Manipulating Stochastically Generated Single-Elimination Tournaments for Nearly All Players

Who Can Win a Single-Elimination Tournament?

“…Suppose that |V | = 4. If |A| ≥ 2, the result follows from [SV11a]. Otherwise |A| = 1, and H = I = ∅ and |J| ≤ 1, which contradicts |V | = 4.…”

“…It is also known that if v is a king and wins against more than half the players or is a "superking" and every w whom v beats indirectly loses to at least log n players whom v beats directly, then v will be able to win an SE tournament. While these results have been useful on their own for showing that tournaments generated by certain random models are likely to have many players who can win [Vas10,SV11a], it is natural to wonder how robust these results are to changes in the exact sufficient conditions. Recent results of [KV15] seem to suggest that the parameters for these structural results are brittle; namely, when the exact parameters of the conditions are relaxed, finding a winning seeding for v (if it exists) becomes NP-Hard.…”

“…In particular, [Vas10] showed that a "superking" -a king v where every player in N in (v) loses to at least log n players from N out (v) -is always an SE winner. [SV11a] showed a generalization they call a "king of high out-degree" -that is, a king with out-degree k, who loses to fewer than k players that have out-degree greater than k -is always an SE winner. This result was the first to generalize the conditions on players who can win SE tournaments.…”

“…Note that the superking result [Vas10] corresponds to the special case where H = J = ∅, while the "king of high out-degree" result [SV11a] corresponds to the special case where I = ∅.…”

On the complexity of bribery and manipulation in tournaments with uncertain information

“…Williams found that, for certain dominant entrants in a cup tournament it is easy to find a schedule such that a particular entrant will win [56]. In a series of papers, Stanton and Williams have also shown that, for deterministic tournaments where the relation between players is generated from a known random prior, it is computationally easy to find a knockout tournament such that a preferred candidate will win [47][48][49]. In the uncertain information case, Hazon et al [24] find that the problem of setting an agenda that maximizes a particular candidate's probability of winning is NP-hard.…”