<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<meta content="text/html;charset=ISO-8859-1" http-equiv="Content-Type">
</head>
<body bgcolor="#ffffff" text="#000000">
Sorry it's taken so long to respond -- that'll teach me to download
mail at a remote location. <span class="moz-smiley-s1"><span> :) </span></span>
<br>
<br>
Steve Eppley wrote:
<br>
<br>
<blockquote type="cite"><br>
Kemeny finds the "best" social ordering by finding
<br>
the ordering that maximizes the sum of sizes of majorities that agree
with it. That makes it vulnerable to clone manipulation. MAM is
equivalent
<br>
to analogous method that finds the "best" social ordering by using the
maxlexmax comparitor rather than the sum comparitor, so it's
independent of clones.
<br>
<br>
Here's an example to consider, involving Kemeny's
<br>
vulnerability to clone alternatives. (I hope I have the example
straight. I whipped it up fairly fast.)
<br>
<br>
9 voters, 3 alternatives:
<br>
4: A>B>C
<br>
3: B>C>A
<br>
2: C>A>B
<br>
<br>
There are 3 majorities:
<br>
6 voters rank A over B.
<br>
7 voters rank B over C.
<br>
5 voters rank C over A.
<br>
<br>
Nearly every method elects A here, yes? Kemeny and MAM pick
A>B>C as the social ordering. Kemeny picks A>B>C because
it agrees with 13 (the majority of 6 who voted "A>B" + the majority
of 7 who voted "B>C"). Note that the social ordering B>C>A
agrees with only 12 (the 7 "B>C" majority + the 5 "C>A"
majority).
<br>
<br>
(There's another definition of Kemeny that finds the social ordering
that maximizes the sum of pairwise preferences that agree with it,
rather than the sum of sizes of majorities that agree with it, but its
principle is nearly equivalent, and a similar example illustrates its
problem with clones.)
<br>
<br>
Now add in one or more clones of C. Each clone added in adds another
majority of size 5 to the Kemeny-best social ordering that ranks C over
A than to the Kemeny-best social ordering that ranks A over C. Thus,
adding a clone of C, say C', causes either B>C>C'>A or
B>C'>C>A to become the Kemeny-best social ordering, changing
the outcome from A to B. On informational grounds this is illogical,
since no information about A or B is gained when voters also rank
alternatives that are nearly identical to C. Worse, it creates
incentives to manipulate the outcome by strategic nomination, and it
could even lead to routine farces where each faction nominates as many
(clone) alternatives as they can find that will pairwise-beat the
alternative(s) that will pairwise-beat their favorite(s).
<br>
<br>
</blockquote>
This is a problem with Kemeny ordering -- and it's definitely illogical
-- but I don't think it's <b class="moz-txt-star"><span
class="moz-txt-tag">*</span>that<span class="moz-txt-tag">*</span></b>
serious. (heh, if that doesn't get a response, nothing will). If
candidate B were to use it to manipulate the election, he would not
only have to support a competitor (C) but a clone of a competitor, both
of which are preferred to A. If B's support were slightly less or C/C'
slightly more the cycle would break and B would lose outright -- and if
C's policies and platform were preferable to A to begin with, why
wouldn't competitor B take them as his own if he is that Machiavellian?
<br>
<br>
Of course if the clones C and C' were to join the race on their own,
they could influence the outcome in favor of B, though generally it
wouldn't be to their benefit so there would be little incentive to run
separate campaigns (though perhaps there might be an argument on who
would be the top of the ticket).
<br>
<br>
On the other hand, according to Michel Truchon (referring to Young and
Levenglick, which you mentioned in a previous email) -- <a
class="moz-txt-link-freetext"
href="http://citeseer.ist.psu.edu/cache/papers/cs/11707/http:zSzzSzwww.ecn.ulaval.cazSzw3zSzrecherchezSzcahierszSz1998zSz9814.pdf/truchon98figure.pdf">http://citeseer.ist.psu.edu/cache/papers/cs/11707/http:zSzzSzwww.ecn.ulaval.cazSzw3zSzrecherchezSzcahierszSz1998zSz9814.pdf/truchon98figure.pdf</a>
-- Kemeny satisfies a weaker version he calls local independence of
irrelevant alternatives (LIIA) <b class="moz-txt-star"><span
class="moz-txt-tag">*</span>and<span class="moz-txt-tag">*</span></b>
satisfies reinforcement (where two distinct groups have the same order,
this is the same as the consensus ranking for both groups together).
According to Truchon, Kemeny is the only rule that satisfies them both.
<br>
<br>
I'm not sure if this is true (not having the math required to prove it
either way), but it would certainly be an illogical result to have two
groups with the same preference order adding up to a different
preference order. I'm not sure if it would be possible to use it for
political manipulation -- some weird form of gerrymandering, perhaps --
but it's certainly a logical conundrum. And of course, according to
Arrow there is no such thing as a perfect system, we just have to pick
which flaws we can live with. <span class="moz-smiley-s1"><span> :) </span></span>
(BTW, I saw your mention of the reinforcement issue, I just think it
would be more difficult to influence an election by adding clone
candidates of a competitor than it would be for politicians to redraw
districts. My personal preference would be to have the voters choose
the electoral map they want at the same time they vote for the
candidate they want, that would keep politicians on their toes)
<br>
<br>
In addition, I like the fact that the Kemeny order is a good method for
reducing spam in search-engine results -- trying to push a website up
the rankings is somewhat analogous to strategic voting. (MAM is
probably pretty good, plus it's easier to compute, but local
Kemenization is computationally cheap as well).
<br>
<br>
BTW, I think I lost the email (or it's on my other computer) but did
you say that MAM satisfies the extended Condorcet criteria? If so, the
difference (in results, at least) between Kemeny and MAM must be very
small, and the major benefit would be the ease of calculation (not an
insignificant factor). It would be kind of fun to test both systems
with real-world examples -- not just voting, but spam prevention,
character recognition, and the like.
<br>
<br>
Mike
<br>
<a class="moz-txt-link-abbreviated" href="mailto:mrouse1@mrouse.com">mrouse1@mrouse.com</a>
<br>
</body>
</html>