# Extending tournament solutions

## Abstract

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.

## Notes

### Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft under Grant BR 2312/7-2, by a Feodor Lynen research fellowship of the Alexander von Humboldt Foundation, and by the ERC under Advanced Grant 291528 (“RACE”) and Starting Grant 639945 (“ACCORD”). We thank Vincent Conitzer, Christian Geist, and Hans Georg Seedig for helpful discussions and the anonymous reviewers and an associate editor for valuable feedback. Preliminary results of this paper have been presented at the 13th International Symposium on Artificial Intelligence and Mathematics (Fort Lauderdale, January 2014) and at the 28th AAAI Conference on Artificial Intelligence (Québec City, July 2014).

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