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<a class="moz-txt-link-abbreviated" href="mailto:election-methods-electorama.com-request@electorama.com">election-methods-electorama.com-request@electorama.com</a> wrote:<br>
<pre wrap="">Message: 5
Date: Thu, 26 Feb 2004 01:47:40 -0500
To: <a class="moz-txt-link-abbreviated"
href="mailto:election-methods-electorama.com@electorama.com">election-methods-electorama.com@electorama.com</a>
From: Adam Tarr <a class="moz-txt-link-rfc2396E"
href="mailto:atarr@purdue.edu"><atarr@purdue.edu></a>
Subject: Re: [EM] Re: Election-methods digest, Vol 1 #517 - 3 msgs
Ken Johnson wrote:
</pre>
<blockquote type="cite">
<pre wrap=""><span class="moz-txt-citetags">></span>...
<span class="moz-txt-citetags">></span>
<span class="moz-txt-citetags">></span>On the other hand, it may be that the optimum-rating function looks more
<span class="moz-txt-citetags">></span>and more like a discontinuity as the number of voters increases, so I
<span class="moz-txt-citetags">></span>could be wrong - maybe CR does reduce effectively to Approval for large
<span class="moz-txt-citetags">></span>voting populations.
</pre>
</blockquote>
<pre wrap=""><!---->
My intuition is that that would be the case. Basically, it comes down to
this. After some, possibly messy, calculations, you will derive some
expected utility for voting for candidate B. If that expected utility is
positive, then you should maximize your utility by giving B 100. If that
expected utility is negative, you should maximize your utility by giving B
0. Only when that expected utility is zero (i.e. your feelings toward B
are exactly equal to your expectation of the election outcome) does a vote
between zero and 100 make sense - of course you can put B anywhere in that
case.
Your mention of "large voting populations" is crucial, since it allows me
to assume that the marginal utility for an additional ranking point given
to candidate B remains constant through the entire 0->100 range. On a
small committee, this may not be the case, so it's possible (I'm not sure)
that intermediate rankings make sense in that case.
...
-Adam</pre>
<br>
<br>
I did some "messy calculations" and convinced myself that the optimum
CR strategy is indeed identical to Approval, at least under two special
circumstances: (1) large voter population, and (2) if I have no
knowledge of how other people might vote. This doesn't entirely address
the issue I raised earlier, which considered situations where I have a
good idea of which candidates are more or less likely to get elected,
but I can kind of see how the more general analysis would turn out.
Basically, what it boils down to is that when the number of voters is
large, each candidate's probability of success is approximately a
linear function of my ballot rating for B. Hence, the statistical
"expectation value" for the winner's rating (based on my sincere rating
standard) is also linear, and if you look for a maximum of a linear
function, naturally you find yourself marching off towards plus or
minus infinity. So the optimum is at one of the limits of the valid
rating range. The point at which the slope of the expectation value
function goes from positive to negative is where my sincere B rating is
midway between my A and C ratings, equivalent to Approval.<br>
<br>
If I have prior knowledge of the candidates' chances of winning, then
the expectation value function will still be linear in my ballot rating
for B, but the optimal threshold dividing my "approved" vs "unapproved"
rating levels will probably not be midway between my A and C ratings.
The strategy gets more complicated, but I would still give B an extreme
rating. However, if there are few voters (or if I am part of a large
voting block), the expectation value function is actually nonlinear,
possibly admitting a maximum between the rating limits.<br>
<br>
Ken Johnson<br>
<br>
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