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<TITLE>Re: [EM] 0-info approval voting, repeated polling, and adjusting priors</TITLE>
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<DIV id=idOWAReplyText30545 dir=ltr>
<DIV dir=ltr><FONT size=2>Let x, y, and z be positive integers such that
x+y+z=N, and max(x,y,z)<N/2, where N is the number of some large population
of voters, and the ordinal preferences are divided into three
factions:</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>x: A>B>C</FONT></DIV>
<DIV dir=ltr><FONT size=2>y: B>C>A</FONT></DIV>
<DIV dir=ltr><FONT size=2>z: C>A>B</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Further assume that the cardinal ratings of the middle
candidate within each faction are distributed uniformly, so that in
the first faction the cardinal ratings of B are distributed evenly between
zero and 100%.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Let (alpha, beta, gamma) be a "lottery" for this
election.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Then the number of voters that prefer A to this
lottery is given by the expression</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2> p(A) = x +
beta*z/(gamma+beta)</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Corresponding expressions for B and C are</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2> p(B) = y +
gamma*x/(alpha+gamma) and</FONT></DIV>
<DIV dir=ltr><FONT size=2> p(C) = z +
alpha*y/(beta+alpha)</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>If we set (alpha, beta, gamma) equal to</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2> (x+y-z, y+z-x, z+x-y)/(2*N)
,</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>then p(A)=p(B)=p(C)=N/2 , which means that none of the
candidates is preferred over the lottery by more than half of the
population.</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Isn't that interesting?</FONT></DIV>
<DIV dir=ltr><FONT size=2></FONT> </DIV>
<DIV dir=ltr><FONT size=2>Forest</FONT></DIV></DIV>
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