  A single-elimination (SE) tournament is a popular way to select a winner in
both sports competitions and in elections. A natural and well-studied question
is the tournament fixing problem (TFP): given the set of all pairwise match
outcomes, can a tournament organizer rig an SE tournament by adjusting the
initial seeding so that their favorite player wins? We prove new sufficient
conditions on the pairwise match outcome information and the favorite player,
under which there is guaranteed to be a seeding where the player wins the
tournament. Our results greatly generalize previous results. We also
investigate the relationship between the set of players that can win an SE
tournament under some seeding (so called SE winners) and other traditional
tournament solutions. In addition, we generalize and strengthen prior work on
probabilistic models for generating tournaments. For instance, we show that
\emph{every} player in an $n$ player tournament generated by the Condorcet
Random Model will be an SE winner even when the noise is as small as possible,
$p=\Theta(\ln n/n)$; prior work only had such results for $p\geq
\Omega(\sqrt{\ln n/n})$. We also establish new results for significantly more
general generative models.
