Do All Tournaments Admit Irrelevant Matches?

We consider tournaments played by a set of players in order to establish a ranking among them. We introduce the notion of irrelevant match, as a match that does not influence the ultimate ranking of the involved parties. After discussing the basic properties of this notion, we seek out tournaments that have no irrelevant matches, focusing on the class of tournaments where each player challenges each other exactly once. We prove that tournaments with a static schedule and at least 5 players always include irrelevant matches. Conversely, dynamic schedules for an arbitrary number of players can be devised that avoid irrelevant matches, at least for one of the players involved in any given match. Finally, we prove by computational means that there exist tournaments where all matches are relevant to both players, at least up to 8 players.