<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:times new roman, new york, times, serif;font-size:12pt"><DIV>Jameson,</DIV>
<DIV> </DIV>
<DIV>This Condorcet-Range hybrid you suggest seems to me to inherit a couple of</DIV>
<DIV>the problems with Range Voting. </DIV>
<DIV> </DIV>
<DIV>It fails the Minimal Defense criterion.</DIV>
<DIV> </DIV>
<DIV>49: A100, B0, C0</DIV>
<DIV>24: B100, A0, C0</DIV>
<DIV>27: C100, B80, A0</DIV>
<DIV><BR>More than half the voters vote A not above equal-bottom and below B, and yet</DIV>
<DIV>A wins.</DIV>
<DIV> </DIV>
<DIV>Also I don't like the fact that the result can be affected just by varying the resolution</DIV>
<DIV>of ratings ballots used, an arbitrary feature.<BR></DIV>
<DIV>I think it would be better if the method derived approval from the ballots, approving all</DIV>
<DIV>candidates the voter rates above the voter's average rating of the Smith set members.<BR></DIV>
<DIV> </DIV>
<DIV>"For strategies which don't change the content<BR>of the Smith set, it does very well on other criteria, fulfilling<BR>Participation, Consistency, and "Local IIA". "<BR></DIV>
<DIV> </DIV>
<DIV>The criteria you mention only apply (as a strict pass/fail test) to voting methods, not <BR>"strategies" (and have nothing to do with strategy).</DIV>
<DIV> </DIV>
<DIV>We know that Condorcet is incompatible with Participation (and so I suppose also with</DIV>
<DIV>the similar Consistency). I don't see how a method that fails Condorcet Loser can meet</DIV>
<DIV>Local IIA.</DIV>
<DIV><BR>"And because it uses Range ballots as an input but encourages<BR>more honest voting than Range,.."<BR><BR>That is more true of the "automated approval" version I suggested, and also it isn't</DIV>
<DIV>completely clear-cut because Range meets Favourite Betrayal which is incompatible</DIV>
<DIV>with Condorcet.</DIV>
<DIV><BR> </DIV>
<DIV>Chris Benham</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>Jameson Quinn wrote (25 June 2009) wrote:</DIV>
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<BR>I believe that using Range ballots, renormalized on the Smith set as a<BR>Condorcet tiebreaker, is a very good system by many criteria. I'm of course<BR>not<<A href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-January/014469.html"><FONT color=#810081>http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-January/014469.html</FONT></A>>the<BR>first one to propose this method, but I'd like to justify and analyze<BR>it further.<BR><BR>I call the system Condorcet/Range DSV because it can be conceived as a kind of<BR>Declared Strategy Voting system, which rationally strategizes voters' ballots for them assuming that<BR>they have correct but not-quite-complete information about all other voters.<BR>Let me explain.<BR><BR>I have been looking into fully-rational DSV methods using Range ballots both<BR>as input and as the underlying method in which strategies play out. It turns<BR>out to be
impossible, as far as I can tell, to get a stable, deterministic,<BR>rational result from strategy when there is no Condorcet winner. (Assume<BR>there's a stable result, A. Since A is not a cond. winner, there is some B<BR>which beats A by a majority. If all B>A voters bullet vote for B then B is a<BR>Condorcet winner, and so wins. Thus there exists an offensive strategy. This<BR>proof is not fully general because it neglects defensive strategies, but in<BR>practice trying to work out a coherent, stable DSV which includes defensive<BR>strategies seems impossible to me.) Note that, on the other hand, there MUST<BR>exist a stable probabilistic result, that is, a Nash equilibrium.<BR><BR>Let's take the case of a 3-candidate Smith set to start with. (This<BR>simplifies things drastically and I've never seen a real-world example of a<BR>larger set.) In the Nash equilibrium, all three candidates have a nonzero<BR>probability of winning (or at least, are
within one vote of having such a<BR>probability). Voters are dissuaded from using offensive strategy by the real<BR>probability that it would backfire and result in a worse candidate winning.<BR>This Nash equilibrium is in some sense the "best" result, in that all voters<BR>have equal power and no voter can strategically alter it. However, it is<BR>both complicated-to-compute and unnecessarily probabilistic. Forest Simmons has<BR>proposed an interesting<BR>method<<A href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011028.html"><FONT color=#810081>http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011028.html</FONT></A>>for<BR>artificially reducing the win probability of the less-likely<BR>candidates,<BR>but this method increases computational complexity without being able to<BR>reach a single, fully stable result. (Simmons proposed simply selecting the<BR>most-probable
candidate, which is probably the best answer, but it does<BR>invalidate the whole strategic motivation).<BR>There's an easier way. Simply assume that any given voter has only<BR>near-perfect information, not perfect information. That is, each voter knows<BR>exactly which candidates are in the Smith set, but makes an ideosyncratic<BR>(random) evaluation of the probability of each of those candidates winning.<BR>That voter's ideal strategic ballot is an approval style ballot in which all<BR>candidates above their expected value are rated at the top and all<BR>candidates below at the bottom. However, averaging over the different<BR>ballots they'd give for different subjective win probabilities, you get<BR>something very much like a range ballot renormalized so that there is at<BR>least one Smith set candidate at top and bottom. (It's not exactly that, the<BR>math is more complex, especially when the Smith set is bigger than 3; but<BR>it's a good enough
approximation and much simpler than the exact answer).<BR><BR>Let's look at a few scenarios to see how this plays out.<BR><BR>First, the typical minimal Condorcet scenario. You have Va voters who say<BR>A>B>C, and think on average that B is b% as good as A compared to C (that<BR>is, if they rank them 70, 60, 20, then b=(60-20)/(70-20)=80%); Vb voters who<BR>say B>C>A with C on average c% as good as B; and Vc voters who vote C>A>B<BR>with A at a%. Without loss of generality, Va > Vb or Vc, but Va < Vb + Vc<BR>(or A would be a Condorcet winner). The renormalized Range tiebreaking<BR>scores are A=Va + a*Vc; B=Vb + b*Va; and C=Vc + c*Vb. What does that mean?<BR>* If all of a, b, and c are 50%, then the candidate with the most<BR>exceptionally strong win or weak loss, wins (that is, if the two strongest<BR>wins are farther apart than the two weakest losses, then the strongest win,<BR>otherwise the weakest loss).<BR>* If one of a, b,
and c is near 100% where the others are near 0%, then that<BR>candidate wins. You could say that the XYZ voters' opinion of Y is acting as<BR>the tiebreaker for Y.<BR>* In general, for honest ballots, the winner is the candidate with the least<BR>renormalized Bayesian regret. Assuming the effects of renormalization are<BR>random, this will tend to be the candidate with the least Bayesian regret<BR>overall.<BR><BR>Note that this could elect a Condorcet loser. For instance, if you had<BR>ballots A>B>D>>>C, B>C>D>>>A, and C>A>D>>>B, (that is, each ballot rates<BR>candidates at 100, 99, 98, and 0) then D is the Condorcet loser but has<BR>higher utility than any other candidate, and wins. But this could only<BR>happen if there is a Condorcet tie for winner. In general, I find the<BR>scenario pretty implausible, and the result still optimal for that scenario.<BR>(Because I think such results are optimal, I advocate
using the Smith set<BR>and not the Schwarz set for renormalizing).<BR><BR>How does this method do on other criteria? It fulfills Condorcet (by<BR>definition) and is monotonic. For strategies which don't change the content<BR>of the Smith set, it does very well on other criteria, fulfilling<BR>Participation, Consistency, and "Local IIA". However, as the content of the<BR>Smith set changes, it can fail all of those latter criteria - but only by<BR>moving *towards* the renormalized utility winner, who is arguably the<BR>correct winner anyway. I believe that, because of its construction, it will<BR>have relatively low Bayesian Regret among Condorcet systems.<BR><BR>How resistant is it to strategy? When the Smith set is unchanged, all useful<BR>strategies are at worst asymptotically on the honest side of semi-honest -<BR>that is, they only require ranking equal (or, for nearly all of the<BR>strategic benefit, nearly equal) candidates who are not honestly
equal.<BR>Moreover, I think that the DSV construction of this system gives it<BR>excellent resistance to real-world strategies. Unless you have more<BR>information than the agent which strategizes your ballot, you simply vote<BR>honestly and allow the system to strategize for you. Thus, you wouldn't be<BR>motivated to use strategy unless you felt you knew not only the exact<BR>possibilities and chirality of the Smith set, both before and after your<BR>strategizing, but also the probable winner from that Smith set. Under<BR>realistic polling information, I think that such scenarios will be rare; if<BR>you know of a possible Condorcet tie, then you will not generally know much<BR>about the likely winners of this tie.<BR><BR>(It may even be provable that if you assume voters exist in some kind of<BR>continuous, unimodal distribution in ideology space, and motivate Condorcet<BR>ties by having non-euclidean but continuous distance measures, then
there<BR>will always naturally exist enough "honest defensive votes" to make any<BR>strategy backfire).<BR><BR>Note that in regard to resisting simple strategies, this method is a serious<BR>improvement over either Range or Approval. Because it is a kind of DSV, it<BR>"does the strategy for you". So a candidate will gain no significant<BR>advantage if their voters are more strategic (that is, more dichotomous and<BR>better at evaluating the expected winners of the election) than other<BR>candidates' voters. It allows for naive votes of many kinds, including<BR>potentially "don't know" votes for certain candidates, simple approval-style<BR>votes, simple ranking-style votes, and others, giving all approximately the<BR>same power. And because it uses Range ballots as an input but encourages<BR>more honest voting than Range, it enables the society to see (as an academic<BR>question) who is the true Range winner, when that differs from the
Condorcet<BR>winner. I believe that, in successive elections, enough voters would<BR>reevaluate such candidates to make one of them win in both senses.<BR><BR>Nonetheless, I will present one scenario where strategy might be employed.<BR>Say the candidates are Nader, Gore, and Bush, and assume (contrafactually)<BR>that this is simple a matter of 1-dimensional ideology and that all voters<BR>agree that the candidates are, left to right,<BR>nader..gore..........................bush (that is, Nader and Gore in this<BR>scenario are considered much closer than Gore and Bush). Assume also that<BR>there is some dearth of center-left voters who prefer Gore to Bush only<BR>weakly. So the honest preferences are something like<BR><BR> Nader Gore Bush<BR>11% 100 99
0<BR>40% 99 100 0<BR>49% 0 1 100<BR><BR>If all of the 11% of Nader voters dishonestly "bury" Gore under Bush, then<BR>Nader wins with 50.6% (that is, 11% + (99%*40%)). However, if even 2 of the<BR>40% Gore voters honestly vote Gore, Bush, Nader - or EVEN if those 2% simply<BR>give Nader 1 point instead of 99 points - then the strategy backfires and<BR>elects Bush. Given the honest utilities assumed, the Nader voters would not<BR>use this strategy if they thought this was even 1% likely. In the real<BR>election, of course, there was in fact a non-negligible minority of honest<BR>Gore>Bush>Nader voters. (Gore voters were probably likelier to support Bush<BR>second than Nader voters, and I've seen polls that
say 1/4 of Nader voters<BR>supported Bush second (!); 1/4 of Gore voters would be around 12%). So this<BR>strategy would never have worked in real life, and in fact it requires an<BR>artificial gap in the voter distribution right near the Gore>Bush>Nader zone<BR>(extending on both sides of that zone, because for instance Gore>>Nader>Bush<BR>votes hose the strategy too).<BR>-------------- next part --------------<BR>An HTML attachment was scrubbed...<BR>URL: <<A href="http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20090625/4b100ca1/attachment.htm"><FONT color=#0000ff>http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20090625/4b100ca1/attachment.htm</FONT></A>><BR><!--endarticle--><!--htdig_noindex-->
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