Talk:Condorcet ranking

If there are pairwise ties or cycles, do Condorcet rankings reflect this? For example, if there is a 3-candidate Smith Set and a 4th candidate, are the Smith candidates considered to be in a tie for 1st, with the 4th candidate in 2nd place? BetterVotingAdvocacy (talk) 22:35, 20 February 2020 (UTC)

I don't think it reflects cycles, though it could reflect ties. Reflecting cycles would be a "Smith ranking", not a Condorcet ranking. So a Condorcet ranking is only defined when there's a sequence of candidates so that A beats everybody, B beats everybody but A, C beats everybody but A and B and so on. A bit like criteria, that a method that always returns a Condorcet ranking doesn't say anything about what happens when there isn't one. Kristomun (talk) 00:25, 21 February 2020 (UTC)

I'd like to build a consensus for a "generalized Condorcet ranking criterion"; a voting method complies with this criterion if it ~~(Edit: here's a more succinct way to put it: if we take any two groups of candidates, with the two groups not necessarily amounting to all candidates together, and any candidate in the first group can pairwise beat any candidate in the second group, then all candidates in the first group must be ranked higher than all candidates in the second group.)~~ (Edit 2: This is actually wrong, since it implies that in a 3-way Condorcet cycle, one candidate must be ranked above another above another above the first, which is impossible.) always says that A is at least as good as B when A pairwise beats or ties B. In essence, if a group of candidates pairwise beat all others, they must be ranked higher than all of them, but if within the group there is a cycle or pairwise ties, then the voting method can either rank some candidates in the group above others or show ties between some of them. From there, we can consider a "generalized Condorcet ranking" one where candidates in a cycle or pairwise ties are considered to be tied in the generalized ranking. The reason I'm proposing this is that it will help us pick out the "good" Smith-efficient methods (since I'd say this is at least one pretty decent standard of deciding what a good order of finish looks like), and further, helps us explain to the public how it is that Smith-efficient Condorcet methods in general go about deciding which candidates are better than others in the order of finish. In general, I just like the idea of clearly showing to people which groups of candidates are better than others based on pairwise preference, rather than solely pairwise preferences between two individual candidates. What do you think?

Edit 3: I think I have the definition down this time, and you're probably right that it should be called a "Smith ranking": take the definition of a Condorcet ranking, but replace the words "Condorcet winner" with Smith set, and make it so that if there is a multi-member Smith set, they need not all be ranked at the same place (1st, 2nd, etc.), so long as they are all ranked higher than the alternatives they dominate. Also, regarding Condorcet losers, I think I have a new analagous concept for that: the "Smith loser set", the smallest group of candidates such that all candidates in the group pairwise lose to all candidates not in the group. I'm discussing that at https://electowiki.org/wiki/Talk:Instant_Pairwise_Elimination as well, but if it is a sensible concept, then it could replace the words "Condorcet loser" in the definition as well.

Edit 4: With regards to the "Smith loser set" idea, I think it's worth mentioning that it is not so simple to extend the idea in the same ways that the Smith set has been extended. For example, the Smith criterion requires that one of the Smith set candidates must win; if we tried to do this with the Smith loser set ("nobody in the Smith loser set should win") then we can end up with all candidates being eliminated; simple example is 1 voter voting A and another voting B. So it seems that a "Smith loser criterion" may have to be something along the lines of "candidates who are in the Smith loser set but not in the Smith set must not win." Also, I'm not sure of what use it might be, but a property analagous to ISDA for the Smith loser set might be "if a candidate is in the Smith loser set but not in the Smith set, then eliminating that candidate shouldn't change the result of the election." ISDA seems to imply this property automatically, and pretty much the only way I can imagine a voting method passing ISDA but not this property is if it is specifically designed to focus on Smith loser set candidates. The only remaining property that it seems would be interesting to extend would be the majority loser criterion to the mutual majority case (maybe "if a majority of voters prefer every other candidate over a group of candidates, nobody in that group of candidates should win." ?) BetterVotingAdvocacy (talk) 00:55, 21 February 2020 (UTC)