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Happiness in Slavery or maybe More than Mind Control/Not Brainwashed — it depends on the person's reason for staying.
If it's a trope about the dramatic "don't save me" moment, it's Unwanted Rescue.
Edited by Chariset
I guess the first ones would be why they refuse the offer, and Unwanted Rescue is closer, but I don't think I explained well enough.
Like, say, Bob the Big Bad kidnapped Alice at some point or another, but later on he has a change of heart and says Alice is free to go, but Alice says, "No thanks, I'm fine here with you."
What's the trope for someone in servitude or they were kidnapped or something in the like and are offered freedom, but they refuse the offer?