<div>Yes, in standard game theory everyone would know the exact utility of the B supporters in each outcome.</div><div><br></div><div>Here, those utilities are hidden, so there is some incentive for the B supporters to lie and say they are indifferent between A and C.</div>
<div><br></div><br><br><div class="gmail_quote">On Mon, Dec 12, 2011 at 4:17 PM, <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Thanks for checking the details.<br>
<br>
In traditional game theory the rational stratetgies are based on the assumption of perfect knowledge, so<br>
the A faction would know if the B faction was lying about its real preferences. Even knowing that the<br>
other faction knew that they were lying they could still threaten to defect, and even carry out their threat.<br>
There is no absolute way out of that.<br>
<div><br>
----- Original Message -----<br>
From: Andy Jennings<br>
</div><div>Date: Monday, December 12, 2011 12:40 pm<br>
Subject: Re: [EM] This might be the method we've been looking for:<br>
</div><div>To: Jameson Quinn<br>
Cc: <a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>, <a href="mailto:election-methods@lists.electorama.com" target="_blank">election-methods@lists.electorama.com</a><br>
<br>
> You're right. I've drawn out the game theory matrix and the<br>
> honest outcome:<br>
> 49 C<br>
> 27 A>B<br>
> 24 B>A<br>
> is indeed the stable one, with A winning.<br>
><br>
> So the only way for B to win is for his supporters to say they are<br>
> indifferent between A and C and threaten to bullet vote "B".<br>
> Then the A<br>
> supporters fall for it and vote "A=B" to prevent C from winning.<br>
> B wins.<br>
><br>
> I wonder if this is sequence of events is likely at all.<br>
><br>
> ~ Andy<br>
><br>
><br>
><br>
> On Fri, Dec 9, 2011 at 2:31 PM, Jameson Quinn<br>
</div><div>> wrote:<br>
> > No, the B group has nothing to gain by defecting; all they can<br>
> do is bring<br>
> > about a C win. Honestly, A group doesn't have a lot to gain<br>
> from defecting,<br>
> > either; either they win anyway, or they misread the election<br>
> and they're<br>
> > actually the B's.<br>
> ><br>
> > Jameson<br>
> ><br>
> > 2011/12/9 Andy Jennings<br>
> ><br>
</div><div><div>> >> Here’s a method that seems to have the important properties<br>
> that we<br>
> >>> 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<br>
> beta rates X<br>
> >>> 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<br>
> 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)<br>
> greatest minimum<br>
> >>> 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<br>
> >>> 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<br>
> result as<br>
> >>> minmax margins, namely C wins<br>
> >>> (by the tie breaking rule based on second lowest row value<br>
> between B and<br>
> >>> 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<br>
> stopping C from<br>
> >>> 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<br>
> minmax into a<br>
> >>> method satisfying the FBC.<br>
> >>><br>
> >>> Thoughts?<br>
> >>><br>
> >><br>
> >><br>
> >> To me, it doesn't seem like this fully solves our Approval<br>
> Bad Example.<br>
> >> There still seems to be a chicken dilemma. Couldn't you<br>
> also say that the<br>
> >> B voters should equal-top-rank A to stop C from being elected:<br>
> >> 49 C<br>
> >> 27 A<br>
> >> 24 B=A<br>
> >> Then A wins, right?<br>
> >><br>
> >> But now the A and B groups have a chicken dilemma. They should<br>
> >> equal-top-rank each other to prevent C from winning, but if<br>
> one group<br>
> >> defects and doesn't equal-top-rank the other, then they get<br>
> the outright<br>
> >> win.<br>
> >><br>
> >> Am I wrong?<br>
> >><br>
> >> ~ Andy<br>
> >><br>
> >><br>
> >><br>
> >> ----<br>
> >> Election-Methods mailing list - see <a href="http://electorama.com/em" target="_blank">http://electorama.com/em</a><br>
> for list<br>
> >> info<br>
> >><br>
> >><br>
> ><br>
><br>
</div></div></blockquote></div><br>