It is a fact about the balls that one ball is physically continuous with the ball previously labeled as mine, while the other is not. It is a fact about our views on the balls that we therefore label that ball, which is physically continuous, as mine and the other not.

And then suppose that one of these two balls is randomly selected and placed in a bag, with another identical ball. Now, to the best of your knowledge there is 50% probability that your ball is in the bag. And if a random ball is selected from the bag, there is 25% chance that it's yours.

So as a result of such manipulations there are three identical balls and one has 50% chance to be yours, while the other two have 25% chance to be yours. Is it a paradox? Oh course not. So why does it suddenly become a paradox when we are talking about copies of humans?

It is objectively the case here that 25% of the time this procedure would select the ball that is physically continuous with the ball originally labeled as "mine", and that we therefore label as "mine".

Ownership as discussed above has a relevant correlate in reality - physical continuity in this case. But a statement like "I will experience being copy B (as opposed to copy A or C)" does not. That statement corresponds to the exact same reality as the corresponding statements about experiencing being copy A or C. Unlike in the balls case, here the only difference between those statements is where we put the label of what is "me". 

In the identity thought experiment, it is still objectively the case that copies B and C are formed by splitting an intermediate copy, which was formed along with copy A by splitting the original.

You can choose to disvalue copies B and C based on that fact or not. This choice is a matter of values, and is inherently arbitrary.

By choosing not to disvalue copies B and C, I am not making an additional assumption - at least not one that you are already making by valuing B and C the same as each other. I am simply not counting the technical details of the splitting order as relevant to my values.

The issue, to me,  is not whether they are distinguishable.

The issues are:

• is there any relevant-to-my-values difference that would cause me to weight them differently? (answer: no)

and:

• does this statement make any sense as pointing to an actual fact about the world: "'I' will experience being copy A (as opposed to B or C)" (answer: no)

Imagine the statement: in world 1, "I" will wake up as copy A. in world 2 "I" will wake up as copy B. How are world 1 and world 2 actually different?

Answer: they aren't different. It's just that in world 1, I drew a box around the future copy A and said that this is what will count as "me", and in world 2, I drew a box around copy B and said that this is what will count as "me". This is a distinction that exists only in the map, not in the territory.

This actually sounds about right. What's paradoxical here?

Not that it's necessarily inconsistent, but in my view it does seem to be pointing out an important problem with the assumptions (hence indeed a paradox if you accept those false assumptions):

(ignore this part, it is just a rehash of the path dependence paradigm. It is here to show that I am not complaining about the math, but about its relation to reality):

Imagine you are going to be split (once). It is factually the case that there are going to be two people with memories, etc. consistent with having been you. Without any important differences to distinguish them, and if you insist on coming up with some probability number for "waking up" as one particular one of them obviously it has to be ½.

And then, if one of those copies subsequently splits, if you insist on assigning a probability number for those further copies, then from the perspective of that parent copy, the further copies also have to be ½ each.

And then if you take these probability numbers seriously and insist on them all being consistent then obviously from the perspective of the original the probability numbers for the final numbers have to be ½ and ¼ and ¼. As you say "this actually sounds about right".

What's paradoxical here is that in the scenario provided we have the following facts:

1. you have 3 identical copies all formed from the original
2. all 3 copies have an equal footing going forward

and yet, the path-based identity paradigm is trying to assign different weights to these copies, based on some technical details of what happened to create them. The intuition that this is absurd is pointing at the fact that these technical details aren't what most people probably would care about, except if they insist on treating these probability numbers as real things and trying to make them follow consistent rules. 

Ultimately "these three copies will each experience being a continuation of me" is an actual fact about the world, but statements like "'I' will experience being copy A (as opposed to B or C)" are not pointing to an actual fact about the world. Thus assigning a probability number to such a statement is a mental convenience that should not be taken seriously. The moment such numbers stop being convenient, like assigning different weights to copies you are actually indifferent between, they should be discarded. (and optionally you could make up new numbers that match what you actually care about instrumentally. Or just not think of it in those terms).

Presumably the 'Orcs on our side' refers to the Soviet Union.

I think that, if that's what he meant, he would not have referred to his son as "amongst the Urukhai." - he wouldn't have been among soviet troops. I think it is referring back to turning men and elves into orcs - the orcs are people who have a mindset he doesn't like, presumably to do with violence.

I now care about my observations!

My observations are as follows:

At the current moment "I" am the cognitive algorithm implemented by my physical body that is typing this response.

Ten minutes from now "I" will be the cognitive algorithm of a green tentacled alien from beyond the cosmological horizon. 

You will find that there is nothing contradictory about this definition of what "I" am. What "I" observe 10 minutes from now will be fully compatible with this definition. Indeed, 10 minutes from now, "I" will be the green tentacled alien. I will have no memories of being in my current body , of course, but that's to be expected. The cognitive algorithm implemented by my current body at that time will remember being "me", but that doesn't count, that's someone else's observations.

Edit: to be clear, the point made above (by the guy who is now a green tentacled alien beyond the cosmological horizon, and whose former body and cognitive algorithm is continuous with mine) is not a complaint about the precise details of your definition of what "you" are. What he was trying to point at is whether personal identity is a real thing that exists in the world at all, and how absurd your apparent definition of "you" looks to someone - like me - who doesn't think that personal identity is a real thing.

"Your observations"????

By "your observations", do you mean the observations obtained by the chain of cognitive algorithms, altering over time and switching between different bodies, that the process in 4 is dealing with? Because that does not seem to me to be a particularly privileged or "rational" set of observations to care about.

 Here are some things one might care about:

1. what happens to your physical body
2. the access to working physical bodies of cognitive algorithms, across all possible universes,  that are within some reference class containing the cognitive algorithm implemented by your physical body
3. ... etc, etc...
4. what happens to the physical body selected by the following process:
   1. start with your physical body
   2. go forward to some later time selected by the cognitive algorithm implemented by your physical body, allowing (or causing) the knowledge possessed by the cognitive algorithm implemented by your physical body to change in the interim
   3. at that later time, randomly sample from all the physical bodies, among all universes, that implement cognitive algorithms having the same knowledge as the cognitive algorithm implemented by your physical body at that later time
   4. (optionally) return to step b but with the physical body whose changes of cognitive algorithm are tracked and whose decisions are used being the the new physical body selected from step c
   5. stop whenever the cognitive algorithm implemented by the physical body selected in some step decides to stop.

For 1, 2, and I expect for the vast majority of possibilities for 3, your procedure will not work. It will work for 4, which is apparently what you care about.

Terminal values are arbitrary, so that's entirely valid. However, 4 is not something that seems, to me, like a particularly privileged or "rational" thing to care about.