I'd said:
If a majority prefer X to Y, that's a majority pairwise preference (MPP). The strength of that MPP is measured by the number of voters who prefer X to Y.
You say:
Well, obviously that's one way to define the strength of a pairwise defeat. Margins provides another way, and cardinal pairwise yet another.
I reply:
No, that obviously is _not_ a way of defining the strength of a pairwise defeat. It's a way of defining the strength of a majority pairwise preference, however.
One way (but only one way :-) of defining a majority pairwise preference is as an instance of a majority praeferring one candidate to another. And one way of defining the strength of that majoritly pairwise preference has to do with how big a majority that is. Completely arbitrary, of course :-)
I'd said:
An MPP for X over Y is outdone if there is a sequence of MPPs from Y to X, consisting of MPPs that are all at least as strong as the MPP of X over Y.
To violate majority wishes means to elect someone who has an MPP against him that isn't outdone.
You say:
Lately, I have been defining majority rule violation as the selection of an option outside the Smith set.
I reply:
The Smith set has nothing to do with majority.
You continue:
This is close to your definition
I reply:
No. It bears no resemblence to my definition.
You continue:
..., except that you get more specific by assuming that there is a single normative definition for "defeat strength"
I reply:
We all have heard that there are other definitions for "defeat-strength". Whether they're normative or not, I've told why they aren't as useful for meeting criteria about guarantees of minimizing strategy need. And I've told why they aren't as good in terms of the goal of not overruling avoidably many voters. If saying that something is normative means saying that it's what should be, then I've told you why wv is normative.
Now you're talking about majority rule, and so now, instead of majority pairwise preferences, it's majority pairwise votes (MPVs). I suggested measuring the magnitude or strength off such a vote by the magnitude of the majority. Call me original :-)
You continued:
, and then proceeding to define majority rule along the lines of what Jobst calls the "uncovered set". (I think that's basically it, right?)
I reply:
You'd have to ask Jobst. But even if my definition of majority rule is the same as Jobst's definition of the uncovered set, then if that means that I defined majority rule along the lines of the uncovered set, that was purely accidental and coincidental.
You continue:
I have to object to the defeat strength part of your definition, in part on the behalf of cardinal pairwise. Winning votes is surely one way to define defeat strength, and it is not a bad one, but I do not accept it as the one and only definition
I reply:
I have never said that wv is the only way to define defeat strength. Perhaps you would like me to define defeat strength in all of the possible ways. Forgive me, but it seemed best to only define it in one way, and, if it's ok with you, I chose wv.
You don't say how you'd rather define the magnitude of a majority pairwise vote. Margins? You'd like to define the magnitude of a majority vote in terms of the majority and the minority? Most of us woiuld define the magnitude of a majority vote by the magnitude of the majority.
In fact, that fundamentally makes more sense than defining the magnitude of a majority vote in terms of the minority. It's normative :-) ...,even though I hadn't said that before.
Note that I don't say that you can't define it differently, but you must accept that it's ok if we don't all define it as you do.
You continue:
, and I doubt that very many other people will either [accept my definition of majority rule]
I reply
But why is that important to you, how many people accept my definition of majority rule.
For me, the value of my definition of majority rule is that it's used in my definition of offensive strategy. But my definition of offensive strategy isn't important, even to me. As you've no doubt heard, defensive strategy, the need for defensive strategy, is my strategy concern.
So maybe you could change what you're saying, and say that you object to my definition of majoritiy pairwise preference (MPP) and say that you want to define it in terms of margins, instead of being so arbitrary as to derfine the magnitude of a majority pairwise preference in terms of the majority that it refers to. What I said about MPV applies to MPP.
But what makes you think that it's important if people accept my definition of the magnitude of a majority pairwise preference, and, hence, my definition of defensive strategy.
The defensive strategy criteria aren't based on my definition of defensive strategy. Only their name is based on that definition. So are you fighting about a name? The matter of whether or not a method meets a criterion is unaffected by the criterion's name.
My definition of defensive strategy isn't something that I ask anyone to accept. It would never have occurred to me to ask anyone to accept it. It's just my way of saying what I mean when I speak of defensive strategy, when I tell what shouldn't be needed in its more drastic forms, and give to it a name.
So, sure, don't accept my definition of the magnitude of majority pairwise preferences, or my definition of defensive strategy. And don't accept my definition of majority rule, which means that you then won't accept my definition of offensive strategy.
Now, of course there's something that you don't like about my definition of majority rule: When Cardinal Pairwise elects Kerry in your example, instead of Bush, CP is violating majority rule, as I define majoritly rule. I stand by that definition and that conclusion that CP violates majorilty rule when it elects someone who has an un-outdone majority defeat.
But I'm not saying that CP shoudn't do that. Or, in your terminology, I'm not saying that it's un-normative for CP to do that :-) It's a good thing when CP elects Kerry in that example, a good thing that CP has and uses information other than the pairwise votes, when that can thwart offensive order-reversal, or when it can resolve a natural circular tie in a way that is fairer.
By the way, AERLO too is useful in natural circular ties. With AERLO, I'd have no strategic reason to not rank all the acceptable candidates in order of preference, instead of equal-ranking all of them in 1st place.
But, though I stand behind my definition of majority rule, and the conclusion that CP violates majority rule (for a good cause) when it elects Kerry in your example, I _don't_ stand by my definition of offensive strategy. According to my definition of offensive strategy, voters in CP who adjust their ratings intentionally to protect Kerry from offensive order-reversal by the Bush voters are themselves engaging in offensive strategy. That means that what they're doing is both offensive strategy and defensive strategy (because they're doing it to protect the CW, in violation of majority rule*). Is that nonsense? Sure. I said previously that I wasn't entirely satisfied with my definition of offensive strategy.
*But note that that isn't a problem for my defensive strategy definition, because defensive strategy is defined in terms of majority wishes, not majority rule.
CP provides a case in which my definition of offensive strategy gives an answer that is the opposite of what we expect offensive strategy to mean. I won't defend that definition of offensive strategy. It would be better to re-write it so that it won't give answers that aren't what we mean by offensive strategy. I might re-write it eventually, and anyone else is welcome to also.
But, for one thing, my definition of offensive strategy usually works fine, in keeping with what we expect offensive strategy to mean--its failure with CP is an exception. For another thing, offensive strategy isn't my main concern anyway. Need for defensive strategy is more important. So, in the meantime, till a better offensive strategy definition is written, I don't consider it a problem that there isn't a better one yet.
My point is that this isn't important.
You continued:
I think that the Smith set is the narrowest majority rule definition we can make that stands some chance of gaining general consensus.
I reply:
...except that the Smith set isn't a majority rule definition.
You continue:
Do you really think that it would be a majority rule violation to choose Kerry rather than Bush in the example above (in part I, that is)? I don't think so at all. But I think it would be if we accepted your recent definition.
I reply:
Yes. But that just means that Cardinal Pairwise (CP) protects the CW in spite of that being a majorilty rule violation due to falsified preferences. I define majoritly rule in terms of votes rather than preference, and that's how we all expect majority rule to be defined. For instance, we all probably agree that Pluralitiy meets the Majority Criterion: A candidate who is voted favorite by a majority should win. And majority rule should be defined in terms of pairwise votes, but not in terms of the preference intensities, expressed or otherwise.
But I agree that my offensive strategy definition is fouling-up when it says that those voters in CP who protect Kerry with their voted preference intensities are using both offensive and defensive strategies when they do so. They obviously aren't using offensive strategy in any meaningful sense.
The offensive strategy definition is wrong in that instance.
Mike Ossipoff
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