  We consider the manipulability of tournament rules for round-robin
tournaments of $n$ competitors. Specifically, $n$ competitors are competing for
a prize, and a tournament rule $r$ maps the result of all $\binom{n}{2}$
pairwise matches (called a tournament, $T$) to a distribution over winners.
Rule $r$ is Condorcet-consistent if whenever $i$ wins all $n-1$ of her matches,
$r$ selects $i$ with probability $1$.
  We consider strategic manipulation of tournaments where player $j$ might
throw their match to player $i$ in order to increase the likelihood that one of
them wins the tournament. Regardless of the reason why $j$ chooses to do this,
the potential for manipulation exists as long as $\Pr[r(T) = i]$ increases by
more than $\Pr[r(T) = j]$ decreases. Unfortunately, it is known that every
Condorcet-consistent rule is manipulable (Altman and Kleinberg). In this work,
we address the question of how manipulable Condorcet-consistent rules must
necessarily be - by trying to minimize the difference between the increase in
$\Pr[r(T) = i]$ and decrease in $\Pr[r(T) = j]$ for any potential manipulating
pair.
  We show that every Condorcet-consistent rule is in fact $1/3$-manipulable,
and that selecting a winner according to a random single elimination bracket is
not $\alpha$-manipulable for any $\alpha > 1/3$. We also show that many
previously studied tournament formats are all $1/2$-manipulable, and the
popular class of Copeland rules (any rule that selects a player with the most
wins) are all in fact $1$-manipulable, the worst possible. Finally, we consider
extensions to match-fixing among sets of more than two players.
