On 22.6.2011, at 2.53, fsimm...@pcc.edu wrote: > I am more convinced than ever that the best way to measure defeat strength in > Beatpath (aka CSSD) is > by giving the covering relation the highest priority
Being uncovered is a positive criterion in the sense that it tries to improve the outcome with sincere votes. Also positive criteria have the problem that all of them can not be met at the same time. I drafted one cyclic example to see how this criterion and another positive criterion, the worst defeat criterion, relate to each others. 33: A>B>D>C 16: A>D>C>B 33: C>B>A>D 17: D>C>B>A Here candidate A is good from the worst defeat point of view in the sense that it is only two votes short of being a Condorcet winner (and having majority of first preferences). The worst defeats of all other candidates are considerably worse. But A is covered by B, and according to the "covering rule" above that would mean that A can not win. The point is thus that although covered candidates may sometimes be less good than others, they may sometimes be also better than others, e.g. from the worst defeat / number of required extra votes point of view. In this example the covered candidate could well be considered to be the best winner. The votes in this example do not have any very obvious mapping to some real life situation. One approximate explanation could be that almost 50% of the voters support A. All those that do not support A prefer both B and C to A. That is why A loses to two candidates (slightly) and becomes a covered candidate. D is a more complex candidate to explain (but some extra candidates are needed to build the required loop). Another example of a covering relation in a loop could be a situation where we have three parties in a loop. At least one of the parties has several candidates. They all beat all candidates of one of the other parties, and are beaten by all candidates of the other one of them. Within our party there is a clear order of preference between different candidates, and therefore the weaker candidates are covered by the stronger ones. In this situation it would make sense not to elect any of the covered candidates. But on the other hand in this kind of scenarios also the worst defeats (and strongest beatpaths) agree with the covering relation. In this example the covering relation is thus a natural argument in favour of the covering candidates against the covered ones, but adding the covering rule does not improve the method since also other criteria agree on which candidates are good and which ones are bad. The votes could be e.g. 33: A1>A2>A3>B1>B2>B3>C1>C2>C3 33: B1>B2>B3>C1>C2>C3>A1>A2>A3 33: C1>C2>C3>A1>A2>A3>B1>B2>B3 The question then becomes if there are situations (examples) where use of the "covering rule" would clearly (or likely) improve the outcome of the method (and where defeat strengths (or defeat strength based beatpaths) would elect some clearly worse candidate). In the first example the "covering rule" may have led to a worse winner (or what do you think). I may try to find one more example where the "covering rule" would improve the results (of other rules). Anyone else, any good candidates? Many good positive criteria tend to give the same winners. One has to pay special attention to cases where they give different results in order to see which ones of those rules should rule in such situations. Beatpath is not perfect, so there is potential for improvements. Winning votes sometimes give strange results with sincere votes. Also Smith set can sometimes be questioned. On my part the jury is still out on if there are situations that justify using (the usually good) covering rule to be included in the method to improve the results with sincere votes. It seems that there are some cases where the use of the covering rule could make the results also worse. I'm waiting for examples that would show that also the reverse is true. (For the sake of completeness I note also that different societies / elections may have slightly different needs, and therefore the fine-tuning of the methods might differ.) Juho
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