[EM] Proportional election method needed for the Czech Green party - Council elections

[fsimmons at pcc.edu]

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.  

Both methods satisfy Monotonicity, and both methods are as clone free as three
slot Range is, based on the idea that the truer the clones, the more likely they
will be rated the same, and when not rated the same they will be rated adjacently.

Both methods satisfy Independence from Pareto Dominated Alternatives, provided
that ties are broken by random ballot or by random ballot probabilities
conditioned on the tied alternatives.

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?

Thanks,

Forest