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The most obvious problem with that would be statistically insignificant representatives, these people would not have floor privileges, only the top few with statistically significant vote counts would be able to speak or attend Congress in person.
What statistically significant representatives there are, would have their time apportioned amongst themselves. In other words, a district/state would be given a certain amount of speaking time, and that time would be sliced up proportionally based upon the number of people a representative had vote for them in an election.
So, in other words, each congressional district would be given 60 minutes of speaking time. If a Democratic representative received 75% of the vote in a given district, they would have 45 minutes to speak on the floor. On the other hand, the Republican received 24% of the vote, they'd get 15 minutes of the time. The 1% representative(s) would not have floor privileges.
It would still be possible to have a majority in Congress, but each district would have a representative from either of the current parties, or another party if it should arise at a future date.
In addition, this system might cause people to pursue direct democracy. As long as someone is qualified to be a representative in the US Constitution, they could run for office and vote for themselves.
A system of electronic, over the internet, voting could be instituted to cast votes. If someone is no longer interested in representing themselves, they would be counted as present, undecided, or some other neutral voting position. At the next election they would be able to vote for another candidate, or not vote and not have representation.
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