Extending tournament solutions

An important subclass of social choice functions, so-called majoritarian (or C1) functions, only take into account the pairwise majority relation between alternatives. In the absence of majority ties-e.g., when there is an odd number of agents with linear preferences-the majority relation is antisymmetric and complete and can thus conveniently be represented by a tournament. Tournaments have a rich mathematical theory and many formal results for majoritarian functions assume that the majority relation constitutes a tournament. Moreover, most majoritarian functions have only been defined for tournaments and allow for a variety of generalizations to unrestricted preference profiles, none of which can be seen as the unequivocal extension of the original function. In this paper, we argue that restricting attention to tournaments is justified by the existence of a conservative extension, which inherits most of the commonly considered properties from its underlying tournament solution.