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. And 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 .
Who gets on the ballot. Divide the population into geographically compact districts of roughly equal population. Any candidate collecting more than n signatures from a district beforehand gets on the ballot for that district. The number n is determined periodically by the state, it applies to all districts, and the goal is to get about 6 people on the ballot per district. 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 cutoff, like if there are over 10 candidates with enough signatures then only the top 10 are listed. (This rule can be gamed by a party fielding 10 candidates and spending enough to give them all the most signatures, there has to be a better way.)
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, and need r representatives, repeat this r times:
Identifying representatives. The election should also publish who each representative represents. The election assigned v/n voters to some representatives, and less than v/n to others. For those voters who voted for them at all, scale their weights until the representative has v/n of the electorate or the voter is fully represented. For the remainder, assign the remaining representatives to remaining voters round robin.
The scheme for identifying representatives unfortunately make it public approximately who voted for whom. It is vital for representatives to know who they represent, so this scheme would be worse without it, but it is a strike against this scheme. Voting for one representative per district for example only requires revealing what district a voter is in. Proportional representation requires revealing a voter's party, which seems as bad to me as revealing approximately who they voted for.
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. So it's better if voters vote on laws directly (and set up rules for how to vote on laws they don't care to vote on directly) than to elect representatives who do 100% of the voting for them.
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 50% 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. A way to distinguish the two is to have people vote for five positions, all coming from the same pool, then people can give the candidates different weights for different position. That seems awfully complicated.
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.
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