Coincidence of Condorcet committees

The Condorcet Committee à la Gehrlein (CCG) is a fixed-size subset of candidates such that each of its members defeats in a pairwise contest any candidate outside. The Condorcet Committee à la Fishburn (CCF) is a fixed-size subset of candidates that is preferred to all other subsets of the same size by a majority of voters. In general, these two types of Condorcet committees may not always exist. Kaymak and Sanver (Soc Choice Welf 20:477–494, 2003) studied the conditions under which the CCF exists under a large class of extensions of preferences. We focus here on the most important members of their class, the lexicographic extension of preferences, and we define more precisely, the conditions under which these committees coincide when they exist. Our results depart from the rather optimistic conclusions of Kaymak and Sanver (Soc Choice Welf 20:477–494, 2003) on the coincidence between the CCG and the CCF. We exhibit profiles for which the CCF is empty while the CCG exists and the preferences are all of lexicographic type. Furthermore, we obtain the same conclusion when we derive preferences on candidates from those on sets of candidates using the separability assumption.