Every candidate has the option of declaring a party affiliation. If a candidate does not declare a party affiliation, then they are a "party of one" for the purposes of the algorithm.
After the election is completed in all N districts, the votes cast for each party in all districts are totalled. Seats are then allocated to each "party" using a typical aportionment scheme (webster's method, for instance).
The winners of the election are found by maximizing the number of voters who voted for the winning candidates, with the constraints that:
1) Each "party" gets their appropriate number of seats
2) Each district elects exactly one candidate.
This will produce some strange results in certain cases. For instance, if the Greens poll at some consistent low percentage all around a state, then they might elect one candidate from a district somewhere, even though that candidate finished a distant third in the election in that district. But that district does get local representation, and proportionality across the state is preserved.
In the case where there are only two parties, then it's easy to see what the result will be. Imagine party A gets 55% of the vote in a 20-seat legislature, while party B gets 45% of the vote. Then the 11 party A candidates who polled highest will be elected, and the other 9 districts will be won by party B. This would be true even if the party A candidate won 75% of the districts due to Gerrymandering.
I don't claim that this is a great, or even an especially good PR method. But it has the distinction that it works using simple plurality voting in single-member districts.
-Adam
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