Gerrymandering is the deliberate drawing of district lines in such a way as to distort representation of subgroups of the population.
Consider Case 1: Suppose that a population is divided 60%/40% over some longstanding, continuing dispute. The A's have 60%, the B's have 40. The B's typically live in urban areas, the A's in suburbs and rural areas.
There are five districts, each of which elects one representative to the legislature.
If the A's are drawing the lines, they can draw the districts to slice through the cities, so that all five have a 60/40 majority of A's. The A's get all the representation, the B's get none.
If the B's are drawing the lines, they can draw two districts that are rural and outer suburbs, essentially all A's, and three districts that are urban and inner suburbs, that will have a substantial majority of B's. The B's will have three representatives and the A's two; the B's will control the legislature.
This seems to me to be independent of which method is used to elect the single winner from each district.
Case 2: Suppose we tried making fifteen districts, each electing a single representative? The A's could still draw long, narrow pie slices through the cities to get all fifteen representatives, and the B's could draw shorter, fatter pie slices surrounded by rural cookies, to get more than half the representatives. Making smaller districts does not solve gerrymandering as long as their shape is not restricted.
Case 3: If, somehow, the drawing of district lines were taken out of interested hands and made immune to meddling by the legislature, having lots of smaller districts might solve the problem. The impartial body drawing the lines could use a computer program to minimize the total length of district lines while keeping all districts of equal population.
Imagine a tray with vertical edges, the shape of your state. Put as many bubbles into it as you need districts, for whatever election you are holding- State legislature, Congress, whatever. The bubbles will arrange themselves by their surface tension, to minimize the length of their walls.
Software could simulate this process, enlarging or contracting bubbles as needed to equalize the population covered by each bubble. The result would be a map of irregular polygons, all convex, with all boundaries between districts being straight lines. The more districts, the more likely it would be that representation would be proportional to the numbers of voters in each camp.
It would be more likely, but not guaranteed. Each district would still be electing one representative, and 51% of the vote would be sufficient to elect them. In each district, 49% of the voters would have wasted their votes, either by voting for a losing candidate ("undervotes"), or by adding their vote to a candidate who already had more than enough to win ("overvotes"). With almost half the vote being ineffective in electing representation, there is a lot of slack in the system; the results could diverge from proportionality to one side or the other unintentionally.
In my opinion, to take the drawing of lines out of the hands of the legislature, and to keep it out of their hands, would require a widespread commitment of the voters to fair play. And if you had such
commitment, it would be simpler to aim at proportional representation directly.
This direct approach would be to draw larger districts with multiple representatives, allocated by some method of proportional representation.
Case 4: Suppose there were still five districts, but each district elected three representatives instead of one. We assume the simplest method of proportional representation, Party List. Any party with more than 25% of the vote in a district will get a seat; any party with more than 50% will get two seats; to win all three would require winning 75% of the vote.
If the A's drew the lines slicing through cities as before, the A's would win 10 seats and the B's 5. If the B's were in charge of drawing the lines, they could do as before, and get 6 seats to the A's 9. So,
with three seats per district, there is still some effect of gerrymandering, but the effect is much less than with one.
The more seats per district, the harder it is to skew the results by gerrymandering; as a rule of thumb, if districts have at least five members, elected by proportional representation, it makes little if any difference where the lines are drawn.
Case 5: Suppose there were three districts of five representatives each. If the A's drew the lines as before, each district would elect three A's and two B's. If the B's drew the lines as before, there would be one district with five A's, and two with two A's and three B's. In either case, the count would be nine A's and six B's, which is proportional.