Aggregation of Coarse Preferences

We consider weak preference orderings over a set An of n alternatives. An individual preference is of refinement l<n if it first partitions An into l subsets of tied alternatives, and then ranks these subsets within a linear ordering. When l < n, preferences are coarse. It is shown that, if the refinement of preferences does not exceed l, a super majority rule with rate 1-1/l is necessary and sufficient to rule out Condorcet cycles of any length. It is argued moreover how the coarser the individual preferences, (1) the smaller the rate of super majority necessary to rule out cycles in probability (2) the more probable the pairwise comparisons of alternatives, for any given super majority rule.