<div dir="ltr"><div>Well, by the definition of a CWs, the CWs is preferred to each other candidate by more people than vice-versa, and that doesn't depend on how many dimensions there are.<br><br></div><div>With n dimensions, of course the dimensions might not all share a common median for the distribution, but, for the purpose of wv-like properties, I assume that they do, and that there's always a CWs.<br><br>But if the voters are uniformly-distributed, or if they're continuously, symmetrically distributed about that common median (I assusme that at least one of those is so) then, not only is the CWs preferred to each of other candidates by more voters than vice-versa, but s/he also has a pairwise preference-majority over each of the other candidates.<br><br></div><div>When I say "the median", I mean the common distribution-median for all of the dimensions.<br><br></div><div>Regarding the line that connects the median to a candidate who is away from the median, and regarding the plane that perpendicularly bisects that line, a majority of the points in the distribution are on the median side of that plane.<br><br></div><div>I suggest that the n-dimensional generalization of a "wing" is the set of candidates on one side of a plane that includes the median.<br><br></div><div>Of course, because the CWs has a pairwise preference majority over each of the other candidates, no candidate can have a preference majority over the CWs.<br><br></div><div>As I was saying, for the purpose of defining wv-like strategy properties, I stipulate that only one wing stratgegizes, and the other wing votes sincerely.<br><br>So, by these assumptions, with wv, or MMPO, or MDDTR, truncation from one side can't take the win from the CWs, because the truncators' candidate has a voted majority against him (The CWs is voted over hir by a majority), and the CWs doesn't have a majority against hir (because no candidate has a pairwise preference majority against hir, and no one is burying).<br><br></div><div>And if some candidate's preferrers bury against the CWs, making a voted pairwise majority against hir, and if all the people preferring the CWs to the buriers' candidate (those people are a majority, for the reason that I stated)<br></div><div>decline to vote the buriers' candidate over anyone, then the burier's candidate can't hava a majoriity over anyone, including the candidate they've insincerely ranked over the CWs.<br><br></div><div>So the wv's, MMPO's & MDDTR's automatic protection against truncation, and their deterrence of burial, apply regardless of the dimensionality.<br><br></div><div>Michael Ossipoff<br><br></div><div><br></div><div><br><br></div><div class="gmail_extra"><br><div class="gmail_quote">On Wed, Nov 9, 2016 at 10:57 PM, Michael Ossipoff <span dir="ltr"><<a href="mailto:email9648742@gmail.com" target="_blank">email9648742@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">(Replying farther down)<br><br><div><div class="gmail_extra"><div class="gmail_quote"><span class="">On Wed, Nov 9, 2016 at 10:07 PM, C.Benham <span dir="ltr"><<a href="mailto:cbenham@adam.com.au" target="_blank">cbenham@adam.com.au</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
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<div class="m_-7563802397097719283m_7872677193537689502moz-cite-prefix"><span>On 11/10/2016 11:48 AM, Michael
Ossipoff wrote:<br>
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<blockquote type="cite">But that doesn't change the fact that all
of my examples of wv's CWs "protection" guarantees had the CWs
preferred from both sides, and supported from one wing, the wing
opposite the truncating or burying wing.<br>
<br>
That's the "wv-like strategy" that I've been referring to.<br>
<br>
...even though wv has an additional anti-burial guarantee, or
even though its anti-burial guarantee is stronger and more
general.</blockquote>
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Mike,<br>
<br>
I'm not completely clear on the exact definition of this
property/criterion that you think is worth giving up compliance
with Mono-add-Plump<br>
and Plurality to have.<span><br>
<br></span></div></div></blockquote></span><div>Good question. When I previously said what I meant by "wv-like strategy", I assumed that no one is indifferent between the CWs and any other candidate.<br></div><div>...which means that the CWs has _lots_ of support from the preferrers of other candidates.<br><br></div><div>In fact, I assumed, without explicitly saying so, that voters & candidates were on a 1D spectrum, with 2 "wings" (sets of voters separated by the CWs), and that the truncation (innocent or strategic) or burial all came from one wing, so that one wing all unanimously ranked the CWs over the other wing's candidates.<br><br></div><div>So the CWs has a preference majority against everyone, and has a voted pairwise majority against all of the candidates of the strategizing wing.<br></div><div><br></div><div>I don't know how well that holds up with more dimensions, with Euclidean or city block distance.<br><br></div><div>Maybe the mathematicians can help with that. Forest?<br><br></div><div>In the meantime, maybe I should just say that "wv-like strategy" is only defined for 1D, with the above-stated assumptions as stipulations.<br><br></div><span class=""><div> <br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div bgcolor="#FFFFFF" text="#000000"><div class="m_-7563802397097719283m_7872677193537689502moz-cite-prefix"><span>
<blockquote type="cite">Yes, in the standard chicken-dilemma
example, MDDTR elects A, and that's a violation of the Plurality
Criterion. Try to forgive MDDTR for electing the most
favorite-popular candidate who isn't majority-beaten :^)</blockquote>
<br></span>
I'm afraid I find the justification "most favorite-popular
candidate who isn't majority-beaten" to be quite oblique and
arbitrary-sounding.<br></div></div></blockquote><div><br></div></span><div>Majority is a familiar notion. Losing to another candidate by a majority is a reasonable enough grounds for disqualification, if not everyone is.<br><br></div><div>Among the non-disqualified candidates, choosing the most favorite one sounds too natural to be called "arbitrary".<br><br></div><div>And the rule to elect the most favorite candidate who doesn't have someone else ranked over hir by a majority has uniquely many of the best properties. ...practical properties that make voting easier & make sincerity safer.<br><br> <br></div><span class=""><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div bgcolor="#FFFFFF" text="#000000"><div class="m_-7563802397097719283m_7872677193537689502moz-cite-prefix">
<br>
"Majority-beaten" can go away if we add a few ballots that just
plump for nobody, so big deal. </div></div></blockquote><div><br></div></span><div>Fine. So then I recommend that, in an MDDT election: If your candidate is particularly in danger of majority-disqualification, you should recruit as many voters as possible to plump for no one.<br></div><div><br></div><div>...or wait...Better yet, tell them to rank the candidates you like (and suggest that they should like too) over the ones you don't like.<br><br></div><div>But, whatever you do, get the vote out. Giving an incentive to get everyone to vote--Is that a bad thing? We'd have a big turnout. <br><br>And then, when one of those people shows up to vote, are they just going to say to to themselves, "He said that it would be in my best interest to come here & plump for no one."? Would that be in their best interest? Or might they realize that, having come to the polling place, it might be even better to preferentially rank the candidates whom they like more.<br><br></div><div>So, by all means, get the vote out.<br><br></div><div>Michael Ossipoff<br></div><div><br></div><div class="m_-7563802397097719283m_7872677193537689502moz-cite-prefix"> <br>
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