Some ideas are:
- You branch off into another timeline and your actions make no difference to the previous timeline
- You’ve already taken said actions but just didn’t know about it so nothing changes
- Actions taken can have an effect (so you could suddenly erase yourself if you killed your parents)
- Only “nexus” or fixed events really matter, the timeline will sort itself out for minor changes
- something else entirely
Are you asking which system I think is the most plausible, or which is the most desirable?
Plausibility:
Well, I’d guess that time travel probably isn’t possible, and if it is, it’s probably under extremely limited conditions that render it impractical for viable exploitation. But if you’re operating under the assumption that it is, I’d say the “your actions do not affect this timeline” or similar type.
Why?
We have had no record of time travel or seen phenomena likely resulting from it. If at time T, time travel is discovered, it seems unlikely that someone after that time wouldn’t have come back in time and done something that we’d have noticed.
And it’s not just us. If self-timeline-affecting time travel is possible, then you consider all the possible civilizations out there in the universe who might discover it at some point in time and want to take advantage of it. Yet we’ve seen nothing from them. It’s the Fermi Paradox on steroids. The Fermi Paradox asks why intelligent aliens, half of whom statistically probably evolved before us and should have colonized the universe if they’re out there, aren’t visible to us. The time to travel over even huge distances, though it is large, is small compared to the time required to evolve a spacefaring civilization. But in the presence of self-timeline-affecting time travel, then even the evolutionary time becomes a non-factor, since civilizations from the future could also show up, and roll back in time with their advanced technology and make use of it from then. The question is no longer just “where is everyone”, but the even harder to explain “where is everyone from all time?”
Okay, that’s the plausibility question. How about the desirability one, which system I’d like to exist?
Hmm. I guess I’d give the same answer, the “no affecting your own timeline” form. I think that if you could affect your own timeline, that probably some kind of incident in the future – only takes one – would be likely to have mucked up things sufficiently to wipe out civilization, and we probably wouldn’t be around to even be pondering the matter.