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<TITLE>Re: [EM] 0-info approval voting, repeated polling, and adjusting priors</TITLE>
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<DIV id=idOWAReplyText41219 dir=ltr>
<DIV dir=ltr><FONT size=2>As usual, Jobst has given us lots of food for
thought.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>First I would like to compare Joe Weinstein's approval
strategy with its marginal version based on tie probabilities.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>The marginal version is to approve alternative X iff X
is more likely to tie for first place with an alternative that you prefer X to,
than with an alternative that you prefer over X.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Weinstein's own suggestion was to approve X iff an
alternative worse than X is more likely to win than a better
alternative.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>It seems to me that the marginal version is the
approval strategy that maximizes your probability of making your approval ballot
pivotal (in the desired direction). In other words, it is the max "voting
power" approval strategy. It is the one that minimizes the probability
that given an election where your ballot could have made a difference, it either
didn't make a difference or it made it in the wrong direction. In other
words, it minimizes the probability of voter regret, given the opportunity of
being tie maker or tie breaker.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>But most elections are not that close, which is why I
prefer to approve precisely the candidates that I believe deserve
my support. And if I like X better than Y and think that Y deserves my
support, then I think that X does also, so an approval cutoff is all I need to
decide, once I have ranked the alternatives. I vote to support,
not under the illusion that my vote will be the tie breaker.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Furthermore, it seems to me that Weinstein's original
suggestion is more robust than its marginal version. By "robust" I mean it
works well even in less than ideal conditions.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Next, what do we mean by probability of
winning?</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Suppose that voter preferences are known
precisely:</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>40 A>B>C</FONT></DIV>
<DIV dir=ltr><FONT size=2>35 B>C>A</FONT></DIV>
<DIV dir=ltr><FONT size=2>25 C>A>B</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>What is the probability that A will win?</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>That depends on the method.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>But what if the method itself incorporates (prior)
winning probabilities as part of the method?</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Where do we get those probabilities?</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>They could be subjective, but is their a better
way?</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>What if we take random samples (with replacement) of
size ten, say, from the ballot set and see how a bootstrap version of the method
works on those sets. Then we get the win and tie probabilities
empirically. The results might be mathematically predictable, so that the
actual samples wouldn't have to be taken, etc.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Gotta Go,</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>FWS</FONT></DIV></DIV>
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