@pemerton @Ovinomancer @Manbearcat - Thank you all for your answers and play examples!
No worries!
The thing I'm wondering about most immediately is how much of enjoyment of the DM and players is what they imagine they're doing, as opposed to what they're actually doing. For example, if a player was enjoying what he thought was going through a dungeon with pre-specified difficulties, how much of that should be lost if it turns out the DCs were being made up on the fly? If a player was having fun thinking that all of the responses hadn't been pre-imagined by the DM and showed up organically in play, how much of that should be lost if they learn the DM had a pre-imagined plot that sometimes dropped in?
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Do the adamant no-dice-fudgers still adjust whether reinforcements all show up or have the villain switch attacks just to save a PC? Do they adjust what magic items were going to be found if a character dies and is replaced by a different class? etc. And I wonder how many players would want the "no-fudge" option when picking a game style, but then want the fudging out at some level (is taking character backgrounds into account when world designing considered fudging to some).
I don't do a lot of
dungeon with pre-specified difficulties play. But I think if I had signed up for that, and then learned that in fact the GM was making it up as s/he goes along, I would feel a bit ripped off. I have had somewhat parallel experiences where, a certain way (eg a few sessions) into a game it becomes clear that nothing we do as players is having a substantial effect on the situations the GM is presenting to us. Generally those games have not been good ones.
When I'm playing Burning Wheel it tends to be fairly apparent to me when the GM is introducing something that was prepared or pre-imagined. This is because (i) I know him fairly well and can recognise his predictions, and (ii) they tend to have a degree of intricacy that suggests some prep rather than spontaneity. I don't really care where he draws his material from, provided that he follows the precepts of the game - that is, framing scenes that speak to player-authored Beliefs, Instincts et al; and narrating consequences (especially failure consequences) that reflect the player intent for action declarations. Because of the role of player authorship, there are going to be practical constraints on how much pre-imagining is feasible and helpful.
There are no secret rolls in the games that I play/GM at the moment, with one exception: Classic Traveller calls for secret rolls by the referee to determine if a branch of the Psionics Institute exists on a world. I have adopted a practice of making that roll at the same time that I make my other world notes as part of the process of world generation. There is also an express permission in Classic Traveller for the referee to use non-random considerations in generating worlds, and so far I have done that once in relation to branches of the Psionics Institute. I can't remember what exactly I said to the players at the time, but it doesn't worry me if they work out that I made a deliberate decision on that occasion: the trigger for it was their declared action to search for a branch on the world they were on, which had an established connection to an ancient psionics-using alien civilisation.
When I was GMing 4e, I would make decisions about reinforcements, which PC a villain might attack, etc based on what seemed sensible and fun at the time. At least one of my players adopted a general policy of not unloading all his big guns early on the basis that "pemerton always keeps something up his sleeve" in an encounter. As far as magic items were concerned in that game, these were mostly gifts or improvements of existing items; but when items were occasionally discovered I was generally going off player-authored "wishlists" to decide what they were.
In our Prince Valiant game there have been two "magic items" so far. One is a silvered dagger blessed in the waters of St Sigobert. It was a gift to one of the PCs from the Duke of York (on the occasion of another PC's wedding to the Duke's daughter Elizabeth), the recipient PC having founded a holy military order The Knights of St Sigobert. The other is a greatsword that one of the PCs (the husband of Elizabeth of York) took from a cursed ghost/haunt that the PCs defeated in the forests of Dacia. It seems likely that the greatsword is cursed in some fashion, and so far the character uses it sparingly.