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Doubling the overlay resolution doesn't really make a heck of a lot of sense, since the monitor is not twice as big as itself.

Doubling the background resolution gives you fake anti-aliasing, which is what most people are after when they're doing this. Tripling or quadrupling it would likely slow the game down too much for most people to bother. And doing things in strange multiples (like 1.5x or something) is just weird and doesn't look as good.

As far as doubling the width and then using that value in both dimensions (2048x2048), instead of doubling the width and then doubling the height (2048x1536), I really don't know why people go with the first option. It may be that on certain hardware, that works better. Or it may be that it used to be required on certain hardware, and now it's just how everyone does it, especially since the resolutions you can set through the provided settings app are square. Or maybe they only want to multiply once instead of twice. Personally, I've always gotten better performance out of just doubling each dimension (2048x1536, or, in my case, 2880x1800). YMMV.