What was your Inquisitor's reason when deciding who should stay behind?

Well then it likely wasn't a thread I posted in, because I do not.

I can only use the Stroud scenario as my example. I have a play with Warden Loghain going, which is my new canon, but haven't gotten there yet to see what he says about it.

In the moment, with what is required, Hawke and the Warden are on equal footing as far as ability; they only need to supply a distraction to allow the other two to escape. Yet a choice has to be made based on something. My Inquisitor sees a benefit to allowing the Wardens a chance at redemption by having Stroud sacrifice himself. That is the primary source of his reasoning.

I have some other meta-level reasoning that I use to supplement the decision, but you asked for the Inquisitor RP choice.

Even if my motivation for a choice is personal as the player, I will always try to come up with a suitable RP reason to justify the decision. If I can't, then I won't go through with it.

Same here. I didn't see why Hawke should feel Corypheus was their responsibility. Actually, considering what the Nightmare demon had done to the Wardens by working with Corypheus to amplify his false calling, it actually seemed rather appropriate that Stroud should be the one to foil its plan to stop us returning from the Fade. It was redemption for the Wardens and retribution on the demon/Corypheus combined.

While Hawke did release Corypheus, s/he only did it for the purpose of killing him.

Was that a stupid idea? Not at all. Corypheus was manipulating the Wardens despite being sealed, so he had to be stopped somehow.

For my canon worldstate, it came down basically to headcanons I have for my personal Hawke and Inquisitor.  My Inquisitor is widowed and has a young daughter prior to the Conclave (although she fails to mention this to anyone until after they get to Skyhold).  Hawke is married to Fenris and they have a baby by the time of Inquisition (Fenris stayed back to protect their child).

When the decision between Hawke and Stroud needs to be made, it needs to be a snap decision: there's no time to weigh the pros and cons of each, and either way someone's going to die.  So she make an emotional one.  Even if Hawke has tried to keep her baby's existence a secret, Lavellan has figured it out - she sees the same look in Hawke's eyes as she watches Skyhold's children that she's sure was in her own eyes as she watched Haven's children.  She doesn't want to leave the child motherless.  She knows the pain she felt when her husband died and didn't want to put Fenris through that same pain.  Stroud stayed.

Mm... yeah, I hadn't considered that. It IS a snap decision, so the Inquisitor doesn't have time to make a rational and throughly thought choice.

Heh, that's true. I always take Solas because Fade shenanigans and all that.

Varric has always known that Hawke had a heroic death in store at some point, though. He's surly for a couple of weeks afterwards, but I think he forgives us.

For my part, my Inquisitor sacrificed Hawke because she thought Loghain would be more effective at keeping the Grey Wardens in line in the aftermath, and was otherwise just better connected as the father of Ferelden's queen and longtime national hero. And Cassandra kept framing Hawke as a potential rival for the Inquisitor position, which annoyed her. Loghain thus had an inherently higher value to the Inquisition in her mind, and her instinct was to preserve him and let a potential challenge to her authority die heroically instead.

She didn't regret the decision, but she may have felt a bit guilty about her reasons after talking with Varric and thinking it over.

I'm still trying to decide which of my Inquisitors will be the canon, but one of my favorites chose to sacrifice Stroud. He knew that Hawke had family and friends waiting, such as Varric. He was also hoppin' mad at the Wardens at that particular moment.

My Inquisitor didn't have any reasoning. He was actually gonna say "Stroud, Hawke, it's kinda stupid to sit here arguing when everyone else is over by the door already anyway, and heck, we already kicked this guy's ass once." But apparently Stroud heard his name and decided to go all "FOR THE WARDENS!" before Quizzie could finish his sentence.

That's really the only logical explanation for why something so dumb happens in that scene.

If it's revealed in DA4 that's it's somehow possible for whoever gets left behind to survive, with the right choices made in the other games, (not that I actually think there's a chance of that happening), then I would change it to my Hawke staying (because I feel it would fit her character).  In that case, the Inquisitor saying "Hawke" wouldn't be so much a "Hawke, thanks for taking the sacrifice for us. Bye!"  as it would be a "Hawke! What the f**k do you think you're doing. Get back here!". 

Nearly all my playthroughs I have left Stroud. He has been a Warden for a long time and he realizes his time is nearing.

It would seem Stroud's odds of survival across multiple playthroughs and multiple players approaches close to 0%

I've come to the conclusion that the Inquisitor doesn't actually decide anything. The fact that there is no approval changes for the companions shows that it isn't a choice at all.