[EM] MMPO revisited

[fsimmons at pcc.edu]

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: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com

> fsimmons at pcc.edu 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?
>