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My suggestion for changing the way officials are elected into congress combines the Single Transferable Vote system that I discussed in my presidential election article with another change to further promote alternate parties. Instead of voting for particular candidates, people would vote for parties. Then the number of votes that a party receives in their state determines how many of that state's seats go to that party.
The legislation would stipulate that the seats must be apportioned fairly. Rather than letting each state decide what a fair distribution of seats mean, the law would contain an explicit description of what a fair distribution is. Consider that every party has some value called a 'seat rate', which is the number of votes they received divided by the number of seats they received. A lower seat rate gives more seat, so it's better for you. It'd be unfair if another party has a seat rate, which, if your party had that rate, would have given your party more seats than it received.
"The democrats had a seat rate of 25,000! If we'd had that seat rate, we'd have received 4 seats in the house instead of 3. That's unfair!"
I'll provide a mathematical statement of that seat rate, as well as an algorithm for determining a fair distribution of seats in another post. It will also describe how the single transferable vote system would play into this new way of electing congresspersons.