If you did the limit by program length, it might converge, but there's no obvious reason that there must be a certain order to count them in. If you can count them in any order, you can alternate between states where the arrow hits and states where it does not, and make it look like it has a 50% chance of hitting.

Well for any finite amount of time that you predict into the future you've also got a finite amount of space to consider, as anything too far away wouldn't be able to travel fast enough to effect the outcome of the thing being predicted about. Each state of the universe would really be the state of this finite area of space which would be expressed in binary. One way to order the states would be in terms of how large a binary number they form, from smallest to largest.

I'm not sure how making it look like the arrow had a 50% chance of hitting would make any difference to anything though?

Given a state, you could add another non-interacting (or just interacting based on one guy's decision) universe where 3^^^^3 people die. This universe is constant complexity, and small complexity compared to 3^^^^3. Let's call the complexity of that part k.

Given a state, you could also add another non-interacting (or just interacting based on one guy's decision) universe where 3^^^^3 people lives are saved. I don't know if this is the right terminology, but it seems to me that when you start adding extra possible universes on, their outcomes become causally decoupled from the original decision to give/not-give the mugger $5.