On Approximately Strategy-Proof Tournament Rules for Collusions of Size at Least Three

A tournament organizer must select one of $n$ possible teams as the winner of a competition after observing all $\binom{n}{2}$ matches between them. The organizer would like to find a tournament rule that simultaneously satisfies the following desiderata. It must be Condorcet-consistent (henceforth, CC), meaning it selects as the winner the unique team that beats all other teams (if one exists). It must also be strongly non-manipulable for groups of size $k$ at probability $\alpha$ (henceforth, k-SNM-$\alpha$), meaning that no subset of $\leq k$ teams can fix the matches among themselves in order to increase the chances any of it's members being selected by more than $\alpha$. Our contributions are threefold. First, wee consider a natural generalization of the Randomized Single Elimination Bracket rule from [Schneider et al. 2017] to $d$-ary trees and provide upper bounds to its manipulability. Then, we propose a novel tournament rule that is CC and 3-SNM-1/2, a strict improvement upon the concurrent work of [Dinev and Weinberg, 2022] who proposed a CC and 3-SNM-31/60 rule. Finally, we initiate the study of reductions among tournament rules.