<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META content="text/html; charset=us-ascii" http-equiv=Content-Type>
<META name=GENERATOR content="MSHTML 8.00.6001.18783"></HEAD>
<BODY>
<DIV dir=ltr align=left><SPAN class=582152418-25062009><FONT color=#0000ff
size=2 face=Arial>I have a hard time reconciling "<FONT color=#000000 size=3
face="Times New Roman">Note that this could elect a Condorcet loser" and "It
fulfills Condorcet (by definition) ". </FONT></FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=582152418-25062009><FONT color=#0000ff
size=2 face=Arial><FONT color=#000000 size=3
face="Times New Roman"></FONT></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=582152418-25062009><FONT color=#0000ff
size=2 face=Arial><FONT color=#000000 size=3 face="Times New Roman">If the first
is true, the second cannot be, by, uhhh,
definition.</FONT></FONT></SPAN></DIV><BR>
<DIV dir=ltr lang=en-us class=OutlookMessageHeader align=left>
<HR tabIndex=-1>
<FONT size=2 face=Tahoma><B>From:</B>
election-methods-bounces@lists.electorama.com
[mailto:election-methods-bounces@lists.electorama.com] <B>On Behalf Of
</B>Jameson Quinn<BR><B>Sent:</B> Thursday, June 25, 2009 12:56 PM<BR><B>To:</B>
election-methods@lists.electorama.com<BR><B>Subject:</B> [EM] Condorcet/Range
DSV<BR></FONT><BR></DIV>
<DIV></DIV>I believe that using Range ballots, renormalized on the Smith set as
a Condorcet tiebreaker, is a very good system by many criteria. I'm of course <A
href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-January/014469.html">not</A>
the first one to propose this method, but I'd like to justify and analyze it
further.<BR><BR>
<DIV>I call the
system Condorcet/Range DSV because it can be conceived as a kind of Declared Strategy Voting
system, which rationally strategizes voters' ballots for them assuming that they
have correct but not-quite-complete information about all other voters. Let me
explain.</DIV>
<DIV><BR>I have been looking into fully-rational DSV methods using Range ballots
both as input and as the underlying method in which strategies play out. It
turns out to be impossible, as far as I can tell, to get a stable,
deterministic, rational result from strategy when there is no Condorcet winner.
(Assume there's a stable result, A. Since A is not a cond. winner, there is some
B which beats A by a majority. If all B>A voters bullet vote for B then B is
a Condorcet winner, and so wins. Thus there exists an offensive strategy. This
proof is not fully general because it neglects defensive strategies, but in
practice trying to work out a coherent, stable DSV which includes defensive
strategies seems impossible to me.) Note that, on the other hand, there MUST
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 simplifies things
drastically and I've never seen a real-world example of a larger set.) In the
Nash equilibrium, all three candidates have a nonzero probability of winning (or
at least, are within one vote of having such a probability). Voters are
dissuaded from using offensive strategy by the real probability that it would
backfire and result in a worse candidate winning. This Nash equilibrium is in
some sense the "best" result, in that all voters have equal power and no voter
can strategically alter it. However, it is both complicated-to-compute and
unnecessarily probabilistic. Forest Simmons <A
href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011028.html">has
proposed an interesting method</A> for artificially reducing the win probability
of the less-likely candidates, but this method increases computational
complexity without being able to reach a single, fully stable result. (Simmons
proposed simply selecting the most-probable candidate, which is probably the
best answer, but it does invalidate the whole strategic motivation).
<DIV><BR></DIV>
<DIV>There's an easier way. Simply assume
that any given voter has only near-perfect information, not
perfect information. That is, each voter knows exactly which candidates are in
the Smith set, but makes an ideosyncratic (random)
evaluation of the probability of each of those candidates winning. That
voter's ideal strategic ballot is an approval style ballot in which all
candidates above their expected value are rated at the top and all candidates
below at the bottom. However, averaging over the different ballots they'd give
for different subjective win probabilities, you get something very much like a
range ballot renormalized so that there is at least one Smith set candidate at
top and bottom. (It's not exactly that, the math is more complex, especially
when the Smith set is bigger than 3; but it's a good enough approximation and
much simpler than the exact answer).</DIV>
<DIV><BR></DIV>
<DIV>Let's look at a few scenarios to see how this plays out.</DIV>
<DIV><BR></DIV>
<DIV>First, the typical minimal Condorcet scenario. You have Va voters who say
A>B>C, and think on average that B is b% as good as A compared to C (that
is, if they rank them 70, 60, 20, then b=(60-20)/(70-20)=80%); Vb voters who say
B>C>A with C on average c% as good as B; and Vc voters who vote
C>A>B with A at a%. Without loss of generality, Va > Vb or Vc, but Va
< Vb + Vc (or A would be a Condorcet winner). The renormalized Range
tiebreaking scores are A=Va + a*Vc; B=Vb + b*Va; and C=Vc + c*Vb. What does that
mean?</DIV>
<DIV>* If all of a, b, and c are 50%, then the candidate with the most
exceptionally strong win or weak loss, wins (that is, if the two strongest wins
are farther apart than the two weakest losses, then the strongest win, otherwise
the weakest loss). </DIV>
<DIV>
<DIV>* If one of a, b, and c is near 100% where the others are near 0%, then
that candidate wins. You could say that the XYZ voters' opinion of Y is
acting as the tiebreaker for Y.</DIV>
<DIV>* In general, for honest
ballots, the winner is the candidate with the least
renormalized Bayesian regret. Assuming the effects of renormalization are
random, this will tend to be the candidate with the least Bayesian regret
overall.</DIV>
<DIV><BR></DIV>
<DIV>Note that this could elect a Condorcet loser. For instance, if you had
ballots A>B>D>>>C, B>C>D>>>A, and
C>A>D>>>B, (that is, each ballot rates candidates at 100, 99, 98,
and 0) then D is the Condorcet loser but has higher utility than any other
candidate, and wins. But this could only happen if there is a Condorcet tie for
winner. In general, I find the scenario pretty implausible, and the result still
optimal for that scenario. (Because I think such results are optimal, I advocate
using the Smith set and not the Schwarz set for renormalizing). </DIV>
<DIV><BR></DIV>
<DIV>How does this method do on other criteria? It fulfills Condorcet (by
definition) and is monotonic. For strategies which don't change the content of
the Smith set, it does very well on other criteria, fulfilling Participation,
Consistency, and "Local IIA". However, as the content of the Smith set changes,
it can fail all of those latter criteria - but only by moving *towards* the
renormalized utility winner, who is arguably the correct winner anyway. I
believe that, because of its construction, it will have relatively low Bayesian
Regret among Condorcet systems.</DIV>
<DIV><BR></DIV>
<DIV>How resistant is it to strategy? When the Smith set is unchanged, all
useful strategies are at worst asymptotically on the honest side of semi-honest
- that is, they only require ranking equal (or, for nearly all of the strategic
benefit, nearly equal) candidates who are not honestly equal. Moreover, I think
that the DSV construction of this system gives it excellent resistance to
real-world strategies. Unless you have more information than the agent which
strategizes your ballot, you simply vote honestly and allow the system to
strategize for you. Thus, you wouldn't be motivated to use strategy unless you
felt you knew not only the exact possibilities and chirality of the Smith set,
both before and after your strategizing, but also the probable winner from that
Smith set. Under realistic polling information, I think that such scenarios will
be rare; if you know of a possible Condorcet tie, then you will not generally
know much about the likely winners of this tie. </DIV>
<DIV><BR></DIV>
<DIV>(It may even be provable that if you assume voters exist in some kind of
continuous, unimodal distribution in ideology space, and motivate Condorcet ties
by having non-euclidean but continuous distance measures, then there will always
naturally exist enough "honest defensive votes" to make any strategy
backfire).</DIV>
<DIV><BR></DIV>
<DIV>Note that in regard to resisting simple strategies, this method is a
serious improvement over either Range or Approval. Because it is a kind of DSV,
it "does the strategy for you". So a candidate will gain no significant
advantage if their voters are more strategic (that is, more dichotomous and
better at evaluating the expected winners of the election) than other
candidates' voters. It allows for naive votes of many kinds, including
potentially "don't know" votes for certain candidates, simple approval-style
votes, simple ranking-style votes, and others, giving all approximately the same
power. And because it uses Range ballots as an input but encourages more honest
voting than Range, it enables the society to see (as an academic question) who
is the true Range winner, when that differs from the Condorcet winner. I believe
that, in successive elections, enough voters would reevaluate such candidates to
make one of them win in both senses.</DIV>
<DIV><BR></DIV>
<DIV>Nonetheless, I will present one scenario where strategy might be employed.
Say the candidates are Nader, Gore, and Bush, and assume (contrafactually) that
this is simple a matter of 1-dimensional ideology and that all voters agree that
the candidates are, left to right, nader..gore..........................bush
(that is, Nader and Gore in this scenario are considered much closer than Gore
and Bush). Assume also that there is some dearth of center-left voters who
prefer Gore to Bush only weakly. So the honest preferences are something
like</DIV>
<DIV><BR></DIV>
<DIV> Nader Gore
Bush</DIV>
<DIV>11% 100 99
0</DIV>
<DIV>40% 99 100
0</DIV>
<DIV>49% 0 1
100</DIV>
<DIV><BR></DIV>
<DIV>If all of the 11% of Nader voters dishonestly "bury" Gore under Bush, then
Nader wins with 50.6% (that is, 11% + (99%*40%)). However, if even 2 of the 40%
Gore voters honestly vote Gore, Bush, Nader - or EVEN if those 2% simply give
Nader 1 point instead of 99 points - then the strategy backfires and elects
Bush. Given the honest utilities assumed, the Nader voters would not use this
strategy if they thought this was even 1% likely. In the real election, of
course, there was in fact a non-negligible minority of honest
Gore>Bush>Nader voters. <SPAN style="FONT-SIZE: x-small"
class=Apple-style-span>(Gore voters were probably likelier to support Bush
second than Nader voters, and I've seen polls that say 1/4 of Nader voters
supported Bush second (!); 1/4 of Gore voters would be around 12%).</SPAN> So
this strategy would never have worked in real life, and in fact it requires an
artificial gap in the voter distribution right near the Gore>Bush>Nader
zone (extending on both sides of that zone, because for instance
Gore>>Nader>Bush votes hose the strategy
too).</DIV></DIV></DIV></BODY></HTML>