Duncan Black's Electoral System is a single winner electoral system where each voter lists all the candidates in order of preference. Then if there is a Condorcet winner (a candidate that wins all pair wise comparison), that is selected. If not the Borda winner is selected.
Instead a Condorcet system is a system such that if there is a Condorcet winner, such is selected, but if there is no Condorcet winner, then there is a loop and one of the candidates of the loop is selected.
On the Black wikipedia page it says that the Black Electoral system is a"Condorcet system", but I cannot seem to find formal proof that in the absence of a Condorcet Winner, the Borda winner would be one of the winners of the winning cycle. It could elect someone else all together. And if this was the case then it would NOT be a Condorcet system.
This would be solved if the definition of Black Electoral System was to apply Borda only among the candidates in the winning cycle, but I find no reference of that.
Any suggestions?
Thanks,
Pietro
As I understand the notion of a "Condorcet system" or a "Condorcet consistent system" it only implies that the Condorcet winner is elected when there is one. In that case, Black's procedure is Condorcet consistent by definition.
Thank you. You are right, I understood the definition of Condorcet method incorrectly. In fact what I understood has (I just discovered from wikipedia) a name in itself: Smith-Efficient. https://en.wikipedia.org/wiki/Condorcet_method#Two-method_systems
Given that, you still pose a very interesting possibility. But, given the work of Don Saari and a lot of other people about all of the good properties of Borda Rule, it might be difficult to prove that Black's method does what you suggest. However, it might be true. I do not know.
In fact I am quite sure we could find a counter example where the Borda winner is in fact not in the Smith Set. We know that Borda rules does not bring necessarily a Condorcet winner. So we start with an example where there is a Condorcet winner and a distinct Borda winner. Now the question is, can I clone the Condorcet winner into three candidates, such that each is a winner respect to any other candidate in a pairwise comparison, while they form a loop, and while the previous Borda winner remains the Borda winner. My intuition is that it is plausible that it is possible. But I would have to write it down to be sure. If this is possible then Black system does not always provide a Smith Efficient result. If not it might always provide a Smith efficient result.
The Borda winner does not necessarily belongs to the Smith set.
Given a voting situation, the Smith set is the smallest non-empty set of candidates such that each member defeats (in a pairwise election) every candidate outside the set. Therefore, when there exist a Condorcet winner, call it x, the Smith set is just {x}. Since we know of examples where, as you say, there is a Condorcet winner and a distinct Borda winner, we can conclude that the Borda winner can be outside of the Smith set.
It seems to me that one can easily construct an example where there is a top cycle (with majority rule), but the Borda winner is not an element of the top cycle.
Maurice Salles is correct. Let there be 8 candidates: a,b,c who will be the top cycle; x who will be the Borda winner; d1, d2,d3,d4 who are dummy (filler) candidates.The three voters have preferences a,b,x,d1,d2,d3,d4,c; b,c,x,d1,d2,d3,d4,a; c,a,x,d1,d2,d3,d4,b. x is the Borda winner with 18 points versus 16 points for a (and symmetrically b,c).