Consider the following example.
In district I, A wins (in most Condorcet methods, anyhow). In district II, (same as district I but the roles of B and D are swapped), A also wins. But in the combined 2-district country, C is the Condorcet winner (beats A by 8:6, beats B and D by 9:5).
Consequently, some Condorcet methods cannot be "counted in Precincts." For example, the Smith,IRV method invented by Woodall is Condorcet but I see no efficient way to count it in precincts if there are a large number of candidates (say 100 candidates, if you want something concrete to think about – see any feasible way to count it other than a centralized count? I don't).
However, the Condorcet methods that depend only on the matrix of "pairwise totals" can be counted in precincts (despite the nonadditivity paradox illustrated above!) in the sense that each precinct can find and publish its pairwise matrix and the summed matrix will be used for the whole country.
Analogous problem for IRV system
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