Kristopher,
Thanks for your interest. Yes, MMTD also elects C in your scenario. But that
is no problem in the
context that I have in mind for MMTD, namely allowing all nominated lotteries
into the competition. In
this case an obvious lottery to include is (A+B)/2 . With this addition the
preferences become
100000: A > (A+B)/2
1: A = C > B>(A+B)/2
1: B = C > A>(A+B)/2
100000: B>(A+B)/2
with (A+B)/2 getting a 50% rating by all voters.
Then the max disappointment of this lottery being elected instead of A, B, or C
is 50000.5 which is the
smallest of the max disappointments. So (A+B)/2 is elected by MMTD.
As a matter of course, the benchmark lottery (random ballot) should be
included, but in this case that
lottery is practically identical with (A+B)/2 .
Here's another example:
x: A
y: B
We throw in the lotteries p*A+q*B for 0<p=1-q<1, in other words all of the
lotteries on the line segment
AB.
It turns out that the MMTD winner is the benchmark lottery (x*A + y*B)/(x+y).,
and the MinMax
disappointment exactly q*x=p*y = x*y/(x+y).
Another way to think of this example is an election concerning where to locate
some facility to be
shared by two communities A and B of populations x and y, resp.
The majority solution would be to locate it at the center of the larger
community.
The MMTD solution would be to locate it between the two communities at the
point
(x*A + y*B)/(x+y),
which minimizes the max disappointment compared with any other possible point,
assuming that
disappointment is proportional to geometrical distance.
Forest
----- Original Message -----
From: Kristofer Munsterhjelm
Date: Friday, April 9, 2010 2:58 pm
Subject: Re: [EM] MMPO revisited
To: [email protected]
Cc: [email protected]
> [email protected] wrote:
> > MMPO minimizes the maximum pairwise opposition, so in some
> sense tries to minimize the total
> > disappointment that results from the MMPO winner being elected
> instead of some other candidate.
> (...)
> > In general no matter who wins, there will be disappointed
> voters. Why not minimize the total
> > disappointment?
>
> MMPO has a horrible Plurality failure:
>
> 100000: A
> 1: A = C > B
> 1: B = C > A
> 100000: B
>
> and C wins.
>
> Does this problem also exist in MMTD?
>
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