Electing congress is different from voting for one decision, because the number of possible congresses is beyond astronomical, so it is not practical to enumerate all the choices. So some heuristics are required. Here are some heuristics (I am making them up, I don't know of standard heuristics for this problem):
The current scheme, one representative per district, is awful. Gerrymandering can have a party with as little as 25% support still have the majority in congress, and since congress votes on bills, whoever holds the majority in congress can ignore everyone else. But not having gerrymandering can be worse: if 51% or more is for one party and you don't gerrymander, then the remaining 49% gets no representatives, so the bills are proposed and revised and voted on with no say from 49% of the people.
Having multiple representatives per district only helps if you associate voters with representatives. For example, if you had 10 positions and 20 candidates (10 party X and 10 party Y) and the 10 with the most votes won, and 51% of voters voted for party X, then all 10 party X candidates would get elected and 49% of voters would have 0% representation. If you associate voters with representatives though then only 5 or 6 party X candidates would get elected and the remainder would be selected by the other 49%.
Here is a proposal. I think it has some good ideas, and some improvements over the current one-representative-per-district, but it also has shortcomings, and is complicated.
The ballot. There would be maybe 6 people listed on the ballot per voter. They may specify write-in candidates too. For each candidate, the voter says what amount they're willing to be represented by that candidate, between 0.0 and 1.0, where 1.0 is highest. The sum of weights for all candidates is allowed to be greater than 1.0. Such a paper ballot would be easy to spoil, but it's easy to make the ballot unspoilable if it is entered on a computer. Alternatively (simpler but less expressive), just list all the candidates, and select the ones you approve of. That is equivalent to rating everyone either 1.0 or 0.0. This ballot is simple and hard to spoil even on paper.
Who gets on the ballot. Divide the population into geographically compact districts of roughly equal population. Major parties would sponsor different candidates in different districts. Fringe candidates would rely on being write-in candidates across many districts. There would be some max m (maybe m=6) of candidates that could go on the ballot per district. If there are more than m candidates with any signatures, pretend that everyone who gave at least one signature cast a ballot. A signature is a weight of 1.0, a lack of a signature is 0.0. Then run the voting scheme below to choose which m candidates go on the ballot.
How to fund elections. Candidates are paid back by the state after the election based on the votes they got. I imagine it is proportional to the number of votes they got, but with some upper bound so that superstars don't collect all the funds. Candidates would have to arrange for loans before the election. Actually getting elected is valuable, so those who are actually elected may get back less than those who did not.
Tallying votes. If you have v voters total cast a ballot, and need r representatives, repeat this r times:
A pretty similar scheme is the fair representation act. I'd vote for it. Both these schemes are blocked by the 1967 single-member district mandate. That mandate was aimed at disallowing at-large elections, which elect the top n vote-getters, even if all the votes came from the same 51% of voters. Which can lead to 0% representation for 49% of the voters. Blocking at-large elections is a good thing. That act should stay. But it should be amended to also allow these other multi-representative schemes that do a better job of giving voters representation than single-representative districts.
Identifying representatives. Optionally, publish who each representative represents, and by what fraction. This is good because representatives can tell if the people they talk to actually voted for them. But this is bad because it makes the voting non-secret. Districts are a compromise: it causes no extra harm for voters to reveal what district they are in, and representatives can tell if voters are from their districts.
This doesn't solve the problem of representatives knowing who they represent very well. Either you tell them, in which case votes aren't anonymous, or you don't tell them, then they only know whether voters are in their district or not.
Congresses propose laws and revise laws. For that purpose, it's sufficient to have one representative representing you. For proposing and revising laws, if you are willing to be partially represented by many candidates, it makes a big difference if the candidates are duplicates of each other or if they are all mixed bags in different ways. This scheme does nothing to distinguish the two. I haven't seen a simple way to solve that.
Congress votes on laws. For that purpose, you want over 50% of congress agreeing with you. Congresses are bad at that in general. Even the best candidates tend to agree with you about 60% of the time. I suspect this scheme comes as close as any to allowing you to pick an ideal combination of candidates; I haven't tested it much. But there is no candidate or combination of candidates to elect who comes anywhere close to you voting for each law yourself. It's better if voters vote on laws directly (and/or set up rules for how to vote on laws they don't care to vote on directly, most likely that representative x votes for these types of issues while candidate y votes for those types) rather than to elect representatives who do 100% of the voting for them. Fixing this would get a lot closer to direct democracy.
Jungle primaries, where you select two candidates for the final election, shouldn't use this. They should pick the two candidates with the most votes, which is what they actually do. Why? Because you end up being represented by one person, not both of them. The final election narrows it down to one person.
Another idea I like is to have a set of tests that candidates have to pass to get on the ballot. Each test has to be approved of by at least 90% of the population. For example, if you wanted a test that they be a Pastafarian, that would be OK if 90% of the population approves of that test, but otherwise not OK. In particular I'd like to try to have a test that representatives can understand balancing a checkbook, since levying and spending taxes are a large portion of their job.
I've been seeing shared posts on Facebook calling for term limits for Congress. A 2013 poll said 75% of Americans favored them, and a 2018 poll said 82% did. Trump has called for them repeatedly.
I'm puzzled.
Term limits for the president are good, both to limit the abuse that could be done by a president given the extraordinary power of the presidency, and to keep the country accustomed to peaceful transfers of that power. But those reasons don't hold up for congress. The makeup of congress, and even its balance of power, changes every two years already. Individual representatives just don't hold all that much power. Term limits also prevent a representative from holding on forever just out of name recognition without doing anything. But, term limits replace that with lame duck representatives, who statistics show have less influence, serve on fewer committees, sponsor fewer bills, and even skip voting more than representatives who need to be re-elected. I just don't see any upside to term limits.
Term limits have a big and obvious downside. Long-term representatives have more experience with legislating, they have more connections, they are higher ranked. Which means they can get more done. This is very good for the people they represent, and moderately good for the country as a whole. There's a very long learning curve, so cutting people off before they can get good at their job guarantees you won't get much done, and what you do get done will be flawed.
No upside, big downside, equals bad. I asked my wife what she thought, and she came to the same conclusion for about the same reasons.
I can see why President Trump would benefit from congressional term limits. It would make congress less experienced and easier to push around. Which increases presidential power. It's also an issue the people agree with that congress does not, which again casts him in a good light.
But I can't see why 75% to 82% of Americans favor them in the first place. I'm clearly in the minority here, but I don't know why. Also, how could nearly half of congress have served over 8 years while 82% of voters disapprove of electing repesentatives to serve more than 8 years? It sounds like a contradiction.
Hm. Another possibility for wanting term limits is that longtime representatives are fairly old (they may have been young when they started but they've been serving a long time, and time passing is what causes being old), and have been living in Washington DC for a long time rather than their district, so they don't represent young people or people of their district well. This brings up again the problem that congress both writes laws and votes on them: they are needed to write them, and longterm experience helps there, but ideally we would do the voting directly.
Another possibility is that popular longtime representatives get to be REALLY old, and often suffer from dementia or other health issues that prevent them from serving as well as they could. Requiring that representatives will be no more than 80 years old (85?) when their term ends would mostly fix that without making experienced representatives impossible.