Iīve re-titled the thread, because Iīd neglected to write down the subject line of the message that Iīm replying to.
Markus is right to point out that, as Iīve defined it, FBC has a problem when thereīs a tie. As Markus pointed out, a single voter can change the outcome only by changing a definite win to a tie, or changing a tie to a definite win.
FBC, as defined so far, has a problem with a tie. If the way that you could get your best outcome is by breaking a tie, and getting a certain outcome, then, if you donīt do so, that same candidate might win the tie anyway, and so what you get when you break the tie isnīt better than anything that you could get wihtout doing so. On the other hand, if, by voting someone over your favorite, you could make a tie, then you arenīt electing with certainty someone who is better than anyone you could otherwise have gotten, since you arenīt electing anyone with certainty.
The obvious and easy way out of this is to replace "voter" with "set of voters who have the same peferences and vote in the same way".
That fixes the problem. Thanks for pointing the problem out, Markus.
I make that change in both my Certainty FBC and my Nonrandom FBC.
...even though Nonrandom FBC probably doesnīt have that problem. It only applies to nonrandom methods, as Iīve defined that term. And it doesnīt specify an outcome that can be gotten with certainty, as does Certain FBC. For instance, in Plurality, if the greater-evil wins with certainty if you donīt vote for the lesser-evil, and you could make a tie between the lesser and greater evils by voting for the lesser-evil, then you can get your best outcome only by voting that lesser-evil over your favorite, and so Plurality fails FBC in that instance, and therefore fails FBC.
I donīt have my definition of best outcome in front of me right now, but what I say in the previous paragraph is probably true.
Anyway, as I said, Iīm hereby changing FBC to replace "voter" with "set of voters who have the same preferences and vote in the same way".
I define a preference as an instance of preferring one candidate to another.
Markus--
You said:
Dear Kevin,
you wrote (1 March 2005):
I think, in short, that the "situation" (of odds distribution) is not relevant to FBC.
I reply:
Yes, I donīt consider those intermediate lotteries to be outcomes. The outcome is the final single winner.
You continued:
As far as I have understood FBC correctly, then it is about individual voters and not about coalitions of voters. However, an individual voter usually only changes the outcome from one decisive situation to an indecisive situation or from one indecisive situation to a decisive situation or from one indecisive situation to another indecisive situation. But an individual voter usually doesn't change an outcome from a situation where candidate A wins decisively to a situation where another candidate B wins decisively.
I reply:
Quite so, and that creates a problem for FBC, at least in my Certain FBC version. For that reason I now define FBC to be about a set of voters who have the same preferences and vote in the same way, instead of one individual voter. That seems the easiest, simplest way to avoid the problem that you have pointed out.
Kevin had said:
I have to interpret "result" to mean "the candidate who actually got the seat,"
I reply:
Yes, thatīs how I mean "result".
You (Markus) said:
I see more than one possible interpretation. Examples:
1. Mike uses the resolute model. (The "resolute model" says that for every possible profile the winner is determined in advance.)
I reply:
As I said, only if you define the resolute model better would I be able to say whether or not I use it. But, as I said, from your definition so far, the U.S. uses the resolute model when chosing its president.
You continue:
2. Mike talks about coalitions of like-minded voters rather than about single voters.
I reply:
For FBC, Yes. Now I do. But, before, I didnīt.
You continue:
But then the question is whether all these like-minded voters have to vote in the same manner or whether they may vote differently.
I reply:
My FBC improvement, described earlier in this reply, replaces "voter" with "set of voters who have the same preferences and vote in the same way".
Kevin wrote:
Pretending Mike agrees with my interpretation (and that he clarifies FBC accordingly), do you think FBC would then be unambiguous?
Markus replied:
Your question is quite hypothetic because Mike will never clarify his definitions.
I reply:
Iīve always answered questions about my definitions. But if you think that one of my definitions is vague or ambiguous, then the burden is on you to tell me exactly which sentence in that definition is vague or ambiguous, and in what way. Or, for any of my criteria, post an example in which that criterion is ambiguous about the matter of whether or not some particular method passes the criterion.
The problem of FBC when thereīs a tie hadnīt occurred to me, but when it was pointed out, I changed FBC to get rid of the problem. Iīve been answering all the clarification questions, both these questions about FBC, and, in general, about any of my criteria. I always answer questions about my criteria.
I even always answer Markusīs questions about my critreria. Of course sometimes the answer is that I donīt know. Thatīs the case about whether itīs possible to find a PC GSFC failure example, a PC SDSC failure example, or whether itīs possible to find a BeatpathWinner FBC failure example.
I also must answer that I donīt know, when Markus asks me if I use the resolute model, till Markus gives a better definition of the resolute model.
Another question to which I must answer "I donīt know" is when Markus asks me for an instance where SFC gives a different answer than a new criterion that Markus has just defined. If Markus doesnīt understand SFC, which was defined long ago,and has been much explained, discussed, and used, how can he expect me to understand his newly-defined criterion? What that amounts to is that Markus is then asking me a question about his criterion, and I must answer that I donīt know.
If we want to find out if SFC and Markus-Non-FBC are the same, or if they can give different answers, then the best way would be, for several particular methods, for Markus to say whether those methods meet Markus-Non-FBC.
For instance, the following methods pass SFC:
All wv Condorcet methods, including PC, BeatpathWinner/CSSD, SSD, SD, Ranked-Pairs, and Smith//PC.
Markus, do any of those methods fail the criterion that you wrote as your possible restatement of SFC?
The following methods fail SFC:
Pluralty, Approval, Margins Condorcet, IRV, Borda, Bucklin, CR...and pretty much everything but wv Condorcet.
Markus, do any of the 8 methods listed above pass your criterion that you wrote as your possible restatement of SFC?
If neither of us undestand the otherīs criterion, then the best way to find out if theyīre different is for you to answer the above questions.
Mike Ossipoff
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