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<blockquote type="cite"
cite="mid20040303145926.18822.62281.Mailman@geronimo.dreamhost.com">
<pre wrap="">Message: 1
Date: Thu, 04 Mar 2004 00:52:52 +1030
From: Chris Benham <a class="moz-txt-link-rfc2396E" href="mailto:chrisbenham@bigpond.com"><chrisbenham@bigpond.com></a>
...
To my mind, Approval does NOT satisfy Independence of Irrelevant Alternatives (IIA), ...
Initial two candidate election (with ratings out of ten).
01: A(9)>>B(1)
99: B(8)>>A(7)
B wins 99 to 1. Now we add a third candidate X ...
Same voters and initial 2 candidates, but with a third candidate added.
01:A(9)>>X(2)>B(1)
99:B(8)>A(7)>>X(1)
A wins 100 to 99. So adding a clone of A, which ALL the voters ranked last, changed A from a
1/100 loser to the winner.</pre>
</blockquote>
<br>
Chris,<br>
<br>
This might be a good argument against Approval, but I don't believe it
demonstrates a conflict with IIA (as defined by Arrow). There appear to
be two very different interpretations of IIA, as illustrated by the
following links:<br>
<br>
IIA, definition 1:<br>
<br>
"<strong><a
href="http://www.artpolitic.org/infopedia/in/Independence_of_irrelevant_alternatives.html"
class="printable" title="Independence of irrelevant alternatives">independence
of irrelevant alternatives</a>:</strong>
if we restrict attention to a subset of options, and apply the social
choice function only to those, then the result should be compatible
with the outcome for the whole set of options."<br>
<a class="moz-txt-link-freetext" href="http://www.artpolitic.org/infopedia/ar/Arrow">http://www.artpolitic.org/infopedia/ar/Arrow</a>'s_impossibility_theorem.html<br>
<br>
See also <a class="moz-txt-link-freetext" href="http://cowles.econ.yale.edu/P/cd/d11a/d1123-r.pdf">http://cowles.econ.yale.edu/P/cd/d11a/d1123-r.pdf</a> :<br>
"The constitution respects independence of irrelevant alternatives if
the social relative ranking (higher, lower, or indifferent) of two
alternatives A and B depends only on their relative ranking by every
individual."<br>
(There is a more formal and explicit definition in Section 2 of the
paper. This definition is specific to rank methods, but can be
generalized to cardinal methods by replacing "their relative ranking"
with "their rating".)<br>
<br>
IIA, definition 2:<br>
<br>
"According to IIAC, introducing another
candidate into an election should not change the winner -- unless that
candidate actually wins."<br>
<a class="moz-txt-link-freetext" href="http://www.electionmethods.org/Arrow.htm">http://www.electionmethods.org/Arrow.htm</a><br>
<br>
Definition 1 relates to how the outcome of a particular election might
change based on the vote-counting procedure employed, whereas
definition 2 relates to how the result of different hypothetical
elections, held under different circumstances with different
combinations of candidates, might differ (taking into account voter
strategy). I believe the definition used in Arrow's Theorem is the
first (although the second might actually be more useful and
meaningful).<br>
<br>
Applying these definitions to the above example, we start out with the
CR/Approval ratings:<br>
<br>
01: A(9) >> X(2) > B(1)<br>
99: B(8) > A(7) >> X(1)<br>
<br>
100 approve A, 99 approve B, and A wins. Following IIA definition 1, we
ignore X and obtain the result,<br>
<br>
01: A(9) >> B(1)<br>
99: B(8) > A(7)<br>
<br>
Still, 100 approve A while 99 approve B (as well as A), so there is no
change. But following definition 2, we consider what the result would
be if the election were held without candidate X. In this case the 99
voters who approve of both A and B would insincerely rate B
"unacceptable" because there are no worse candidates on the ballot:<br>
<br>
01: A(9) >> B(1)<br>
99: B(8) >> A(7) (insincere)<br>
<br>
So B wins, demonstrating the susceptibility of Approval to insincere
voting. However, if the vote split were really 99:1 the B supporters
would probably know in advance that they are a majority, so they would
also vote insincerely in the three-way contest,<br>
<br>
01: A(9) >> X(2) > B(1)<br>
99: B(8) >> A(7) > X(1) (insincere)<br>
<br>
B would win in either case, consistent with both majority rule and
sincere CR. Thus, it is not clear that this example shows an egregious
susceptibility to insincerity, and in any case it does not appear to
demonstrate violation of Arrow's (narrowly defined) IIA premise.<br>
<blockquote type="cite"
cite="mid20040303145926.18822.62281.Mailman@geronimo.dreamhost.com">
<pre wrap="">
One of my fundamental standards is that a method should perform reasonably when all the voters
vote sincerely (taking no account of how any other voters might vote).
A method should be able to cope with insincerity, but to perform reasonably it definitely shouldn't
DEPEND on insincerity.
</pre>
</blockquote>
This certainly seems like a reasonable criterion (though not one of
criteria of Arrow's theorem). In the above example sincere Approval
would select A with or without candidate X, whereas insincere Approval
would likely select B (as would sincere CR and majority rule). But
given that A and B both have similar, high ratings (7 and 8), all
voters would be satisfied with either outcome, so Approval appears to
perform "reasonably" in this circumstance. <br>
<br>
Consider also the following alternative scenario:<br>
<br>
49: A(10) > B(9) >> C(0)<br>
51: C(10) > B(9) >> A(0)<br>
<br>
Advance polling clearly identifies the two factions' preference
ratings, but does not clearly indicate who has the majority:<br>
<br>
49 +/-3: A(10) > B(9) >> C(0)<br>
51 +/-3: C(10) > B(9) >> A(0)<br>
<br>
Thus, under majority rule voters would insincerely rank B highest to
ensure that one of their preferred candidates wins:<br>
<br>
49: B > A > C (insincere)<br>
51: B > C > A (insincere)<br>
<br>
In this circumstance, Approval, CR, and insincere majority rule select
the candidate who has the overwhelming approval of all voters (B),
whereas sincere majority rule results in an outcome that satisfies only
about half the voters (C).<br>
<br>
A fundamental question with all of these types of comparisions is, by
what standard can you say that one outcome is more "reasonable" than
another? Relying on preference rankings to define the standard is
problematic because the statement "A > B" is so ambiguous. It could
mean that A has a sincere cardinal rating 1 point higher than B, or 10
points higher, or anything in between. In my view, "sincere CR" (e.g.
on a scale of 0 to 1, no quantization of rating levels) provides a
meaningful standard for DEFINING the objective of elections, even
though the complication of strategy makes CR an impractical election
method itself.<br>
<br>
Ken Johnson<br>
<br>
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