Who decides that something is at stake? The GM or the player(s)?
Both.
If the GM inclines to say "yes", the player is at liberty to explain why a roll should be made. If the GM is calling for a roll, the player is at liberty to seek clarification over what the GM thinks is at stake. Generally it's pretty clear to everyone, because the GM will have presented a situation that speaks to some or other player-determined priority.
Responsibility for pacing ultimately lies with the GM. But it's not ungoverned by rules (more on this below).
Seriously. In this scene, did a player say "I think whatshisname should have to roll to see if he can find a cup because that's really important"? If this is the case, then this is ripe for abuse, particularly in a game where the PCs are not a tightly-knit party with a single goal. If Player 2 didn't want Player 1 to succeed at something, like giving the naga blood, then this would be a great way to stop him. It would be terribly meta and antagonistic, but it would be acceptable by RAW and RAI.
As I just posted, it is generally clear.
But there is nothing
antagonistic about other players following the play, and noting what is at stake and hanging on the dice rolls. And the fact that another player is invested in a given player's PC
failing is, in itself, typically going to be sufficient to show that something is at stake.
Upthread I posted that "it seems that you are not really recognising the reason why rolls are called for in Burning Wheel." I still have that feeling.
Having to make a test is not some sort of fail-state. It's the core of play. As the rules say (p 11), " The GM presents the players with problems based on the players’ priorities. The players use their characters’ abilities to overcome these obstacles. To do this, dice are rolled and the results are interpreted using the rules presented in this book."
In the particular example of play I described, the problem based on the player's priority for his PC is
You need the blood of this mage as a sacrifice for your dark master; but he's just been decapitated, and his blood is flowing out of his body into the floor. The player declares how his PC is going to overcome this obstacle: "I look around for a vessel!" The dice are rolled, and the results interpreted - in this case, the test is a success and so the PC achieves what the player wanted for him, that is, he finds a suitable vessel.
Suppose that I, as GM, had narrated a vessel in the room. Then the player's action declaration probably would have been different - say, "I grab the jug to catch the blood". In that case the test would probably be on Agility rather than Perception. That wouldn't be a problem: the game is not intended to ensure any sort of strict correlation between
events in the fiction and
the rolling of tests at the table - as the rules say (p 72), "Unless there is something at stake in the story you have created, don’t bother with the dice. Keep moving, keep describing, keep roleplaying. But as soon as a character wants something that he doesn’t have, needs to know something he doesn’t know, covets something that someone else has, roll the dice."
Once the player has spotted the vessel, is there any need for further tests to successfully catch the blood? Or has the matter that was at stake been resolved, so that now we "don’t bother with the dice. Keep moving, keep describing, keep roleplaying"? There is no abstract answer to this question: it's a matter of participant judgement in the moment of play.
The rules have something to say about it, though (pp 30, 32, 44):
Characters who are successful complete actions in the manner described by the player. A successful roll is sacrosanct in Burning Wheel and neither GM nor other players can change the fact that the act was successful. . . .
One of the most important aspects of ability tests in game play in Burning Wheel is the Let It Ride rule: A player shall test once against an obstacle and shall not roll again until conditions legitimately and drastically change. Neither GM nor player can call for a retest unless those conditions change. Successes from the initial roll count for all applicable situations in play.
A GM cannot call for multiple rolls of the same ability to accomplish a player’s stated intent. Nor can a player retest a failed roll simply because he failed. Tests must be distilled down to as few rolls as possible. The successes of those rolls ride across the entire situation, scene or session. . . .
Soliciting Tests (Test Mongering)
Tests are very important to the game, but badgering the GM for them is very bad form. Can I test? Can I? Sometimes, a player will wish to have his character roll dice for something at an inappropriate juncture in play. It is the GM’s role to pace events and keep play flowing evenly. Therefore, he can have a player hold off on making a test until the appropriate time or have him stay his hand entirely.
Test mongering also involves pestering the GM for a particular test. “I need a routine. Can I make a routine test?” “Dude, it’s a dragon. I don’t think there are going to be any routine tests.” The GM’s job here is to say flat out: “No.” Let the difficulty of the tests arise organically, not at the player’s request so his character can advance. It makes for a much more interesting game.
Torchbearer 2e states similar rules (Scholar's Guide, pp 35, 216-7):
Fun Once
Don’t make the same tests for the same obstacle twice. Either the players have bypassed the original obstacle or it stands but they have to find another way - testing another ability or skill
against another obstacle. . . .
Bypass the minutiae; focus on what’s important. Highlight exciting actions. . . . In this game, skills are very broad and the condition rules are punishing. If you focus too closely on the fine-grained details, you’ll crush the players.
In both games, players
want to test, because this is how they advance their PCs. The GM is expected to manage pacing, and framing, in accordance with the core principles that I've set out.
In the case of Tru-Leigh spotting the vessel to catch the blood, that is that - his intent, to be able to save the blood from being lost on the floor of the sick-room, is achieved. He has no other Belief (say, about his deftness with vessels) that creates a reason to think that more is at stake in the scene than what has been resolved. So "say 'yes' or roll the dice" and "let it ride" apply.
then it's up to the GM to decide. Which means that the GM chooses to call for a roll. They just have say that something is at stake. They decided to not say "yes" to the question of "is there a cup"
If I were the GM in this instance, and I know that the most important thing is getting the blood back to the naga, and I also know that the PC is in a room that is more than likely to have a vessel in it (since it's a room for someone to recover in, meaning cups and bowls of water for drinking, washing wounds, and wetting dressings, plus containers that store meds, herbs, and bandages, plus a chamber pot, plus any number of other things), then what makes it high stakes to roll to see a cup? What makes it wrong or against BW principles for me to say "there's a cup on the bedside table"? What about that makes it low-stakes or low-tension or non-dramatic or whatever, when there's plenty of other uncertainty in the scene?
I feel I've already answered this question, multiple times. But here it is again.
The character has a Belief -
I will bring Joachim's blood to my master. I, as GM, have presented a situation in which that Belief is put under pressure - Joachim has been decapitated, and his blood is flowing away. The player of the character declares an action in response - "I look around to see if there's a vessel in which to catch the blood!" It is the fact that the action declaration is an attempt to overcome the challenge the situation poses to the Belief that means that
something is at stake and thus
a roll is called for. (As the rules say,
what ability is to be tested depends on the task being undertaken. Here, the task is
looking around. Hence, it is Perception that is tested.)
when you're saying that this is the One True Way, that the GM has to do it this way because otherwise it's wrong somehow--that's when I call shenanigans.
I have no idea what you mean by the One True Way. I'm talking about what is involved in playing Burning Wheel.
I determined the priorities for my PC when I played AD&D 2e, and again when I played D&D 3e, and yet again when I play D&D 5e. I also determined the priorities for my PC when I played GURPS, Fate, Call of Cthulhu, Star Wars d6, Mage: the Ascension, Werewolf: the Apocalypse, In Nomine, Masks, and every other system I've ever played. Yeah, in many of those games, particularly the ones I played in the 90s, there was a push to keep the group together, but I've almost never had GMs who wouldn't let me at least attempt accomplish my own goals.
<snip>
OK, so do us a favor and tell us what you mean when you say "the players determine priorities." And when I say "tell us" I mean tell us. Do not quote from your play notes or the book. Use your own words.
I've given many examples: Beliefs, Instincts, Traits (primarily character traits; also some die traits), Relationships, Affiliations, Reputations are the main ones in Burning Wheel.
I don't think BW is especially unique in having players create priorities; although it systematises it more than many RPGs do. What I have pointed out, which contrasts BW with many RPGs (including many approaches to D&D) is that
the GM's principal role is to "present[] the players with problems based on the players’ priorities."
So the GM chooses! And in this case, went against the PC's priorities because their Beliefs and Instincts indicated that Aedhros would have no problem with cold-blooded murder.
The scene presented the player - me - with a problem based on the priorities that I had authored for Aedhros. Namely he had the opportunity to fully
repay hurt with hurt in the context of
never admitting that I am wrong, by murdering the innkeeper. I declared an action for my character, namely, that Aedhros stabs the innkeeper with his black-steel blade Heartseeker. This is cold-blooded murder. And so, as the rules tell the GM to do (given the task, the intent, and the stakes), the GM called for a Steel test.
This is a particular operationalisation of the rule, quoted above, that "The players use their characters’ abilities to overcome these obstacles. To do this, dice are rolled and the results are interpreted using the rules presented in this book." My roll of the Steel test failed, and so Aedhros did not succeed at his attempt at cold-blooded murder.
This:
...makes it look like it was resolved by a Duel of Wits. That is, by some die rolls.
Yes. I've posted multiple times upthread, including in reply to you, that there was a Duel of Wits between Thurgon and Aramina. I scripted for Thurgon; the GM scripted for Aramina.
Did the player explain, in-character, why Alicia should mend the armor, and really roleplay the scene out before making some die rolls? Or did the player just say "I want Alicia to mend my armor first" and then make some die rolls?
If it's the latter, then the scene had no weight to it. It's just rolling dice.
I already quoted the relevant rule for DoW, in the post to which you're replying; and also posted about the scene, in reply to you, what I have reposted below:
I am not going to post the full rules for Duel of Wits. It used to be available as a free download, but that seems no longer to be the case. It broadly resembles other scene-oriented complex resolution systems (such as are found in HeroWars/Quest, Maelstrom Storytelling, 4e D&D skill challenges, Prince Valiant, etc). Actions are declared and resolved by those who are part of the argument, and this progresses towards victory for one or the other party. There is also a mechanism for generating compromise (if the loser has nevertheless made some progress towards victory).
Because the core resolution framework of BW is (as I have posted upthread already, and as is set out the free download) is
intent and task, the player has to say what their PC is doing - that is, identify the task and make clear the intent. In the case of Duel of Wits, this is called "speaking the part" (p 103 of Revised):
When scripting these maneuvers, players must speak their parts. Spitting out moves in a robotic fashion is not a viable use of these mechanics. The arguments must be made. Of course, no one expects us all to be eloquent, so just the main thrust or a simple retort usually suffices (but a little embellishment is nice).
Keep it simple and to the point. Say what you need to in order to roll the dice. A multipoint statement should be broken down into multiple actions across the exchange.
The back and forth of the argument is also key to establishing the content of compromises.
I'm finding that quite small things, of little consequence for the universe (actual or in-game) as a whole, can take on a high degree of importance for me as a player when they matter to my PC, and I know that my own choices are what is bringing them to the fore and shaping them (eg repairing the armour; laying the dead to rest; not fighting the mad skeleton knight of my order). I'm not going to say that it's Vermeer: the RPG, but the stakes don't have to be cosmologically high in order to be personally high - provided that they really are at stake.
I think this provides the answer to your question.