It is a long-standing objective to ease the computation burden incurred by the decision-making problem under partial observability. Identifying the sensitivity to simplification of various components of the original problem has tremendous ramifications. Yet, algorithms for decision-making under uncertainty usually lean on approximations or heuristics without quantifying their effect. Therefore, challenging scenarios could severely impair the performance of such methods. In this paper, we extend the decision-making mechanism to the whole by removing standard approximations and considering all previously suppressed stochastic sources of variability. On top of this extension, we scrutinize the distribution of the return. We begin from a return given a single candidate policy and continue to the pair of returns given a corresponding pair of candidate policies. Furthermore, we present novel stochastic bounds on the return and novel tools, Probabilistic Loss (PLoss) and its online accessible counterpart (PbLoss), to characterize the effect of a simplification.