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It seems as though there are myriad other voting schemes that do not require representatives to run for a "district". For example, representatives for the state could all be on the ballot for the entire state, and the top n in the state (where n is the number of available seats) get seats.

Do all multi-member states draw districts? If so, are they required to do so? If not, then why do they not adopt other voting schemes?

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Q: Are states required to divide into congressional districts?

Yes.

2 U.S. Code § 2c - Number of Congressional Districts; number of Representatives from each District

In each State entitled in the Ninety-first Congress or in any subsequent Congress thereafter to more than one Representative under an apportionment made pursuant to the provisions of section 2a(a) of this title, there shall be established by law a number of districts equal to the number of Representatives to which such State is so entitled, and Representatives shall be elected only from districts so established, no district to elect more than one Representative (except that a State which is entitled to more than one Representative and which has in all previous elections elected its Representatives at Large may elect its Representatives at Large to the Ninety-first Congress).

Occasionally, a bill will be introduced to change to multi-member districts. However, the bills are never brought to the floor for a vote. One such example is H.R.3863 - Fair Representation Act from the 117th Congress. That bill would have allowed for some states to have districts with three to five members. And, others to have up to six at-large members. It would have also also required ranked choice voting for "all elections for Senators and Members of the House of Representatives".

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    It should be noted that there's also a requirement, in states with a sufficiently high non-White populations, to create districts that ensure the election of racial minorities.
    – dan04
    Sep 29, 2023 at 20:02
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    @dan04 this was why I asked, actually, since a number of non-district-based voting schemes would make that whole problem moot. Viz, suppose the population is 70% X group and 30% Y group, and group X always votes party Xparty and never Yparty, group Y always votes Yparty never Xparty. If there are 10 seats available, 20 candidates, 3 of which are Yparty and 7 of which are Xparty, then the seats will be divided 7 and 3, exactly along demographic lines. Of course the real world is more complex, but the district-based system is probably the worst possible system for this problem.
    – Him
    Sep 30, 2023 at 0:11
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    For example, if the 10 districts above were created to include population uniformly at random, then in all probability 0 seats would go to Yparty.
    – Him
    Sep 30, 2023 at 0:12
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    That depends on how the voting is conducted. If there are 10 completely independent at-large seats to be elected, in each of the 10 races the vote will be 70% X and 30% Y, and the outcome will be that there are ZERO seats going to Party Y; this is why we have districts.
    – arp
    Sep 30, 2023 at 1:49
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    @OscarSmith: I think a court would be pretty skeptical of the argument "it's not one district with ten members, it's ten identical overlapping districts with one member each." It seems too cute by half. Oct 1, 2023 at 12:43

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