is the biggest drop off in terms of support for your candidates.

So you say to yourself "I'd rather not punish people for voting for as many candidates as they like, but I don't want one party to win all the seats if they have a slight majority support." Well okay, let's assign seats sequentially then using Sequential Proportional Approval Voting. Voters pick any number of candidates and the votes are added up. Top candidate gets the first seat. Then, for every ballot with W winners on it, its value is assigned to 1/(W+1). For the first round no ballot can have any winners, so all ballots count the same. For the second round, some have weight 1 and others weight 1/2. In the third round 1, 1/2, and 1/3. Then 1, 1/2, 1/3, and 1/4. You get the idea. The aim here is to allow voters to support c andidates that are unlikely to win, since the number of votes they cast doesn't impact their vote weight. But, as a voter gains more and more representatives in office, their ballot is weighed less and less, since they should be more and more satisfied with their representation.

As a simple example, if there were three parties with R=45%, G=43%, and B=12% support, and all voters voted for only their party, the seats would be awarded as follows:

1. R (R=100% G=0% B=0%)
2. G (R=50% G=50% B=0%)
3. R (R=67% G=33% B=0%)
4. G (R=50% G=50% B=0%)
5. B (R=40% G=40% B=20%)
6. R (R=50% G=33% B=17%)
7. B (R=43% G=43% B=014%)
8. And so on and so forth

As with any proportional method the more seats you award the more true to the correct proportion the awarded seats become. This method makes it much more reasonable to vote for everyone you like, but it also discourages insincere votes because if that candidate gets awarded a seat all the rest of your votes just became worth less in all the following rounds.

Honestly any reasonable proportional method with a big enough seat pool is just about as good as any other. I like using the approval method since the voter is presented with a ridiculously simple ballot (you literally can't fill in the bubbles wrong) and they can functionally ignore the counting method knowing that whatever the election is, voting for everyone they like is a pretty reasonable strategy. Slightly more optimal strategies exist if you know the polling data and what kind of election you're voting in, but "everyone you like" is still a very good one.