Gerrymandering

Gerrymandering is the practice of splitting voters into certain districts in such a way as to maximize the chances of a certain set of candidates winning. For example, if there are 5 districts, with the population being 60% loving Vanilla ice cream, and 40% loving Chocolate ice cream, it may be possible, if all of the Vanilla voters live in the same areas, to waste Vanilla voters' votes by putting them all in two districts, and spreading the Chocolate voters out so that a majority of the remaining 3 districts are Chocolate voters, so that the majority of the winners are Chocolate-loving representatives.

Gerrymandering can lead to very disproportional outcomes. Mattingly and Vaughn found that if North Carolina were randomly districted, the 2012 election results would on average give 7.6 Democratic representatives, in contrast to the 4 Democratic representatives who were actually elected that year.

Proportional representation almost or completely eliminates the possibility of gerrymandering.

Cardinal methods
It has been argued by some that among single-winner or bloc voting methods, Approval voting and Score voting best resist gerrymandering, in part because they pass the consistency criterion, and in part because they are claimed to result in the election of consensus candidates. Here is a counterexample: Isn't the problem inherent with gerrymandering though less distortion of the overall political centroid itself and more an exploitation of majority mechanics within the legislature, which are basically impossible to get rid of?

Using the example I've given you before: 10 L voters, 5 C voters, 10 R voters, 5 seats. We suppose all voters of the same faction vote the same, and L voters cast L:5 C:2 R:0, C voters cast L:0 C:5 R:0, and R voters cast L:0 C:2 R:5. We suppose that the winners of each district sit precisely on the centroid for their district, so they'll vote as an average not merely of those who elected them (which would skew this further) but of the district members as a whole; letting a L-voter represent -1, C-voter 0, and R-voter 1; so a representative for a 3 L, 1 C, 1 R district would be value (3*-1 + 0 + 1)/5 = -.4.

We then gerrymander so that the districts are Now, clearly under these suppositions, the overall political centroid for the entire population is properly 0; and the average for the elected body as a whole is indeed this. However, because of our gerrymander, The first three seats can form a majority whose weakest members are of value -0.6, which is clearly not representative of the entire populace. Also: Next: it doesn't neuter gerrymandering at all. Take that standard WaPo graphic example, except now you've got 10 Red, 5 Purple, 10 Blue. Drawing 5 districts, make two {4 R, 1 P} and then two {1 R, 1 P, 3 B} and then finally one {1 P, 4B}. Let's generously suppose that Score will return the average for each district, where each R is a -1, a P is a 0, and a B is a +1. So the first two are -.8, the next two are +0.4, and the last one is a +.8. You can then form a majority using the 2 +0.4's and the +.8.

So yeah, supposing that Score works somehow better than it probably will in reality, you've changed the result in the legislature from a +1 to a +.4. It's a big improvement, but I'd hardly call it "neutering".

PR returns the better result here IMO, since P (the dead-center of the district) determines what passes or not.