<div dir="ltr">I always thought the "standard" way of defining runner-up was that it lived in the definition of the voting rule itself. In Arrow's definition, a social welfare function determines a single social preference ordering from a set of individual preference orderings, so it's whoever's second in that social ordering. Maybe that's begging the question, but the answer for most single-winner methods is pretty obvious / well-defined.</div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Jan 26, 2020 at 4:27 PM Forest Simmons <<a href="mailto:fsimmons@pcc.edu">fsimmons@pcc.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>Richard,</div><div><br></div><div>Thanks for your enthusiastic reply. I think it is a very good idea in the general context of "how do we define second choice?"</div><div><br></div><div>But what I'm looking for is in the limited context of a single winner election how o we decide who came closest to beating the actual winner? In other words, who turned out to be the greatest rival of the winner for the single seat of a single winner election?</div><div><br></div><div>We're not saying that this greatest rival should be the next candidate to be seated in a multi-winner election. <br></div><div><br></div><div>For example. in an approval election the candidate with the second greatest approval would be the chief rival of the approval winner by any reasonable standard, but would probably not be the winner of the next round in the multi-winner context because voters who approved this runner-up would have the weight of their ballots cut in half for the second round.</div><div><br></div><div>So it's not exactly what I was looking for, but very good related information!<br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Jan 26, 2020 at 10:57 AM VoteFair <<a href="mailto:electionmethods@votefair.org" target="_blank">electionmethods@votefair.org</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">On 1/25/2020 3:43 PM, Forest Simmons wrote:<br>
> We need a standard way of defining a second place candidate or<br>
> "runner-up" for single winner elections.<br>
> 1. One way is ....<br>
> 2. ....<br>
> 3. ....<br>
> 4. Any other ideas?<br>
<br>
Yes to number 4.<br>
<br>
Please take a look at:<br>
<br>
<a href="https://electowiki.org/wiki/VoteFair_representation_ranking" rel="noreferrer" target="_blank">https://electowiki.org/wiki/VoteFair_representation_ranking</a><br>
<br>
I'm not sure what your word "standard" means. But hopefully you intend <br>
to mean "fair." If so, that's what VoteFair representation ranking is <br>
all about.<br>
<br>
Specifically "VoteFair representation ranking" looks deeply into the <br>
ballot info to correctly identify which candidate/choice is most popular <br>
among the voters who are not well-represented by the winner of the first <br>
seat.<br>
<br>
And it does so in a way that appropriately reduces the influence of <br>
well-represented voters to the extent they exceed a 50% threshold.<br>
<br>
Forest, thanks for asking this important question.<br>
<br>
Richard Fobes<br>
<br>
<br>
On 1/25/2020 3:43 PM, Forest Simmons wrote:<br>
> Now that we have "de-cloned Borda" by changing the rankings to what we<br>
> could call "pseudo ratings," we need just one one more ingredient for<br>
> our PR framework:<br>
><br>
> We need a standard way of defining a second place candidate or<br>
> "runner-up" for single winner elections.<br>
><br>
> 1. One way is to re-run the election with the winner removed from the<br>
> ballots to see who the new winner is.<br>
><br>
> 2. Another is to see which loser gave the winner the greatest pairwise<br>
> opposition.<br>
><br>
> 3. Another is to see which candidate needs the fewest "plump" votes to<br>
> become the winner.<br>
><br>
> 4. Any other ideas?<br>
><br>
><br>
><br>
> Kristofer,<br>
><br>
> Thanks for your constructive comments. That first version left a lot of<br>
> room for improvement, so here goes a second attempt:<br>
><br>
> As you mentioned methods like PAV based on approval ballots and<br>
> versions of<br>
> Proportional Range Voting based on cardinal ratings style ballots<br>
> are not<br>
> naturally adapted to ordinal ranking style ballots.<br>
><br>
> In my first attempt at a general framework I basically said if you<br>
> want to<br>
> convert rankings to approval style ballots, just use implicit approval.<br>
> Obviously that leaves much to be desired. So the next approximation<br>
> would<br>
> be to give "equal top" rank full approval, while only half approval is<br>
> given to rankings strictly between top and bottom, which is what I<br>
> used in<br>
> my first attempt (although omitted from the partial quote below ?).<br>
><br>
> So I want to use this message to take care of this problem, i.e. how to<br>
> approximate ratings from rankings:<br>
><br>
> First let's review why Borda is inadequate. Borda assumes that ranked<br>
> candidates are equally spaced in utility. But this assumption is<br>
> incompatible with clone independence:<br>
><br>
> 40 A>B>C>D>E<br>
> 60 E>A>B>C>D<br>
><br>
> Assuming equal spacing (as in non-parametric statistics) we have<br>
><br>
> 40 A(4)>B(3)>C(2)>D(1)<br>
> 60 E(4)>A(3)>B(2)>C(1)<br>
><br>
> So A is the winner with a score of 4*40 + 3*60, beating out the<br>
> Condorcet<br>
> winner E whose total score is only 4*60, tied with the Pareto dominated<br>
> candidate B!<br>
><br>
> The Pareto dominated candidates B, C, and D artificially prop up<br>
> candidates<br>
> A and B to the point of taking the wind out of the ballot CW.<br>
><br>
> How do we fix this?<br>
><br>
> First we tally first place preferences or "favorite" scores for all<br>
> of the<br>
> candidates. In the above example A gets forty, E gets 60, and the<br>
> other<br>
> candidates get zero each.<br>
><br>
> Then we use these tallys to construct the random favorite probability<br>
> distribution: P(A)=40%, P(B)=0=P(C)=P(D), and P(E)=60%.<br>
><br>
> On any given ballot our estimated rating for candidate X will be R=L<br>
> /(H+L), where L is the probability that (on this ballot) a random<br>
> favorite will be ranked Lower (or unranked)) than X, and H is the<br>
> probability that a random favorite will be ranked strictly Higher<br>
> than X on<br>
> this ballot.<br>
><br>
> Notice that the highest ranked candidate will have H = 0. so that its<br>
> rating will be L/(0 + L) which is 1, or 100 percent. Similarly any<br>
> bottom<br>
> candidate on a ballot will have a value of L equal to zero, so its<br>
> estimated rating will be 0/(H + 0), which is zero.<br>
><br>
> If some candidate X on a ballot has the same values for L and H, which<br>
> means that a random favotite is just as likely to be ranked below X as<br>
> above X, then the estimated rating is given by L/(H+L) = L/(L+L), which<br>
> equals 1/2 or fifty percent.<br>
><br>
> So on any ballot from the first faction the estimated ratings of the<br>
> reaspective candidates are given by R(A) = 60/(0 +60), which equals 1 or<br>
> 100 percent. While R(E) = 0/(60+0), which equals zero. And R(B)<br>
> =R(C)=R(D)<br>
> which are all equal to 60/(40+60) or 60 percent.<br>
><br>
> Similarly on any ballot from the second faction in our example the<br>
> estimated ratings are given by R(E) =40/(0+40) or 100 percent, and<br>
> R(A) =<br>
> 0/(60+0) = 0, and R(X) for the remaining candidates is given by<br>
> 0/(100+0) =<br>
> 0.<br>
><br>
> So the score totals (over all ballots) are T(A) = 40*100% + 60*0, T(E) =<br>
> 40*0 + 60* 100%, and T(X) = 40*60% +60*0, (for each of the clones<br>
> of A).<br>
><br>
> In sum, E wins with a total of 60, followed by A with a total score<br>
> of 40,<br>
> and finally the (near) clone candidates that are Pareto dominated by A,<br>
> with 24 points.each.<br>
><br>
> (I say "near" clones because in this context where equal first and equal<br>
> bottom are allowed, if a candidate falls into one of those extremes on a<br>
> ballot and a clone doesn't, then that clone is only a near clone IMHO.)<br>
><br>
> In my next messaage I'll fix the other problems with my first<br>
> attempt at a<br>
> generalized frameworrk for adapting single winner methods to multiwinner<br>
> elections satisfying Proportional Representation.<br>
><br>
><br>
><br>
> Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de" target="_blank">km_elmet@t-online.de</a><br>
> <mailto:<a href="mailto:km_elmet@t-online.de" target="_blank">km_elmet@t-online.de</a>>> wrote:<br>
><br>
> > On 1/22/20 12:05 AM, Forest Simmons wrote:<br>
> > ><br>
> > > The Multiwinner Method I have in mind chooses the winners<br>
> sequentially.<br>
> > > It is based on the idea that ballots have an initial weight of<br>
> one, and<br>
> > > that as candidates supported by a ballot are added to the winners'<br>
> > > circle, the weight is reduced according to some rule designed to<br>
> > > diminish the influence of the voters who have already achieved some<br>
> > > level of satisfaction.<br>
> > ><br>
> > > At each stage in the election the new seat is filled by the<br>
> candidate<br>
> > > picked by the single winner method applied to the entire ballot<br>
> set with<br>
> > > the current ballot weights in force.<br>
> > ><br>
> > > How, in general, do we diminish the weight of a ballot? Perhaps the<br>
> > > simplest way is to make the current weight 1/(1+S) where S is the<br>
> > > current satisfaction obtained by comparing the ballot preferences<br>
> > > (whether ratings or rankings) with the winners elected so far.<br>
> As long<br>
> > > as the current satisfaction is zero, the weight remains at one since<br>
> > > 1/(1+0) is just one.<br>
> ><br>
> > A quick reply (been a bit busy lately): Approval methods need to<br>
> pass a<br>
> > weaker proportionality criterion than ranked methods. For<br>
> Approval, you<br>
> > just need to give X a seat if enough voters approve X, but Droop<br>
> > proportionality is nested: a vote can contribute to multiple solid<br>
> > coalitions at once.<br>
> ><br>
> > Thus I'm not sure basing a ranked proportional method on Approval will<br>
> > lead to a good outcome, at least not if that's not explicitly<br>
> taken into<br>
> > account.<br>
> ><br>
> > E.g. consider the "D'Hondt without lists" proposal from 2002. It<br>
> > combined reweighting with pairwise matrices, but I'm pretty sure it<br>
> > fails the DPC.<br>
><br>
><br>
><br>
> ----<br>
> Election-Methods mailing list - see <a href="https://electorama.com/em" rel="noreferrer" target="_blank">https://electorama.com/em</a> for list info<br>
><br>
</blockquote></div></div>
----<br>
Election-Methods mailing list - see <a href="https://electorama.com/em" rel="noreferrer" target="_blank">https://electorama.com/em</a> for list info<br>
</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail_signature"><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><b style="font-size:12.8px">Greg Dennis, Ph.D. :: Policy Director</b><div style="font-size:12.8px">Voter Choice Massachusetts</div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">e :: <a href="mailto:greg.dennis@voterchoicema.org" target="_blank">greg.dennis@voterchoicema.org</a><br><div>p :: <a href="tel:617.863.0746" value="+16177848993" style="color:rgb(17,85,204)" target="_blank">617.863.0746</a><br></div></div><div style="font-size:12.8px">w :: <a href="http://voterchoicema.org/" style="color:rgb(17,85,204)" target="_blank">voterchoicema.org</a></div><div style="font-size:12.8px"><br></div><div style="font-size:12.8px">:: Follow us on <a href="https://www.facebook.com/voterchoicema" style="color:rgb(17,85,204)" target="_blank">Facebook</a> and <a href="https://twitter.com/voterchoicema" style="color:rgb(17,85,204)" target="_blank">Twitter</a> ::<br></div></div></div></div></div></div></div></div></div></div></div>