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<DIV dir=ltr><FONT face=Arial color=#000000 size=2>Dear Adam (and other
interested parties),</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>The John and Jane dialogue must have given
you the wrong impression that my other discussions of the possibilities were
limited to the three candidate case. Please read them again with the idea
in mind that they apply to any finite number of candidates.</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>I won't quibble about whether an exhaustive
consideration of cases is more thorough than a course of simulations. They
are both valuable and complement each other, because each can give insights to
improve the next version of the other.</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>Before the dialogue, I considered two
cases: (1) Compromise is sure to beat Favorite. (2) Compromise is
not sure to beat Favorite.</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>So far these two cases exhaust the
possibilities, but that won't stop us from considering some subcases
later.</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>In case (1) DMC gives no incentive at all
for Favorite Betrayal, because approval of Compromise already
reinforces the Compromise>Favorite defeat to the max.</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>However, in the same case under Shulze,
Ranked Pairs, or River (whether margins or wv) it sometimes helps to betray
Favorite by ranking Compromise strictly ahead of Favorite. I'll give an
example below, for those that have never seen this before.</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>Near the end of my previous message
(in a part not quoted by Adam) I showed that (in the Bubble Sorted Approval
formulaton of DMC) only in (what I called) case 2d would Favorite Betrayal
payoff, and that case 2d is not only unlikely, but also very difficult to
trust the polls on. </FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT face=Arial size=2>(I would appreciate a similarly
thorough analysis of cases (1) and (2) from the wv
folks.)</FONT></DIV>
<DIV dir=ltr><FONT face=Arial size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Here is case 2d.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Favorite has more approval than Compromise (so is
seeded above Compromise), and Favorite gets defeated by someone seeded
above Compromise before being challenged (but not defeated) by Compromise.
</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>At this point we might regret that we
helped Favorite block<BR>Compromise from a chance of proceeding to the
top.</FONT></DIV>
<DIV dir=ltr><FONT size=2><BR>However, given that Compromise had less approval
than Favorite and that<BR>Favorite beat Compromise, we shouldn't expect
Compromise to have great<BR>chances of beating all the guys ahead of Favorite,
notwithstanding<BR>pre-race polls to the contrary.</FONT><BR></DIV>
<DIV dir=ltr>My further comment on 2d was this:</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr><FONT size=2>This last case (2d) would be a pretty flimsy excuse
for voting Compromise<BR>over Favorite. I suppose it could happen if your
preference of Favorite<BR>over Compromise were extremely weak, or if you had
extremely precise,<BR>detailed information in advance about voter
preferences that gave Compromise a sure<BR>chance of beating every candidate
that could beat Favorite despite<BR>Compromise having less approval than
Favorite and being unable to beat<BR>Favorite.</FONT><BR></DIV>
<DIV dir=ltr>The John/Jane dialogue only served to dramatize the difficulty of
convincing someone to rank Compomise over Favorite under case 2d (with three
candidates for simplicity). </DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Now here's the promised example (in tournament form) of a case
where wv voters would regret not having ranked Compromise ahead of Favorite,
even though Compromise did end up beating Favorite, pairwise.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>First the wv pairwise results without the Favorite betrayal.
Here F stands for Favorite, and C stands for Compromise, and the defeats
are expressed in the form A(65)D, meaning that A beat D by 65 to
something:</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>C(55)F(70)A(65)D(75)F together with A(60)C(20)D</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>After weakest defeats in their cycles, i.e. the ones of strength
20, 55, and 65, respectively, we are left with a beatpath D(75)F(70)A(60)C
. Candidate D wins under Beatpath, River, and Ranked Pairs.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Now suppose that 25 voters go back in a time machine and reverse
their F>C marks to C>F.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Then the C(55)F defeat becomes C(80)F, and all of the other defeats
retain their directions and wv strengths.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>This time the weakest defeats in their respective cycles are the
ones with strengths 20, 60, and 65.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Already by the time we remove the defeats of strength 20 and
60, candidate C is undefeated.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>If we proceed as in Ranked Pairs the respective defeats of
strengths 80, 75, 70, and 20 are set in place, with C as winner.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>In River we set in place the defeats of strength 80, then 70, and
then 65, which coincidentally yields the strongenst beatpath C(80)F(70)A(65)D
through the candidates.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>If I am not mistaken, River, Beatpath, and Ranked Pairs all agree
on C as winner.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>I hope that this earns some respect for DMC's Favorite Betrayal
resistance :-)</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>My Best,</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Forest</DIV></DIV>
<DIV dir=ltr><BR>
<HR tabIndex=-1>
<FONT face=Tahoma size=2><B>From:</B> Adam Tarr
[mailto:ahtarr@gmail.com]<BR><B>Sent:</B> Fri 9/9/2005 7:18 PM<BR><B>To:</B>
Simmons, Forest <BR><B>Cc:</B> election-methods-electorama.com@electorama.com;
Condorcet@yahoogroups.com<BR><B>Subject:</B> Re: [EM] RE: FBC comparison: WV,
margins, MMPO, DMC<BR></FONT><BR></DIV>
<DIV>On 9/9/05, <B class=gmail_sendername>Simmons, Forest</B> <<A
href="mailto:simmonfo@up.edu">simmonfo@up.edu</A>> wrote:<BR><BR>
<DIV>
<BLOCKQUOTE class=gmail_quote
style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">Adam
and Kevin have argued that Winning Approval tends to encourage Favorite
Betrayal more than Winning Votes does. <BR><BR>But a thorough examination of
the possibilities does not seem to support that view.</BLOCKQUOTE>
<DIV><BR>A "thorough examination" would really involve some sort of simulation
or aggregation of a whole host of reasonable scenarios; something neither you
nor I have done in any form. All we have are our gut instincts about the
likeliness of various scenarios, which disagree. Kevin has discussed
running a simulation to test this, which seems like a great idea.<BR></DIV><BR>
<BLOCKQUOTE class=gmail_quote
style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">When
winning votes is used as a measure of defeat strength, you know that if your
compromise does beat your favorite, then you can still help your compromise by
strengthening that defeat. In order to do this, you have to rank
your compromise strictly ahead of your favorite.</BLOCKQUOTE>
<DIV><BR>Let's examine the case you're hypothesizing here.<BR><BR>A priori,
assume our sincere preference is A>C>B. You said that compromise
beats favorite, so C>A. Since compromise doesn't already win, we know
B>C. Since the case is trivial (and hopeless) if B is the Condorcet
winner, we assume the cycle A>B>C>A.<BR><BR>Since you state a need for
favorite betrayal ("helping your compromise") here, we can further assume that
the weakest defeat is A>B, causing B to be elected.<BR><BR>So strengthening
C>A does nothing for our cause. What we actually need to do is WEAKEN
B>C or C>A, so that A>B is no longer the weakest defeat.<BR><BR>But a
priori, C>B and A>C are our sincere preferences. We are already
doing everything in our power to weaken those defeats.<BR><BR>So order-reversal
cannot help our case here. Winning votes is immune to this particular
favortie beytrayal scenario. Your scenario does not make sense, as I
understand it. Maybe I made a mistake in my analysis; if you believe so,
then please present an example. </DIV><BR>
<BLOCKQUOTE class=gmail_quote
style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">Summary:
This conversation shows that it would take a lot of confidence in a set of
unlikely (but possible) conditions to entice a rational voter into Favorite
Betrayal under DMC even if condition (3) is thought to be
likely. Furthermore, if condition (3) is known to be false, then
there is no Favorite Betrayal incentive at all.<BR><BR>Does that
help?</BLOCKQUOTE></DIV><BR>What would be more convincing to me would be to show
a reasonable scenario where wv has favorite betrayal incentive, but DMC does
not. I believe I have shown the opposite.<BR></DIV></BODY></HTML>