  We consider the manipulability of tournament rules which map the results of
$\binom{n}{2}$ pairwise matches and select a winner. Prior work designs simple
tournament rules such that no pair of teams can manipulate the outcome of their
match to improve their probability of winning by more than $1/3$, and this is
the best possible among any Condorcet-consistent tournament rule (which selects
an undefeated team whenever one exists) [Schneider et al., 2017, Schvartzman et
al., 2020]. These lower bounds require the manipulators to know precisely the
outcome of all future matches.
  We take a beyond worst-case view and instead consider tournaments which are
"close to uniform": the outcome of all matches are independent, and no team is
believed to win any match with probability exceeding $1/2+\varepsilon$. We show
that Randomized Single Elimination Bracket [Schneider et al., 2017] and a new
tournament rule we term Randomized Death Match have the property that no pair
of teams can manipulate the outcome of their match to improve their probability
of winning by more than $\varepsilon/3 + 2\varepsilon^2/3$, for all
$\varepsilon$, and this is the best possible among any Condorcet-consistent
tournament rule.
  Our main technical contribution is a recursive framework to analyze the
manipulability of certain forms of tournament rules. In addition to our main
results, this view helps streamline previous analysis of Randomized Single
Elimination Bracket, and may be of independent interest.
