Time travel loops are... weird. For example, generally, it creates ex-nihilo events with no apparent causation.
Well, not quite. Not if you take the right perspective. Ahem.
Ways to Think About Stable Time Loops: A Rant
Consider playing a chess game against someone. Suppose you could move a figure to capture a pawn. However, you know that if you do so, it would let your adversary execute a sequence of moves that leaves you at a bigger disadvantage than the benefits of capturing that pawn. So you let this apparent opportunity to hurt your enemy pass, for
no immediate/present-time reason to speak of! The actual board-state is such that there's an immediate move available to you that leaves your opponent with fewer figures; if you were a greedy optimizer trying to pick actions that minimize the enemy's forces at every point, you would make that move. But instead, you take actions in the present that are dependent on some potential
future board-state. You time-travel! (Indeed, from a certain perspective,
all agents are basically time-travel devices.)
So you reject the idea of taking that figure, and instead generate some other potential move you could make. You simulate its future consequences, and perhaps reject it as a bad one as well.
You do that a few more times, and finally hit upon a candidate move that's predicted to lead to good consequences. So, at last, you execute it in reality.
Now consider:
what caused this move? It was generated as the output of that process of iterative generation and evaluation of candidate moves. And the move that you actually ended up
taking basically represents some "stable solution": a move such that, if its consequences are extrapolated into the future, those consequences
lead to that move being taken in the past. I. e.,
a stable time loop.
Obviously real-life agents aren't
actually oracles. They only do their best to approximate this process within the bounds of their computational resources. If we model the whole thing mechanistically, it's all just "normal" causality, atoms bouncing around. But these atoms are doing their best to violate said causality.
"True" stable time loops can be modeled/framed/simulated as the limiting case of this approximation as the available computational resources go to infinity. They're not downstream of "ordinary" causality within the universe, but they're downstream of a process of iterative generation and rejection of potential consistent timelines, which produces the output the moment it finds a stable solution.
And in some situations, we can jump straight ahead to that stable solution: solve for the equilibrium, rather than doing blind trial-and-error.
The counter-argument here would be to note that there are still free parameters. For one, there isn't necessarily a
unique stable solution. Much like there isn't necessarily only a single chess move that advances your position, there may be several consistent timelines featuring time-travel: e. g., one where Hazou sends back a letter in a green envelope and one where the envelope is red.
There's two ways to address that:
First, we can fix the
sequence in which the iterative-generation process evaluates the candidate solutions. In that case, the first stable solution in the sequence wins. By analogy, when you're playing chess, you're not evaluating moves
at random – you have some prior for what a good move looks like, which you've learned from the experience of playing chess. When you're generating candidate moves, you're sampling from this prior.
Similarly, you can define some "universal prior" for which stable solutions are most likely/in what sequence they're generated. And this prior doesn't need to be
random – you can postulate a prior that makes sense within the context of the setting. (For example, perhaps the fictional world is being dreamed up by a timeless Lovecraftian monstrosity whose cognition is unbounded (letting it simulate
true time-travel), and which likes e. g. dramatic irony, so it picks dramatically ironic solutions.)
Second, we can define some optimization objective/utility function by which all candidate solutions are rated, then pick the stable solution that maximizes it. You'll note that it's actually just a variant on the first idea above, but with the sequence of candidate solutions ordered by these ratings. It's a pretty good way to make the "prior" nonrandom/well-compressible: just pick some simple function to compute the ratings!
And it makes sense within the computationally bounded context of chess as well. If you're playing to e. g. win in the shortest number of moves possible, there is certainly some
objectively correct set of move-sequences that achieves this. (E. g., all sequences that win in four moves against the specific opponent you're playing.) By sampling from your prior of "good chess moves", you're just trying to approximate generating this best solution.
So you can pick a
unique stable solution to time-travel by defining some highly sensitive simple optimization objective, then find the stable solution that maximizes it.
The most "natural" choice would be some manner of simplicity prior. Intuitively, the most simple stable solution is "no time travel", which, well, is why there is no time-travel around us. But if you perturb the initial conditions somehow in your fictional context (e. g., by an act of eldritch entity
forcibly introducing time-travel into the universe), the next-best solution may be "minimal time-travel subject to the perturbation". Hence, you'll most likely end up observing universes in which time-travel is difficult, costly, often lost, kept hidden, kept to a limited number of users, used infrequently, et cetera.
And if you're an agent embedded in a universe like this, and you have a time-travel device in your hand?
Well, you don't want the universe to outcome-pump you into dying, do you? So you'll
cooperate with its optimization objective; you'll precommit to engage in time-travel as little as possible and to avoid proliferating it as much as possible. That's how you end up in a universe in which you're
reluctantly allowed to engage in some time-travel, after all. If you
responsibly self-regulate, then the universe won't feel the need to regulate you manually by hastily dropping an asteroid on your head the moment you're in the vicinity of a time-travel device. Or by steering you away from said devices to begin with.
Or, if you're not
that sensible but the universe doesn't have anything on-hand to easily kill you with, maybe your first attempt at exploiting time-travel results in your receiving a letter with "DO NOT MESS WITH TIME" written on it in your handwriting.