[EM] MMPO revisited

[Kristofer Munsterhjelm]

fsimmons at pcc.edu wrote:
> 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 .

Could lotteries be used to unify single-winner and multiwinner methods? 
The logic would be that in the long run, a council where each bloc is 
present in a proportion equal to the probability value, would reach the 
same decisions as a single-winner position where the candidate holding 
it is regularly changed according to those probability values.

That reasoning fails to account for dynamic effects, though, but it is 
interesting, because it's sort of the flipside to the idea of assembly 
by sortition.

If you're judging (A+B)/2 as 0.5(value of A) + 0.5(value of B), then the 
outcome will be centrist, however. Compared to the outcome of Random 
Ballot, you would get candidates close to the CW more often than 
candidates not... which I suppose would be useful for the lottery, but 
not so much for a multiwinner version.

The link between a lottery and a council would simply be that the value 
of (A+B)/2 is equal to the value of a council housing A and B. For 
MMPO/MMTD itself, the logic breaks a bit at number of winners = 1 
because you can use my scenario to get an undesirable result.

As there are different multiwinner methods, there would also be 
different lotteries. Some could have the logic of minmax Approval: 
"choose the council/lottery which pleases whoever it pleases the least, 
the most". That's not very proportional, but could have its uses. On the 
other end, you could have a majoritarian one which clusters around the CW.