The constant-trace-value condition in the definition of a symmetric informationally complete positive operator-valued measure (SIC-POVM) is often overlooked, since it may give the impression that this condition is redundant. We show that this condition is necessary, otherwise the resulting mathematical object is no longer a SIC-POVM. This observation has led us to define a broader class of measurements which we call the semi-SIC POVMs. In dimension two we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open. Majorization relations have played an important role in quantum Shannon theory. We construct families of games of chance whose pay-off gives rise to three different types of majorization relations: the standard vector majorization, conditional majorization, and channel majorization. We show that our definition of conditional majorization and channel entropy is consistent with previous literature and find a channel entropy that reduces Shannon entropy and asymptotically continuous.