On 15.9.2012, at 13.02, Kristofer Munsterhjelm wrote: > On 09/15/2012 09:55 AM, Juho Laatu wrote: >> On 15.9.2012, at 6.05, Jeffrey O'Neill wrote: >>> You can also now save Condorcet results in HTML format but still >>> working on the best graphics to visualize Condorcet results. >> One solution is to support minmax(margins). With that method you can >> simply draw a histogram that indicates how many new (first preference) >> votes each candidate would need to win (or tie) with the winner / >> current leader. > > That also works with minmax(wv), and similar approaches can work for least > reversal methods and those that use sums of victories rather than > minimum/maximum ones. However, those visualization methods are still linked > to specific voting methods, and thus don't provide a general visualization of > the pairwise results.
For minmax(margins) the number of required additional votes to beat (or at least tie with) all other candidates is an exact definition of the method, and gives one number per candidate that can be used in the histogram. All methods that can do the same thing, i.e. give one numeric value that both indicates the winner and indicates a natural measure of distance of each candidate to winning the election, can use histogram level visualization effectively. One would need also an easy to understand verbal description of what the histogram values mean. I wonder which other Condorcet methods have such intuitive function available. Do you have such functions for some of the methods that you proposed as candidates? The histogram results of minmax(margins) do not give all the information that is present in the pairwise comparison matrix. The histogram indicates only the distance to the worst competitor, which is always the current leader. Pairwise comparison results to all others are lost. (Information on 49-50 vs. 48-49 against the current leader could be included in the histogram if needed. Probably better to leave this to the matrix.) But on the other hand the difference of the hstogram values of any two candidates still indicates exactly how much closer to victory the better one of those candidates is. For complete analysis full matrix is needed, but for practical information to regular voters, especially during the counting process, the histogram may well be all that is needed. (All pairwise comparisons could be presented to allow speculation on "what if there had been only these two candidates".) > > I suppose one could use something similar with Kemeny as well: use integer > programming to find the pairwise sum of scores for the best transitive > ordering that puts X first. That is X's score. Then do it for Y and Z. > Assuming X wins, all other scores will be lower than X's. The relationship > between additional votes and Kemeny scores might not be obvious to the end > user, though. > >> - I have seen 2D graphs that show all parwise wins, e.g. in Debian. They >> are however quite difficult to draw automatically, and the positioning >> of candidates in the drawing space (e.g. higher, lower) may not be >> neutral (unless they are in one row or circle). > > These can be "decluttered" by showing graphs where candidate X has an arrow > to candidate Y iff X beats Y pairwise, otherwise there is no arrow. Then > graph visualization programs can be used to arrange the candidates in a way > so that candidates with more pairwise victories (or stronger ones, or > whatnot) are closer to the top, or so that Smith set members always appear > above non-Smith set members. Drawing Smith set members above non-Smith set members takes in a way position on if Smith set members are all better than all other candidates. That may make sense in methods that elect always from the Smith set since at least the winner will be drawn close to the top. But also in those (Smith set based) methods some members of the Smith set may be further from winning the election than some candidates outside the Smit set. I.e. the graph does not indicate how close different candidates are to winning the ongoing counting process. > > If one wants to visualize Ranked Pairs, it'd be easy to simply color the path > throughout the graph to correspond with the pairwise relations/defeats picked > by Ranked Pairs. > Yes, that approach could work for path based methods. Ranked Pairs and methods that "break all loops" or "make the opinions transitive" could also show the candidaes ordered on a line. (But they would again lose the "approximity to winning" information.) I can't draw any clear conclusions from this on how good Condorcet methods are in visualizing the results or an ongoing counting process. The measure of number of voters to change the result seems to be quite natural measure of "distance to victory". Another approach to visualizing the results could be to try to point out "how good winner each candidate would be". In minmax(margins) these measures coincide (measured as additional votes). In Smith set based methods I guess the intended message is that Smith set candidates are the best winners, although that does not correlate with distance to victory (if measured as number of voters that may make someone win). Each Condorcet method has its own philosophy and measures, and probably visualization too (unless some generic / method independent visualizations are used). Juho ---- Election-Methods mailing list - see http://electorama.com/em for list info
