The first thing I recommend to people interested in anthropics and the Sleeping Beauty Problem is, If a tree falls on Sleeping Beauty by ata. The whole article is excellent, I recommend reading in full, but the basic argument is simple:

• There are different bets we can make regarding the coin. For example, we can pay out Sleeping Beauty each time they are woken up or only once.
• If we pay them out each time they are woken up and then convert this betting structure into an equivalent probability, we obtain the thirder solution, whilst if we only pay them out once, we obtain the halver solution.
• Therefore, the problem is underspecified. In order to be able to solve the problem, we need to explain which bet structure we are using.
• Asking "Which is the tree bet structure?" is the same kind of confusion as asking, "If a tree fell in a forest, would it make a sound?". Well, what exactly do you mean by "sound"? Answer that, and you'll have the answer to your question.

I found this extremely insightful, but at the same time, I can understand why some people might find it unsatisfactory.

After all, when we ask, "What is the probably of the coin landing heads?", we don't have a particular bet structure in mind. And, I don't just mean that we didn't have an explicit bet structure in mind, even when asked to clarify I don't think many people will immediately say, "Oh, I meant that bet structure!".

What we have in the back of our minds isn't some bet structure, but a whole lot of associations about what we mean and even though we can look for the bet structure that best matches these intuitions, the explanation given in "If a Tree Fell on Sleeping Beauty" won't be satisfactory unless we bring these into the story. In this case, I suspect that there's actually a single intution that does most of the work, so I'll just focus on that.

The "Objective" Perspective

I suspect that most halvers have a strong intution that we're trying to measure something objective about the coin. If we say, "Well, the probability of a coin landing on heads isn't something intrinsic to it, but depends on the environment we put it in", they'll still try to make it as objective as possible by specifying the environment as precisely as they can and insisting on the most reliable-seeming way of measuring it. 

In particular, to the extent that a particular observer affects the observation, something has gone wrong. Sleeping beauty gets woken up twice? "So what?", they'll say, 'That's a distortion, not something we're trying to remember? Obviously our experimental protocol should adjust to remove this distortion". From this perspective, it's as though we had a malicious experimentor who recorded tails twice every time it occurs. It's outside of the protocol, so we halve in order to make it as if they followed it correctly. Or we, can imagine someone measuring the width of a door. Asking, "Well, what is the width of the door if we accidentally record it in inches rather than centimeters?". It seems like a pointless question.

The "Subjective" Perspective

Some people have the intuition that we're trying to measure something subjective. The appeal of this perspective is that it helps you in whatever situation you are in. They might consider those who want more objectivity to be unpragmatic. Their only question is, "What should I expect?". Once you tell them that if you keep running the experiment, then they'll be told that the coin came up head approximately a third of the times, that's the matter is settled. If every time they're asked about the coin they act on the basis of the one third probability - including making bets - then the result will be what they expected.

"What does it mean to say that the probability is really half?", they may ask, "Pragmatically, I observe it a third of the time and if it doesn't make a difference to the observations then it isn't real or, even if is real, it's completely irrelevant! Maybe I have been woken up twice, but I can't know that, so I can't adjust for it! Remember that probability is relative to the observer. Given a normal coin in normal circumstances, I'd assign p=0.5 to the coin being heads, but once I see the coin, my probability becomes p=1. Nothing about the coin has changed, just my epistemic position".

The "Objective" Response

Since I'm trying to make both sides sound reasonable, I think I need to respond to provide an "objective" response to the "subjective" position. The "objective" position can be clarified as follows: that the claim is only that given a particular epistemic position regarding the coin toss, we can imagine an "objective" experiment corresponding to that which fits our experimental assumptions and not that our epistemic position is independent of our agent. That is, our knowledge of ourselves as an agent is only relevant in so far as it allows us to correct for the distortions we bring and is not part of what we are measuring.

Next we consider the notion that being doubled is irrelevant because we can't observe it. We can respond that even if we can't observe it, we can imagine an external observer outside the room who can. A good analogy is if we imagine the following experiment: Sleeping Beauty is asked to measure the height of an object, then put to sleep. When they are woken up, they have no memory of the first room apart from the figure they produced. They are given another object which is twice as tall and another ruler, but with the ruler having the marks twice as far apart. They are then asked whether the two objects were the same height.

Someone supporting the "subjective" view might say that for them the two objects are the same height for the subject, as the subject has no way of knowing that the height is different. As far, as I am concerned, this is a pure word game and the fact that an external observer (say someone like us who has the situation described to them) can see what happened means that the heights are different, even if we were to say Sleeping Beauty lacks the language to say what happened (which I don't think is true).

My Position

As I've already indicated, my position that that a certain strong version of the "subjective" perspective - that presents the "objective" perspective as meaningless - is implausible. However, I'm not trying to dismiss the softer version which merely uses claim that it is more convenient to use "probability" to refer to the more subjective version. I happen to think both sides have a point and a reasonable case for why it makes more sense to define probability their way. And, I'm in agreement with Ata that the choice we ultimately make is merely a matter of convention. So, this ends up being about how draw the map and not what the territory is like.