How to combine preferences

In the rank aggregation problem, we consider a set of voters that each state their preference order on a given set of candidates. Our goal is to give a ranked list of the candidates that "best represents" the preferences of the voters. In Kemeny rank aggregation, this goal is formalized as finding a ranked list of the candidates that minimizes the number of pairs that are out of order with respect to the input lists. Finding an optimal Kemeny ranking is NP-hard, and we will discuss two approximation algorithms.
We then consider popular rank aggregation, in which we want to find a ranked list of the candidates so that there is no other solution that a majority of the voters prefer. We show some preliminary results in this setting.