Aggregation of coarse preferences

We consider weak preference orderings over a set A_n of n alternatives. An individual preference is of refinement ℓ ≤ n if it first partitions A_n 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 (within non-abstaining voters) 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.