<div dir="auto">Like the taxi cab metric it is asymmetrical ... the democrqtic cost of convering finish order FO into its opposite FO' Will in general be greater in onf the two directions ... just li like the cost of traveling from A to B may be greater in time or dollqrs than that of the return trip.<div dir="auto"><div dir="auto"><br></div><div dir="auto">There is a classic monograph on this tópic by a Soviet Mathematician entités "What Is Distance?"</div><div dir="auto"><br></div><div dir="auto">The Kendall-tau metric is insensitive to direction ... so is of no use to us ... besides its defective clone dependence.</div><div dir="auto"><br></div><div dir="auto">Using the top and bottom counts to de-clone Kendall-tau gave us the bonus of directional sesitivity.</div><div dir="auto"><br></div><div dir="auto">The simplest example is</div><div dir="auto"><br></div><div dir="auto">60 A>B</div><div dir="auto">40 B>A</div><div dir="auto"><br></div><div dir="auto">The two possible finish orders are the two faction orders A>B and B>A.</div><div dir="auto"><br></div><div dir="auto">The swap cost of converting A>B to B>A is the a×b' ... the For count of A times the Against count of B ... i.e. 60×60.</div><div dir="auto"><br></div><div dir="auto">The democratic cost of reversing B>A is b×a' the For count of B times theAgainst countvof A ...b40×40.</div><div dir="auto"><br></div><div dir="auto">Since 60^2 > 40^2 the democratically preferred order is A>B ... the one that is the most costlybor democratic difficult to reverse.</div><div dir="auto"><br></div><div dir="auto">Over the last dozen years I have given various uses for this swap cost metric ... and always the proof of its clone independence. Then I had better vision ... much easier to type these messages.</div><div dir="auto"><br></div><div dir="auto">I'll just give you the algebraic identity that is the essence of proof:</div><div dir="auto">(Sum a_k)(Sum b'_j) is equal to</div><div dir="auto">DoubleSum [(a_k) (b'_j)]</div><div dir="auto"><br></div><div dir="auto">This is in the context where candidates A and B are replaced by respective clone sets {A_k} and {B_j} and</div><div dir="auto">a=Sum a_k and b'=Sum b'_j</div><div dir="auto"><br></div><div dir="auto">Search theM archive under Swap Cost for more applications.</div><div dir="auto"><br></div><div dir="auto">There is no guarantee that this VFA application will prove useful ... I'm just giving you the opportunity to get in on the experiment if you want to.</div><div dir="auto"><br></div><div dir="auto">I'm 76 years old ...my memory and eyesight are fading ... I hope the EM Archives can survive the degeneration of civilization that seems almost inexorable in some ways.</div><div dir="auto"><br></div><div dir="auto">-fws</div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Feb 2, 2024, 1:34 PM Joshua Boehme <<a href="mailto:joshua.p.boehme@gmail.com">joshua.p.boehme@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><br>
Two questions came to mind...<br>
<br>
1. The cost function isn't symmetric. How do you translate it into a distance between two orderings, then? For example, if A is 70% for/30% against and B is 60%/40%, then the cost of going from A>B to B>A is 28% and vice versa is 18%.<br>
<br>
2. Once you narrow it down to the two most-distant orderings, how do you pick the final one? Yes, it's probably "obvious" to a human, but you can't write a law or program that way. Once we have a rule to formalize that, though, why wouldn't we just directly use it to create a "better than" relationship for all the orderings instead of relying on a two-step process?<br>
<br>
<br>
On 2/1/24 17:18, Forest Simmons wrote:<br>
> VFA abbreviates "Vote For and<br>
> Against"<br>
> <br>
> More than a decade ago I thought of using VFA syle ballot information to<br>
> declone Borda, Copeland, and Kememy-Young ..but none of them was suitable<br>
> for public proposal.<br>
> <br>
> For example, VFA information allows us to construct a clone independent<br>
> "Swap Cost Mettic" that can replace the clone dependents Kendall-tau<br>
> metric, thereby resulting in a clone free version of Kemeny-Young ...<br>
> which, in turn transforms it into an election method with acceptable<br>
> properties ... but only ncorporatimg the VFA information to create the<br>
> clone independent metric without being able to dispense with the voter<br>
> rankings ... a gasoline/ electric hybrid that cannot exist as a standalone<br>
> power source, so to speak.<br>
> <br>
> However, last night it occurred to me that we can use the soap cost metric<br>
> on the space of possible candidate finish orders to make a comceptually<br>
> simple (vsry simple in comparison with Kememy-Young) standalone election<br>
> method:<br>
> <br>
> Choose the finish order most distant from its opposite.<br>
> <br>
> This makes simse when you realize that the cost of a single trasposition of<br>
> a pair of candidates in the finish order is a measure of the combined<br>
> mental anguish that would result from reversing their first and last place<br>
> prferemces.<br>
> <br>
> To be precise, the seap cost of reversing a single preference of X>Y to.Y>X<br>
> in the finish order is the number of nallots that voted for X times the<br>
> number that voted against Y.<br>
> <br>
> If you were toouble the number of voters that voted for X or the number<br>
> that voted against Y,, you would double the total number of voters<br>
> disappoimted by the seap of the positions (in the finish order) of the<br>
> respective candidates A and B ... so the "cost" in seap cost is measured in<br>
> voter disappoiment.<br>
> <br>
> In our context the greater the voter disappointment cost of reversing a<br>
> proposed finish order FO o its opposite FO', the better FO is in contrast<br>
> to FO' ... which is why we chopse the finish order that would result in the<br>
> most disappointment if it were replaced by its opposite..<br>
> <br>
> An app that uses this method for making group decisions based solely on<br>
> first and last choices might catch on at the grass roots level. If so, it<br>
> would work even better in large groups where statistical laws would enhance<br>
> performance.<br>
> <br>
> What do you think?<br>
> <br>
> Is there a VFA organization somewhere in need of revitalization?<br>
> <br>
> -fws<br>
> <br>
> <br>
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</blockquote></div>