As others have noted, proportional representation (PR) is the best solution to gerrymandering. (Not arguing that we shouldn't also support second-best solutions like nonpartisan redistricting and court challenges to the worst gerrymanders.)
The three PR methods in common use are STV-with-multimember-districts; mixed-member proportional (MMP); and open (or, yuck, closed) list systems. All of these (except closed lists) are decent, but have downsides. STV leads to very complex ballots; MMP leads to "two classes" of representatives; open list focuses your voting power on the partisan choice, but doesn't give you much power to help set the direction of your favorite party. And all three can lead to extreme party fragmentation and thus excess "kingmaker" power for splinter parties, unless there are rules against that.
It is, however, possible to design a method without any of these downsides. Perfection is impossible, but the Pareto frontier is, and none of the above methods are on it. Here's one that is:
Do you have a more detailed desription about how this method works? Your links do not describe it in much detail but it is interesting.
Even without the details my sense is that this method suffers from some very serious problems. Picking the two candidates with the highest initial votes seems like a bad idea; this can remove all of the most widely supported candidates if there are enough similar candidates (i.e. if 70% of the voters are split between 10 candidates and 30% between two candidates, but the 70% consider the two that the 30% would elect to be least favored).
In general, strategic voting is likely to have a huge effect. I don't understand what the write-in option does, but if it allows you to vote for candidates in a different district than that would allow votes to be used to disloge more influential members of an opposing party against the wishes of the district that candidate actually represents.
In any case, always interesting to learn about possible voting methods :).
IMO, limiting the effectiveness of strategic voting is one of the most important things a voting method should do, while also not electing anyone who would heavily lose in two candidate races vs. any of the other candidates. I don't think it needs to be strictly Condorcet, but the further away from that a method gets (at least when geographic representation is involed) the less reasonable it seems in a lot of ways IMO.
I've been thinking about the possibility of a parallel system where the grographical representation part is non-partisan. I think this might work if geographical representatives are allowed and expected to allocate their vote on a given issue proportionally to how the people they represent would vote on that issue.
The three PR methods in common use are STV-with-multimember-districts; mixed-member proportional (MMP); and open (or, yuck, closed) list systems. All of these (except closed lists) are decent, but have downsides. STV leads to very complex ballots; MMP leads to "two classes" of representatives; open list focuses your voting power on the partisan choice, but doesn't give you much power to help set the direction of your favorite party. And all three can lead to extreme party fragmentation and thus excess "kingmaker" power for splinter parties, unless there are rules against that.
It is, however, possible to design a method without any of these downsides. Perfection is impossible, but the Pareto frontier is, and none of the above methods are on it. Here's one that is:
http://wiki.electorama.com/wiki/Geographic_Open_List/Delegat...
Here's an article about why it's good:
https://medium.com/@jameson.quinn/make-all-votes-count-part-...