[EM] WMA

[Kevin Venzke]

Hi Forest,

--- En date de : Jeu 29.4.10, fsimmons at pcc.edu <fsimmons at pcc.edu> a écrit :
> Three slot Bucklin is also known as
> MCA (Majority Choice Approval): if no
> alternative is "preferred" on more than fifty percent of
> the ballots, then the
> approval cutoff is lowered to include the middle slot.
> 
> Compare this with three slot WMA (Weighted Median Approval)
> based on the same
> ballots:
> 
> (1) The random ballot probabilities are computed from the
> submitted ballots.
> 
> (2) For each ballot b, if the total probability of the
> preferred candidates on
> ballot b is no more than fifty percent, then the approval
> cutoff is lowered to
> include the middle slot on ballot b.
> 
> In the three candidate case WMA and MCA give identical
> approvals.  But when
> there are four or more alternatives, WMA is less of a blunt
> instrument compared
> to MCA.  When there is no majority preferred
> alternative, under WMA the approval
> cutoff is only lowered on the ballots where there is not a
> good chance that the
> winner will come from among the preferred alternatives on
> those ballots.
> 
> As Chris Benham pointed out, this version of WMA satisfies
> the Participation
> Criterion, whereas MCA does not.  On the other hand,
> MCA is efficiently summable
> by precinct, whereas WMA is not.  
> ...
> MCA satisfies the FBC (Favorite Betrayal
> Criterion).   I'm not sure if WMA
> satisfies the FBC.  In other words, could raising
> one's true favorite from the
> middle slot to preferred status change the winner from
> someone else (compromise)
> with preferred status to a third alternative besides the
> recently raised favorite?

I'm having troubles seeing how WMA can satisfy FBC, monotonicity, or
participation. It seems to me that any mucking about you do with your
top slot (including just showing up to vote) could have completely
unpredictable consequences regarding how far down other voters are
considered to approve.

For monotonicity and participation: If X wins and then you make X
stronger, you could be strengthening other candidates. There's no
promise you're really "strengthening" X at all (at least with 
monotonicity).

For FBC: Suppose your preference order is X>Y and you change your vote
from X=Y to Y. It's possible that the loss of support for X causes
other Y-top ballots to stop having their threshold lowered, allowing
Y to win when previously someone else won.

Kevin Venzke