If I were going to display intermediate results in a Condorcet election I think this is how I would do it. It presents all the information the voters need to see how their candidate is doing compared to all of the others. (I usually convert all of the count: A>B>C style examples on this list to this format anyway, because it is easier for me to spot the patterns of blocks of like-minded voters).
Well, its interesting, but I do not think that showing a pairwise matrix, and especially a list of all ballot combinations, is going to be the appropriate output for most people.
I think they want something more distilled and that instantly communicates, as does a bar graph of scores. If I take a quick look at a vote matrix, it doesn't really communicate very much to me. This is not because I am stupid or don't understand what the matrix represents (obviously I do), its just that a table of numbers is not very easy to take in in any meaningful way to visually oriented people. If I have trouble instantly digesting a matrix, I expect that mainstream uses will have *much* more trouble.
Many people on the list have questioned whether there is any way to simplify the output to a set of scores, and whether such a thing is useful. On the first question, I understand that finding a reasonable way of assigning a 1-d set of scores from a 2-d matrix is a difficult problem, but then again, isn't picking a single winner from a 2-d matrix a similar, and equally difficult, problem?
As I think we all agree, if you can pick a single winner, you should by straightforward extension be able to rank all the candidates. In ranking the candidates we have, then, linearized the matrix. If it can be linearized in a reasonable way, I believe it can be done such that each candidate has not only an order, but a scalar dimension, i.e. a score -- in an equally reasonable way, that does not conflict with the ordering. Maybe this is a naive leap of logic (or maybe intuition) on my part, but I have yet to see an argument which leads me to believe otherwise.
As for the utility of a graph of scores: such a graph has less information than a pairwise matrix, but that doesn't mean it is useless. I tend to look at the various "outputs" like this:
Full set of ballots -- all information Pairwise matrix -- lots of information One score per candidate -- some information Ranking -- little information Single winner -- least information
Looking at a matrix, you cannot tell, for instance, whether Nader voters were likely to prefer Gore over Bush, as you would see if you looked at a count of all ballot combinations. Likewise, looking at a set of linear scores, you can't be tell whether McCain beat Bradley in a pairwise election. But it's a matter of finding the right amount of information to display. I say that a matrix is too much information for most people, and a simple ranking of candidates is too little.
-rob
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