  A number of problems in physics, mathematics, and philosophy involve
observers in given situations which lead to debates about whether
observer-specific information should affect the probability for some outcome or
hypothesis. Our purpose is not to advocate for such observer selection effects,
but rather to show that any such effects depend greatly on the assumptions
made. We focus on the debate about the existence of a `Doomsday
effect'---whether observer index information should cause one to favor
possibilities with fewer observers, which has been argued to have implications
for models of cosmology. Our central goal is to reconcile the apparent
inconsistencies in the literature by introducing a formalism to lay bare
assumptions made and address a key issue that has not been clearly articulated
in such problems: whether the observer is selected by *picking from* or *being
in* a set of worlds. In the former there generally are observer selection
effects, and in the latter there generally are not. This leads us to
differentiate what we call *inclusive* from *exclusive* selection, and how they
relate to the concept of a multiverse. Then we relax the assumption that all
observers are equally typical, and consider the problem of Boltzmann brains,
showing that typicality can play a role in solving the problem. We then stress
the need for scale-invariant questions, which causes us to analyze J. Richard
Gott's approach to the problem. This all allows us to analyze the Doomsday and
Universal Doomsday arguments. We find that there is no Doomsday effect, absent
a set of assumptions we find somewhat unreasonable. Then we use our formalism
to resolve a debate in the philosophy community called the `Sleeping Beauty
Problem.' Finally, we conclude with a heuristic summary, free from equations,
and point to possible future directions of this line of research.
