You create a paradox, it just keeps looping (you go back in time and kill your grandfather -> you don't exist -> you don't build the time machine and go back -> your grandfather lives -> you exist -> you build the time machine etc). But every time around the loop there is a certain amount of actual randomness.
Perhaps that atom decays this time around, or that transistor erroneously conducts due to shot noise.
So all we see is the final "fixed point". The iteration where everything ends up going just right to avoid a paradox - perhaps the gun fired prematurely, or the time machine didn't work, or a passing airplane dropped an engine on you, or...
From the perspective of any time traveler, it ends up being as though the universe is conspiring against you if you try to do anything that would cause a paradox.
This sounds like the Novikov self-consistency principle.
Say you have a wormhole on a billiard table. It curves around and goes three seconds backward in time. You roll a ball into the wormhole, aimed such that after it exits the wormhole, it will knock its earlier self off the path so that it never enters the wormhole. Paradox.
Except when you try it, instead of emerging along the pathway you aimed, it emerges along a slightly different path, and strikes its earlier self only a glancing blow. And why did it emerge along a different path? Because it was struck a glancing blow.
David Deutsch also proposed a similar resolution for closed timelike curves, which say that if you have a probability distribution of the present, all that's necessary is that when you travel back in time, you get the same probability distribution. In short, if you flip a (true!) random coin to before shooting your grandfather, no paradox exists:
The problem with this is the universe is no longer causal. Events can happen which cause themselves. It requires you to try every possible universe and then "throw away" the ones that aren't consistent. And if you simulate a universe, doesn't it cause that universe to exist, even if you later decide you don't like it? Who is to say we don't exist in one of the many "throw away" universes vs the one where the particle causes itself to be consistent.
If your theory was the correct one, could we use that effect to perform computation - perhaps condensing 1000s of years of computation into a few minutes, with the result encoded in which steady state it settles on?
But you'd need absurd reliability. Say you have something that condenses 1000 years of computation into a minute. That's ~500 million / 1 time compression.
What are the chances that if you ran 500 million copies of the machine with no time compression for a minute none of them would malfunction? Relatively small, right? Well, the single machine with time compression would end up doing effectively the same thing - the chance that it gets the right answer is probably much smaller than the chance that it goes up in a puff of smoke because the power supply freakishly died or something.
In order to do practical computation with such a system you'd need absurdly reliable computers, in absurdly safe environments. If you have a time compression ratio of 1 billion to 1, well, how many 1-in-a-billion things are there to go wrong? Better hope you're not near a faultline...
Well, to be fair, any time machine is probably fairly complex. The chances of the time machine being the thing that breaks the loop by breaking/etc is probably non-negligible.
You create a paradox, it just keeps looping (you go back in time and kill your grandfather -> you don't exist -> you don't build the time machine and go back -> your grandfather lives -> you exist -> you build the time machine etc). But every time around the loop there is a certain amount of actual randomness.
Perhaps that atom decays this time around, or that transistor erroneously conducts due to shot noise.
So all we see is the final "fixed point". The iteration where everything ends up going just right to avoid a paradox - perhaps the gun fired prematurely, or the time machine didn't work, or a passing airplane dropped an engine on you, or...
From the perspective of any time traveler, it ends up being as though the universe is conspiring against you if you try to do anything that would cause a paradox.