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<TITLE>Re: A class of ballot set with "unbeaten in mean lotteries."</TITLE>
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<DIV id=idOWAReplyText27794 dir=ltr>
<DIV dir=ltr><FONT size=2>The following lottery method is easier to explain in
terms of ratings (range ballots), but can (and should) be adapted to rankings
(ordinal ballots) by modifying the following definition.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Definition 1: Lottery L1 beats
lottery L2 on a given range ballot</FONT></DIV>
<DIV dir=ltr><FONT size=2> </FONT></DIV>
<DIV dir=ltr><FONT
size=2>
iff </FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>the L1 weighted average of the ratings (on the given
ballot) is greater than the L2 weighted average of the ratings on that
ballot.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Definition 2: The lottery L1 pairwise beats
the lottery L2 </FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT
size=2>
iff </FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>the number of ballots on which L1 beats L2 is greater
than the number of ballots on which L2 beats L1.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>The method:</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>1. Let alpha be the set of approval values of
the alternatives (candidates).</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>2. For each x in alpha let L(x) be the random
ballot lottery over the set of alternatives that have an approval of x or
greater.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>3. Let x be the smallest number in alpha such that for
all y greater than x, the lottery L(x) is not beaten pairwise by
L(y).</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>4. The winning lottery is L(x).</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr>
<DIV dir=ltr><FONT size=2>The idea is to have a random ballot lottery based on a
top segment of the approval list, and to have that segment extend down
the approval list as far as possible without having a pairwise preference for a
shorter such segment.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV></DIV>
<DIV dir=ltr><FONT size=2>By convention no lottery is pairwise beaten by an
undefined lottery, so if no other lottery wins, then L(MaxApproval) wins, i.e.
in that case the winner is chosen by random ballot from the alternatives tied
for most approval.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>I believe that this method is monotone, clone-proof,
and Independent from Pareto Dominated Alternatives.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>I'll post some examples later, when I get the
time.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Forest</FONT></DIV>
<DIV dir=ltr> </DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV></DIV>
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