A fundamental mechanical problem I have with d20/D&D/PF

[icefractal]

That's why there aren't a lot of opposed stat checks in D&D. They're very swingy, they ignore level, and as a result they yield a lot of strange results. So ... don't use them. Personally, I would do something like arm wrestling with a smaller and more curved RNG - 3d4, perhaps.

For skills, the difference is generally more significant, although even then, I think that rolling the standard way only works for circumstances that are somewhat chaotic. In a controlled contest, perhaps both sides taking 10 would be appropriate, or rolling off X times and seeing who won more often.

A quick hack, for cases where you don't want to make it a lengthy contest, and want to keep things similar to standard, is for both sides to roll 3d20 and take the middle one. Sort of a middle-ground between rolling and taking 10, it could be appropriate for semi-threatened situations.