<div>You're right. I've drawn out the game theory matrix and the honest outcome:</div><div><span class="Apple-style-span" style>49 C</span><br style><span class="Apple-style-span" style>27 A>B</span><br style><span class="Apple-style-span" style>24 B>A</span><br style>
</div><div><font class="Apple-style-span" color="#222222" face="arial, sans-serif">is indeed the stable one, with A winning.</font></div><div><font class="Apple-style-span" color="#222222" face="arial, sans-serif"><br></font></div>
<div><font class="Apple-style-span" color="#222222" face="arial, sans-serif">So the only way for B to win is for his supporters to say they are indifferent between A and C and threaten to bullet vote "B". Then the A supporters fall for it and vote "A=B" to prevent C from winning. B wins.</font></div>
<div><font class="Apple-style-span" color="#222222" face="arial, sans-serif"><br></font></div><div><span class="Apple-style-span" style="color:rgb(34,34,34);font-family:arial,sans-serif">I wonder if this is sequence of events is likely at all.</span></div>
<div><span class="Apple-style-span" style="color:rgb(34,34,34);font-family:arial,sans-serif"><br></span></div><div><span class="Apple-style-span" style="color:rgb(34,34,34);font-family:arial,sans-serif">~ Andy</span></div>
<div><br></div><br><br><div class="gmail_quote">On Fri, Dec 9, 2011 at 2:31 PM, Jameson Quinn <span dir="ltr"><<a href="mailto:jameson.quinn@gmail.com">jameson.quinn@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
No, the B group has nothing to gain by defecting; all they can do is bring about a C win. Honestly, A group doesn't have a lot to gain from defecting, either; either they win anyway, or they misread the election and they're actually the B's.<div>
<br></div><div>Jameson<br><br><div class="gmail_quote">2011/12/9 Andy Jennings <span dir="ltr"><<a href="mailto:elections@jenningsstory.com" target="_blank">elections@jenningsstory.com</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div><div class="h5">
<div class="gmail_quote"><div><div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Here’s a method that seems to have the important properties that we have been worrying about lately:<br>
<br>
(1) For each ballot beta, construct two matrices M1 and M2:<br>
In row X and column Y of matrix M1, enter a one if ballot beta rates X above Y or if beta gives a top<br>
rating to X. Otherwise enter a zero.<br>
IN row X and column y of matrix M2, enter a 1 if y is rated strictly above x on beta. Otherwise enter a<br>
zero.<br>
<br>
(2) Sum the matrices M1 and M2 over all ballots beta.<br>
<br>
(3) Let M be the difference of these respective sums<br>
.<br>
(4) Elect the candidate who has the (algebraically) greatest minimum row value in matrix M.<br>
<br>
Consider the scenario<br>
49 C<br>
27 A>B<br>
24 B>A<br>
Since there are no equal top ratings, the method elects the same candidate A as minmax margins<br>
would.<br>
<br>
In the case<br>
49 C<br>
27 A>B<br>
24 B<br>
There are no equal top ratings, so the method gives the same result as minmax margins, namely C wins<br>
(by the tie breaking rule based on second lowest row value between B and C).<br>
<br>
Now for<br>
49 C<br>
27 A=B<br>
24 B<br>
In this case B wins, so the A supporters have a way of stopping C from being elected when they know<br>
that the B voters really are indifferent between A and C.<br>
<br>
The equal top rule for matrix M1 essentially transforms minmax into a method satisfying the FBC.<br>
<br>
Thoughts?<br></blockquote><div><br></div><div><br></div></div></div><div>To me, it doesn't seem like this fully solves our Approval Bad Example. There still seems to be a chicken dilemma. Couldn't you also say that the B voters should equal-top-rank A to stop C from being elected:</div>
<div>49 C</div><div>27 A</div><div>24 B=A</div><div>Then A wins, right?</div><div><br></div><div>But now the A and B groups have a chicken dilemma. They should equal-top-rank each other to prevent C from winning, but if one group defects and doesn't equal-top-rank the other, then they get the outright win.</div>
<div><br></div><div>Am I wrong?</div><span><font color="#888888"><div><br></div><div>~ Andy</div><div><br></div><div><br></div></font></span></div>
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</blockquote></div><br>