[EM] Some thoughts on Condorcet and Burial (re-send with small correction)

[Kristofer Munsterhjelm]

On 11/12/23 01:40, C.Benham wrote:
> 
> Kristofer,
> 
> I'm sorry for being a bit tardy in replying.

No problem; I haven't been the quickest lately either.

>> What is your definition of UMDT?
> 
> My definition of MDT is that if a set S of candidates are voted together 
> above all outside-S candidates on more than one third
> of the ballots, and all the members of S pairwise-beat  all the 
> outside-S candidates, then the winner must be a member of S.
> 
> My definition of UMDT is that if the winner T is a member of S, then it 
> must not be possible to make some outside-S candidate X
> the winner just by altering some ballots that already vote X above T.
> 
> Perhaps to be a bit more strict we can replace "altering" with ' further 
> down-ranking T on'.
> 
>>  1: A>C>B
>>  1: B>C>A
>>  1: C>B>A
>>
>>  C is the CW, then
>>
>>  1: A>C>B
>>  1: B>A>C <- burying C under A
>>  1: C>B>A
>>
>>  is a perfect tie and every candidate has equal chance of winning.
> 
> This example clarified for me that MDT  (and UMDT) refers to *more than* 
> a third (rather than exactly a third).
> 
> It is clear to me that MDT was meant to be analogous with Mutual 
> Majority (rather than "Mutual Half").
> 
> (A problem is that in English there is no word that means "more than a 
> third".)

Yes. What I had in mind here was {B, C} as the DMT set since it has 2/3 
support. C wins. Then a voter who prefers B to C buries C under A and 
B's chance of winning increases from zero to 1/3.

So clearly we can't protect against burial changing the winner from 
someone in the DMT set to someone else. My thought was that there may 
exist more subtle versions (with more voters) where the honest vote is 
something like X>B>C>A, and then when burying to X>B>A>C, that changes 
the winner from C to B even though that voter wanted X to win.

Your criterion handles that pretty well because it doesn't fail the 
method just because the buriers got someone they preferred. The winner 
has to change to someone not initially in the DMT set.

Do IRV and the Smith-IRV hybrids pass the stronger criterion you 
suggested, where it's impossible to make X the winner by any kind of 
modification, not just burial, by voters who prefer X?

-km