I haven't often had time to check the list, but I've just noticed Richard's reply about FBC, and I'm replying to it now:
Richard quoted my definition of FBC: > >Favorite Betrayal Criterion (FBC): > >By voting another candidate over his/her favorite, a voter should >never gain an outcome that s/he prefers to every outcome that s/he >could get without doing so. Richard then said: This may be nit-picking, but that wording is ambiguous. Say there are five candidates: A, B, C, D, and E. Without voting another candidate over her favorite, a voter might get any of the following results: A, B, C, D, or E. When voting another candidate over her favorite, she might also get any of the following results: A, B, C, D, or E. None of the choices she could get by voting another candidate over her favorite is preferred to all of the choices she could get by not doing so. Therefore any method (provided it is capable of electing any one of the candidates) could pass FBC under this (unintended) interpretation. I reply: Richard, in order to pass a criterion, it isn't enough that you can find an example in which the method doesn't fail. In order to pass, there must not be any example in which the method fails to meet the criterion's requirement, when the criterion's premise conditions are met. So, if, for some particular method, you've found an example in which a certain voter could elect any of the 5 candidates by voting someone over his favorite, and could also elect any of the 5 candidates without voting someone over his favorite, that isn't enough to allow you to say that the method meets FBC. Richard continues: A better wording would be: A method passes FBC if there is no scenario in which, by voting another candidate over his or her favorite, a voter could gain an outcome he or she prefers to any of the outcomes he or she could gain in the same scenario without doing so. It's a seemingly subtle change but it removes that ambiguity. I reply: Ok, so you're interpreting my wording to say that by voting another candidate over his favorite, a voter should never gain an outcome that he prefers to every outcome that he could get in that or some other election without doing so. Actually, an outcome means an outcome of that election. Electing candidate A in a different election at some later time isn't the same outcome as electing candidate A in today's election. In that future election, the candidates who might win if A doesn't win might be different, the social and other conditions in the country might be different, and A's own policies might be different. It isn't reasonable to treat "outcome" as meaning something other than "outcome of that election". And although "any of the outcomes..." sounds good in everyday conversation, "all of the outcomes..." has a more definite meaning. "I can beat _anyone_ at chess. Can you beat anyone at chess?" Don't think I don't appreciate your effort to find a problem with FBC. Without efforts like that I wouldn't be able to say that the criterion has stood up to discussion. >What I mean by voting one candidate over another: > >A voter votes A over B if s/he votes in such a way that one could >contrive some configuration of other people's votes such that, >if we delete from the ballots every candidate but A & B, A is >the unique winner if & only if we count that voter's ballot. > >[end of definition] > >I suppose one could add to that "...and no one can contrive a >configuration of other people's votes such that, if we delete from >the ballots every candidate but A & B, the unique winner is B if >& only if we count that voter's ballot." > >I don't think that addition is necessary, but it could be added >if someone devised a reasonable example where it seemed necessary. Richard continues: Forest and I had an off-list discussion some time ago about defining monotonicity, and the prerequisite definition of "changing a ballot in a way that favors candidate X". Making such a definition generally applicable (beyond fully ranked methods) is trickier than one would think. For instance, in CR, if candidate X's rating is increased from 25 to 30, does this favor X? Yes, but what if candidate Y's rating is increased by 10 points at the same time X's rating is increased? We never came up with a completely satisfactory resolution. I reply: I too have noticed that wording Monotonicity precisely is trickier than one might at first expect. But I think it's reasonable to assume that when we refer to changing X's rating, that doesn't include changing someone else's rating too. I have a wordier Monotonicity definition that is probably satisfactory, though I haven't finished work on Monotonicity definitions. Richard continues: I haven't studied it carefully enough, but I hope the above definition of "voting one candidate over another" doesn't suffer from similar problems. I think it might be OK, since it involves reducing the ballots to only two candidates, but I just wanted to point out that there are sometimes hidden "gotchas" in some of these definition attempts. I reply: I don't know quite how to reply to that...He's saying that maybe it has a problem, but he hasn't read it carefully enough to know. I suggest that it doesn't have a problem until someone finds a problem with it. >it's brief & simple enough to use in public >discussion. Of course something more universally-applicable can be >useful in mathematical discussion. Richard continues: Yes, I agree there is value in both mathematically rigourous definitions and in colloquial definitions. I reply: Even definitions that use mathematical language and substitute symbols for some words, while defining the symbols in terms of those words, still usually need some English, and so those definitions don't necessarily avoid the issues of what English words mean. What you call "colloquial" is a definition that is entirely in English. I'd call it colloquial if it contains words which the dictionary lists as colloquial. We should resist the temptation to elevate mathematics to a priesthood, without whose language no definition is valid. One thing different about Forest's definition was that it covers more kinds of examples, and, as I said, that's a good thing. But let me add that, when I said that my definition isn't universally applicable, I _didn't_ mean that it isn't applicable to all methods. I meant that it can't be tested in certain kinds of examples, such as examples in which a voter has more than one equally favorite candidate. Richard continues: Of course, for the general public, it might be sufficient to have the colloquial FBC definition, without defining "voting one candidate over another". Presumably anyone interested in voting knows what that phrase means. I reply: If I didn't define it, you or someone would point out that I hadn't defined it. No, I don't tell the definition every time I use a criterion that speaks of voting one candidate over another. Yes, it's necessary to have a definition of that in case someone asks. Mike Ossipoff _________________________________________________________________ Send and receive Hotmail on your mobile device: http://mobile.msn.com
