What One Should Do VS What One Would Do
When talking about how a decision is made, there are two approaches. Perhaps the more typical one is to reason as the decision-maker. From this first-person perspective, we consider all possible actions, imagine their respective outcomes, then choose the one deemed optimal according to some objective. This seems to fit what usually meant by decision-making: what one should do.
Alternatively, we can take an outsider perspective (or a God's eye view if you prefer) and analyze the decision-maker itself by reductionism. For example, this can be done by physically studying the decision-maker's brain and build a model to deduce its outcome. In contrast to the first type of analysis, this is about what one would do
My position is that perspective is a primitive reasoning starting point. Therefore the two approaches above, each basing on a different perspective, must not mix. As that is also the cause for anthropic paradoxes. However, even if you do not agree with that, it is still very reasonable to question the compatibility of the two approaches.
For starters, they are based on very different premises. The first-person approach considers the decision-maker as the agent, the outsider approach considers the decision-maker part of the environment, one regards it as the analyzer while the other regards it as the analyzed.
The first-person approach assumes the decision-maker could select different choices. There are alternative actions I can take that will lead to different outcomes. In comparison, the outsider approach regards the decision-maker as nothing more but a complex machine. There is no sense in talking about alternatives as the decision is simply an output of the said machine.
Mixing the two would lead to some troubling problems. For example, the first-person approach makes decisions by evaluating the respective outcomes of all choices. We can use the outsider approach to reductively analyze the decision-maker to deduce its output. So the choices other than the one being ultimately taken are simply never happening. Using this result in the first-person analysis would produce a contradiction: what is the outcome of an action if that action is not taken? It bumps into the principle of explosion, making the evaluation impossible. (See action-counterfactuals for a related discussion.)
However, we never seem to encounter the above problem in real life. That is due to two reasons. First, once we carry out the outsider analysis and deduced the output, we typically won't conduct the first-person analysis. Second, even if we wish to also conduct the first-person analysis, we will ignore the outsider approach's conclusions. I.E. we simply evaluate the outcome of all choices without minding which one is the deduced output, as if the outsider analysis never happened. This means in practice, at least in obviously conflicting situations like this, we intuitively know not to mix the two approaches.
Newcomb's paradox is problematic because it does mix the two. Its formulation takes the outsider's approach: Omega would analyze the decision-maker like a machine to deduce its outcome. Yet it wants us to take the first-person approach to answer it: what should you do when facing the two boxes. In short, the question is inconsistent. What should have been asked is "how to design a machine (the decision-maker) for this situation", which would also be answered using the outsider approach.
Designing such a machine is uncontroversial: make it take one box only. So Omega would study it and put a million dollars in for the machine to grab. The contradiction only happens if we take the first-person approach: imagine ourselves in the shoes of the decision-maker. When the two boxes with predetermined content are right in front of me, obviously I should take them both. Doing anything different would need to make some serious and dubious modifications to the common notion of causality. Yet the very act of taking the first-person perspective is inconsistent with the question's setup. For there is no sense in contemplating what I should do when what I would do is part of the problem.