On the complexity of schedule control problems for knockout tournaments

Knockout tournaments provide a common and important framework for structuring sporting competitions, worth billions of dollars every year in the global economy. Knockout tournaments also model a specific type of election scheme: namely, sequential pairwise elimination elections. The designer of a tournament typically controls both the shape of the tournament (usually a binary tree) and the seeding of the players (the assignment of players to leaves of the tree). In this paper we investigate the computational complexity of tournament schedule control: the problem of designing a tournament that maximizes the winning probability a target player. We start with a generic probabilistic model consisting of a matrix of pairwise winning probabilities, and then investigate the problem under two types of constraint: constraints on the probability matrix, and constraints on the allowable tournament structure. The various constraints we consider – all naturally occurring in practice – serve to make the control problem either easy (polynomial time computable) or hard (NP-complete). In the hard cases, we present a number of computationally efficient heuristics for control, and evaluate their performance using real world data. We show that they perform well experimentally.