[EM] This might be the method we've been looking for:

[Jameson Quinn]

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.

Jameson

2011/12/9 Andy Jennings <elections at jenningsstory.com>

> Here’s a method that seems to have the important properties that we have
>> been worrying about lately:
>>
>> (1)     For each ballot beta, construct two matrices M1 and M2:
>> 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
>> rating to X.  Otherwise enter a zero.
>> IN row X and column y of matrix M2, enter a 1 if y is rated strictly
>> above x on beta.  Otherwise enter a
>> zero.
>>
>> (2)     Sum the matrices M1 and M2 over all ballots beta.
>>
>> (3)     Let M be the difference of these respective sums
>> .
>> (4)     Elect the candidate who has the (algebraically) greatest minimum
>> row value in matrix M.
>>
>> Consider the scenario
>> 49 C
>> 27 A>B
>> 24 B>A
>> Since there are no equal top ratings, the method elects the same
>> candidate A as minmax margins
>> would.
>>
>> In the case
>> 49 C
>> 27 A>B
>> 24 B
>> There are no equal top ratings, so the method gives the same result as
>> minmax margins, namely C wins
>> (by the tie breaking rule based on second lowest row value between B and
>> C).
>>
>> Now for
>> 49 C
>> 27 A=B
>> 24 B
>> In this case B wins, so the A supporters have a way of stopping C from
>> being elected  when they know
>> that the B voters really are indifferent between A and C.
>>
>> The equal top rule for matrix M1 essentially transforms minmax into a
>> method satisfying the FBC.
>>
>> Thoughts?
>>
>
>
> 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:
> 49 C
> 27 A
> 24 B=A
> Then A wins, right?
>
> 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.
>
> Am I wrong?
>
> ~ Andy
>
>
>
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