Strategic Nominee Selection in Tournament Solutions

Tournament solutions provide methods of selecting winners of a competition based on the results of pairwise comparisons. These methods have been studied in-depth from the perspective of social choice theory, where a comparison between two candidates indicates which of them is preferred to another by the majority of voters. In this paper we study the party setting, in which groups of candidates select their representatives. We consider the Uncovered Set tournament solution, in which a candidate i is selected if no other candidate beats all the options defeated by i, and contrast it with the Condorcet Winner rule, in which either Condorcet winner is chosen or no selection is made. We show that checking if a Nash equilibrium exists is NP-complete for both of these rules. Moreover, from the perspective of Uncovered Set, it is also NP-complete to check if a party has a potential winner.