  Consider a round-robin tournament on n teams, where a winner must be
(possibly randomly) selected as a function of the results from the ${n \choose
2}$ pairwise matches. A tournament rule is said to be k-SNM-${\alpha}$ if no
set of k teams can ever manipulate the ${k \choose 2}$ pairwise matches between
them to improve 2 the joint probability that one of these k teams wins by more
than ${\alpha}$. Prior work identifies multiple simple tournament rules that
are 2-SNM-1/3 (Randomized Single Elimination Bracket [SSW17], Randomized King
of the Hill [SWZZ20], Randomized Death Match [DW21]), which is optimal for k =
2 among all Condorcet-consistent rules (that is, rules that select an
undefeated team with probability 1).
  Our main result establishes that Randomized Death Match is 3-SNM-(31/60),
which is tight (for Randomized Death Match). This is the first tight analysis
of any Condorcet-consistent tournament rule and at least three manipulating
teams. Our proof approach is novel in this domain: we explicitly find the
most-manipulable tournament, and directly show that no other tournament can be
more manipulable.
  In addition to our main result, we establish that Randomized Death Match
disincentivizes Sybil attacks (where a team enters multiple copies of
themselves into the tournament, and arbitrarily manipulates the outcomes of
matches between their copies). Specifically, for any tournament, and any team u
that is not a Condorcet winner, the probability that u or one of its Sybils
wins in Randomized Death Match approaches 0 as the number of Sybils approaches
$\infty$.
