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<TITLE>Re: [EM] 0-info approval voting, repeated polling, and adjusting priors</TITLE>
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<DIV dir=ltr>Jobst wrote ...<BR></DIV>
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<P><FONT size=2>Dear Forest,<BR><BR>I'm not sure what you mean by the red marble
thing or how it clarifies<BR>the meaning of the priors in zero-info
strategy.<BR><BR></FONT></P>
<P><FONT size=2>I reply:</FONT></P>
<P><FONT size=2>I'm not sure either. </FONT><FONT size=2>I'm stabbing around in
the dark looking for some idea to make the probabilities more definitely
meaningful, and as a child I enjoyed playing with marbles ;-)</FONT></P>
<P><FONT size=2>I think that if we want your convergence proof to work, we have
to assume that the voters only know who won each round, and not the approval
amounts. At any rate, if we cannot solve the problem with this simplifying
assumption, then there is no hope in the general case.</FONT></P>
<P><FONT size=2>Here's a method where the probabilities have more definite
meaning:</FONT></P>
<P><FONT size=2>A whole number N > 1 is announced before the
process begins, and an empty bag is made available.</FONT></P>
<P><FONT size=2>The number N is the number of approval polls to be
taken before deciding the winner.</FONT></P>
<P><FONT size=2>After each of the polls, a marble with the name
of the candidate with the most approval in that poll is placed
into the bag. [I had to get marbles into this somehow.] Approval
scores are kept secret. If two candidates tie in approval, the tie is broken by
a coin toss before announcing the poll winner. The occurence of a tie
is not mentioned.</FONT></P>
<P><FONT size=2>After all N of the approval polls are completed, a marble is
drawn at random from the bag to determine the grand winner.</FONT></P>
<P><FONT size=2>Before we treat the case N=100, how about the case N=2
?</FONT></P>
<P><FONT size=2>Of course you would use your prior probabilities for the
first poll. And once you knew the winner, A , of the first poll,
you would adjust (upward) the probability of candidate A being the winner
of the next (and final) poll..</FONT></P>
<P><FONT size=2>It seems like Bayes should tell us how much to adjust this
probability.</FONT></P>
<P><FONT size=2>And how about the prior probabilities of ties? Should
these be adjusted even though we have no information about ties or relative
approval scores from the first poll?</FONT></P>
<P><FONT size=2>On the one hand we might want to adjust (downward) candidate j's
chances of being in a tie. On the other hand we don't know for sure that
the first round wasn't decided by a coin toss.</FONT></P>
<P><FONT size=2>Can we solve even this simple case? Throw in independence
if it will help.</FONT></P>
<P><FONT size=2>Forest</FONT></P>
<P><FONT size=2></FONT> </P>
<P><FONT size=2></FONT> </P></DIV>
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