Abstract We study the tournament fixing problem (TFP), which asks whether a tournament organizer can rig a single-elimination (SE) tournament such that their favorite player wins, simply by adjusting the initial seeding. Prior results give two perspectives of TFP: on the one hand, deciding whether an arbitrary player can win any SE tournament is known to be NP-complete; on the other hand, there are a number of known conditions, under which a player is guaranteed to win some SE tournament. We extend and connect both these lines of work. We show that for a number of structured variants of the problem, where our player is seemingly strong, deciding whether the player can win any tournament is still NP-complete. Dual to this hardness result, we characterize a new set of sufficient conditions for a player to win a tournament. Further, we give an improved exact algorithm for deciding whether a player can win a tournament.