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<TITLE>Re: [EM] 0-info approval voting, repeated polling, and adjusting priors</TITLE>
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<DIV id=idOWAReplyText33935 dir=ltr>
<DIV dir=ltr><FONT size=2>Here's a two line description of the simplest method
that Jobst's proof applies to:</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>1. Ballots are ordinal with approval
cutoffs.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>2. In each round (and on each ballot) the approval
cutoff is adjusted by moving it next to (without moving past) the approval
winner from the previous round.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>This method converges because the approval winner of
the previous round is on the same side of the approval cutoff as in the
previous round, so his approval is the same in the new round, so the new
approval winner (if different from the old) must have greater
approval, which can only happen finitely many times.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>For practical purposes (i.e. not worrying about tied
ranks or tied approvals at some stage) this method is summable in
an (N+1)^3 array, where N is the number of real candidates, and the
"plus one" represents the virtual approval cutoff candidate (whether or not the
ballot is presented that way). </FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>The contribution of a ballot to the (i,j,k) entry of
this array is unity if candidate k is ranked ahead of the j side of
candidate i on the ballot, else the contribution is zero.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>This is the same array that is used in batch style
Approval DSV (Declared Strategy Voting) with strategy A, but it is used
differently.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>In both versions the first index i represents the
current approval winner. In DSV the second index j represents the current
runner-up. But in this new application, the second index represents the
approval winner of the previous round.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>That's why we need entries for the virtual approval
cutoff candidate: this candidate is taken as the previous winner in the
first round.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>This adds a new feature to the method:</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>If the methods converges on the virtual candidate,
then the winner should be picked by random ballot from among the set of
candidates that won at least one round.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>If no real candidate wins even one round, then ....
?</FONT></DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Another thought: why not use the "all winners lottery" even if the
limit of the sequence of winners is
not the virtual candidate?</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Is monotonicity too much to hope for?</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>It seems like Rob LeGrand, Kevin Venzke, and I briefly considered
this method a few years back, but then we didn't have the benefit of Jobst's
proof of convergence. Rob had a whole list of strategies A, B, C, D, E,
... for DSV Approval, and I think that this was one of them.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>This method is a limiting case of Jobst's idea where (in a mixture
of Jobst's and my formulation and notation) lambda is infinitesimal in the
weighted average</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr> L_(k+1) = lambda*L_k + (1-lambda)*L'_k
.</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>The other limiting case, where (1-lambda) is infinitesimal might
also be interesting to study, but I don't think it will be summable.
Furthermore, it will have to deal with ties in probabilities (and therefore more
frequent ties in ranks and approvals).</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Thanks to Jobst for the great proof insight under this subject
heading!</DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr>Forest</DIV>
<DIV dir=ltr> </DIV></DIV>
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