  We consider two-person sports where each rally is initiated by a
\emph{server}, the other player (the \emph{receiver}) becoming the server when
he/she wins a rally. Historically, these sports used a scoring based on the
\emph{side-out scoring system}, in which points are only scored by the server.
Recently, however, some federations have switched to the \emph{rally-point
scoring system} in which a point is scored on every rally. As various authors
before us, we study how much this change affects the game. Our approach is
based on a \emph{rally-level analysis} of the process through which, besides
the well-known probability distribution of the scores, we also obtain the
distribution of the number of rallies. This yields a comprehensive knowledge of
the process at hand, and allows for an in-depth comparison of both scoring
systems. In particular, our results {help} to explain why the transition from
one scoring system to the other has more important implications than those
predicted from game-winning probabilities alone. Some of our findings are quite
surprising, and unattainable through Monte Carlo experiments. Our results are
of high practical relevance to international federations and local tournament
organizers alike, and also open the way to efficient estimation of the
rally-winning probabilities, which should have a significant impact on the
quality of ranking procedures.
