Hello James,
I wrote a long mail. Sorry about that. No need to reply on everything word in it. I however felt that it is worth writing all the text, just in case it would trigger some useful thoughts. Simple answer "thanks but I'm not convinced of the merits of non-Smith-set candidates yet" is also enough :-).
Best Regards, Juho
About your "least additional votes" method: Correct me if I'm wrong, but
I think that your method is equivalent to minimax (margins).
Correct. Didn't want to reveal that right away. Fresh thinking requested. :-)
If the maximum defeat margin of some candidate is n, she needs exactly n+1 additional votes to become a Condorcet winner.
Just clarifying... When you say a larger majority, you seem to mean a majority with a wider margin.
Yes, sorry about not being more exact. All my terms in the mail (e.g. "large majority") referred to margins. (And yes, term "narrowest majority" is more appropriate than "smallest majority".)
I tend to see margins as "natural" and winning votes as something that deviates from the more natural margins but that might be used somewhere to eliminate strategic voting. (not a very scientific description but I don't have any better short explanation available :-) )
A large majority who is unhappy with the change? I would say that there
is a large *minority* (201 voters) that is unhappy with the change.
I meant that when X was the captain people wanted to change him to A, B or C with a small margin of votes. But later when e.g. C became the captain people wanted to change him to B with a large margin. Only a minority wanted to change C to X. But the point is that people (majority of them) are now "less happy" or "more mutinous" because of the problematic B>C relationship. I'm thus just measuring general happiness and risk of mutiny without paying attention to whom people would elect as captain in the current situation. The election method could be so clever that it would elect X (the candidate with least risk of mutiny) even if the people would not make the switch themselves in a mutiny. (Are election methods allowed to be more clever than the voters when picking up the winner?)
it's hard to say, because we only have ordinal information here, nothing about utility, strength of preference, 'approval', etc.
I think it is the nature of (basic) Condorcet methods not to take into account how strong the preferences are. Taking also strengths into account would be wonderful but I guess it is the problems with strategical voting that have kept us away from this ideal target. When talking about Condorcet based methods I tend to limit myself to this "order only" mode.
(I however think that your cardinal pairwise method adds something to this plain Condorcet tradition without taking all the rating related problems in => worth another discussion.)
Violating an unambiguous majority preference (e.g. A>X) is a *larger* problem than violating a majority preference that is contradicted by another majority preference (e.g. A>C), because the former is always avoidable, and the latter is unavoidable in the case of a majority rule cycle.
I think all the majorities are unambiguous (because that is what the voters told us). A>X could be called "loopless", if we want to describe how it is different from the others. Both electing X and electing A violate a majority opinion. One can avoid violating A>X by not electing X (= select one of the Smith candidates). But one can also avoid violating e.g. A>B by not electing B. All of the individual preferences are thus avoidable. And all the Smith loop violations can be avoided by electing X. I guess the key target of my pirate example is to demonstrate that in some (rare) situations violating A>X could be a smaller crime than violating one of the Smith loop preferences. (And my thinking is not based that much on paths but on utility of each captain candidate separately.)
In your pirate example, there are no compromise candidates; the pirate electorate is very badly polarized.
I agree. The basic setting is four parties of about equal size. I think this situation is quite normal. What is exceptional is the strength of the loop. My understanding however is that strong loops may occur also in real life - considerably stronger than ones between three parties of equal size as a result of some random votes. Also sincere, not only as a result of a voting strategy.
But what if multiple dissatisfied groups arose at once?
Then the X supporters would be in desperate trouble (as would supporters
of any other candidate in a similar situation).
Yes. I think now we come to the question if "mutiny" is the "only correct" real life criterion or if there are also others. I claim that "mutiny" is one well defined criterion that is useful is some situations and directly points out the correct voting method (MinMax with margins).
Mutiny of everyone against one is one candidate for another real life criterion. I think mutiny to replace one with one is however the most useful and typical case (both in the ship and in politics). This "mutiny for anyone else" would also give support to sticking to the Smith set when electing the winner. I'm however afraid that these majorities can not be summed up (=> not a strong case to support sticking to the Smith set). (Note also that a Condorcet winner that has not been the #1 favourite of any of the voters has a risk of yet another type of mutiny ("everyone thinks he is not the best").)
There may be also other real life criteria that could be used to characterise (or define) different alternative single winner election types / needs. I think the current paradigm is that there are only one type of single winner elections (i.e. rules are the same irrespective of what the context is (e.g. captain, president, holiday resort or favourite fruit election); one method serves all single winner needs). I tried to prove that eliminating risk of mutiny is one such need, but I don't have any evidence that it is the only one that is needed.
I began to feel that there was
something logically inconsistent about insisting on the Condorcet winner,
but abandoning all strict majority rule requirements when no Condorcet
winner exists.
In my mind one important borderline is cycles. Newtonian physics (=linear ordering) applies as long as there are no loops. Condorcet winner is a very natural concept in this world. But when loops emerge the ideal linear ordering is broken. To me the linear order of Smith sets (the top cycle + bigger ones) (or "loop groupings") is not the most tempting way forward since it seems to hide possibly bigger group preferences inside the "loop groupings" than what the strength of preferences between the groupings are. As I mentioned earlier, there is a risk of trying to make the group opinions look like linear (personal) preferences. I think the drawing (and imagining) techniques may lead us to false conclusions. As a result I have been interested also in criteria that simply evaluate candidates one by one.
I would say none of the above candidates are a sufficient compromise
candidate. I would suggest that the pirates try a bit harder to find a
compromise captain (perhaps someone who has spent several years in each
country). Failing that, I would suggest that the four factions should go
their separate ways, finding boats that are manned by more like-minded
seafarers.
That is not allowed :-). We had an election with four candidates. And elections are not supposed to cause countries to break into separate smaller countries. The best single winner election method must be capable of electing one (the best) of these candidates. Since you say that you want to stick to the Smith set, I guess your answer must be A, B or C. That would violate the "least risk of mutiny" criterion. Does this mean that there is no need for election methods that try find the optimal candidate by minimizing the risk of mutiny.
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