Adaptive Manipulation for Coalitions in Knockout Tournaments

Knockout tournaments, also known as single-elimination or cup tournaments, are a popular form of sports competitions. In the standard probabilistic setting, for each pairing of players, one of the players wins the game with a certain (a priori known) probability. Due to their competitive nature, tournaments are prone to manipulation. We investigate the computational problem of determining whether, for a given tournament, a coalition has a manipulation strategy that increases the winning probability of a designated player above a given threshold. More precisely, in every round of the tournament, coalition players can strategically decide which games to throw based on the advancement of other players to the current round. We call this setting adaptive constructive coalition manipulation. To the best of our knowledge, while coalition manipulation has been studied in the literature, this is the first work to introduce adaptiveness to this context. We show that the above problem is hard for every complexity class in the polynomial hierarchy. On the algorithmic side, we show that the problem is solvable in polynomial time when the coalition size is a constant. Furthermore, we show that the problem is fixed-parameter tractable when parameterized by the coalition size and the size of a minimum player set that must include at least one player from each non-deterministic game. Lastly, we investigate a generalized setting where the tournament tree can be imbalanced.