If it was two artifacts, each belonged to separate PCs, neither artifact drove the plot or was driven by the plot, and it was a direct contest between the two artifacts...
1) I start with thematic effects. If one artifact's thematic stuff trumped the other (light beats dark, water beats fire, etc) I'd stop there.
2) Then I'd use their refresh costs, the more costly artifact trumping the other.
3) If the refresh costs were the same, close enough, or the players weren't satisfied by any of the above, then I would have each player use their artifact's refresh cost (or maybe some skill if it seem appropriate) and have them use that as their defense and attack stat and have the two go at so the rolls can solve the problem.
4) When nothing else seems to fit right, my default would be that the two artifacts are completely unaffected (or equally effected) by the other.
I ran into similar situations in my first DF campaign (which had a lot of "divine" artifacts) and option 3 is what I used most of the time. I had them roll their artifact's refresh cost, but one artifact was a sentient lamp with a high lore so I allowed its lore skill to be rolled instead when it applied.
I used option 3 in a few instances, the water beats fire being the best example, and the final two uber artifacts used option 4 since the artifacts were basically evenly matched and the two players skills were really irrelevant, them just being tools of the weapons (it was their "side-kicks" actions that decided the battle).
