I have been thinking about anthropics for quite some time. And in the process also noticed its connection with embedded agency and its connection with freewill like the Newcomb problem.
My conclusion is perspectives are axiomatic. e.g. You inherently know that "I am this person (Antoine de Scorraille)", just like I know "I am this person (dadadarren)". It is purely based on the fact that the only subjective experience available is due to this physical body. "Which person/thing is the self?" has no underlying logical reason or physical explanation. It is a starting point of logical reasoning.
This means none of the anthropic assumptions, such as SSA or SIA- which regard the self as a certain sort of sampling outcome, is correct. Furthermore, the existence of the self as a particular physical agent cannot be used as evidence to support specific theories over others. And the Sleeping Beauty problem's answer should be double-halving.
Recognizing perspective as axiomatic also means we cannot regard physics as a faithful description of the reality of the entire world from an all-knowing "view from nowhere". Instead, any physical description ought to originate from a given perspective, based on the interactions between itself and the environment. Therefore, the perspective center - or the self, is not physically analyzable. If we want to study something, we have to reason from the perspectives of some other things that interact with it. (There is an obvious disposition to certain quantum interpretations here. I will skip it since it is not directly related to the topic at hand.)
So, in the case of Emmy, there are two approaches we can make the decision. One way is to reason from the first-person perspective of Emmy, and think as the decision-maker herself. From this viewpoint, I inherently know who I am (this person). I can make whatever decision I deem the best, meaning I have free will. I will be an A6-er or A2-er or whatever based on my choice.
Alternatively, we can take the perspective of someone/something else and physically analyze Emmy. From such viewpoints, no different than studying the output of a machine, we can physically deduce what kind of action Emmy would take. She has no free will nor any "choice" so to speak. Whether she is the A6-er or A2-er is a matter of physics.
Both of these approaches are fine on their own. BUT, and that is a big but, they cannot be mixed and matched together. Because they are formulated from different perspectives, or, based on different axioms. The Newcomb paradox is a prime example of such a mix, as well as the anthropic paradoxes. There is no sense asking "Among all the A-ers, which physical person am I?" Because that question already mixed the perspectives and cannot be answered by either approach. From Emmy's first-person perspective, "Which person I am" is a primitive fact that cannot be reasoned. From an outsider's perspective, though the A-ers can be analyzed, the meaning of "I" in the question is lost. Since the self is no longer Emmy.