On the difficulty of computing the winners of a tournament

In voting theory, the result of a paired comparison method as the one suggested by Condorcet can be represented by a tournament, i.e., a complete asymmetric directed graph. When there is no Condorcet winner, i.e., a candidate preferred to any other candidate by a majority of voters, it is not always easy to decide who is the winner of the election. Different methods, called tournament solutions, have been proposed to define the winners. They differ by their properties and usually lead to different winners. Among these properties, we consider in this survey the algorithmic complexity of the most usual tournament solutions: for each one of these methods, we give its complexity.