A few interesting rules questions and situations came up on Sunday. One of them was this:
Luck and other's defense rolls. One of the PCs attacked the pudding during that encounter, and hit. I rolled a 4 on the Dodge for the pudding - a critical success. This sent the PC to the Critical Miss Table (per p. B381). The player wanted to re-roll the pudding's active defense, arguing that getting sent to the Critical Miss Table doesn't feel lucky/isn't lucky/should be something Luck can avert.
I told him he could re-roll the CMT result, but that was it. No forcing a defender to re-roll defenses with Luck, period. He took the re-roll on the CMT after scoring a "crippled your own arm" result.
This isn't really a ruling, though. It's flat-out how Luck is written up on p. B66. You can force an attacker to roll three times and take the worst against you. It doesn't say you can force defenders to re-roll three times because their success (or degree of success) makes bad things happen to you.
Beyond that, though, it's a terrible idea to allow it, even if just to avert a critical defense sending you to the CMT. First, it means Luck isn't a positive stroke in your favor with your rolls or rolls that directly affect you - it's fate conspiring against your enemies, too. Imagine it as a player - you need a 3-4 to Dodge and roll a 4 anyway, and it's a critical. GM says, no, this guy is Lucky, re-roll that and take the worst of the three.
Imagine how potent it would be if you could force a dangerous foe with an iffy defense to roll three times and take the worst.
And dialing it back down to "it should just avert the bad results of a critical failure," well, Luck doesn't do that. It's three rolls, not a change from one condition (I was faked out) to another (he just defended.) You can't change "he hit me and the damage roll was horrible" to "he missed" or "he rolled minimum damage" - you have to let the dice fall three times and pick the best result for you. So even saying, "Spend a use of Luck to convert a critical success into a normal success" is a big upgrade, because that's a new addition.
That's putting aside situations where a person can only critically succeed or fail, no in between, which would mean you'd essentially have to add a possibility to have Luck do that. Adding, "it's still a critical success but not a critical failure for me" is also a big addition - that's a potent thing to have, knowing you can't critically fail because of an opponent's rolls.
So, the answer was no. By the rules, and by the implications of what expanding them to a greater definition of "lucky" that includes rolls that have spill-on negative effects on you and converting roll results into specific non-rolled results.