[EM] WMA

[fsimmons at pcc.edu]

Kevin wrote …

>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.

I reply:  it probably doesn’t satisfy the FBC, but I think it would be difficult and risky to take advantage of 
any potential violation of FBC.

As for Monotonicity, let’s prove it in a more general setting to get more for our money:

The three slot version of WMA can be generalized to other Cardinal Ratings ballots as follows:

Let f be a continuous non-decreasing function that takes the interval [minRange, maxRange] onto the 
interval  [0,100].  For the case of WMA include the condition f(middle slot) = 50.

On each ballot b, for each rating level r in the interval [minRange, MaxRange], approve the alternatives on 
ballot b that are rated at level r iff the alternatives rated strictly above r on ballot b account for no more 
than
 f(r) percent of the random ballot probability.

Elect the alternative approved on the greatest number of ballots, breaking ties by invoking random ballot 
(or else the random ballot probabilities conditioned on the tied alternatives).

Proof of Monotonicity:

Suppose that only the winner W is rated more highly on some ballots.  Then if the random ballot 
probabilities change, only W has increased probability.

In particular, on each ballot b the probability of an alternative rated strictly above W’s rating cannot 
increase, so W will not decrease in approval.

Also, if  r  is less than the rating of  W, then the total probability of alternatives rated strictly above r 
cannot decrease, since W’s increase in probability completely absorbs decreases in probability from 
any of the  other alternatives.  Therefore, no alternative other than W can have an increase in approval.

Note that if desired, each ballot can have a different function f and a different lottery method L, as long as 
each f is a continuous non-decreasing function that takes the interval [minRange, maxRange] onto the 
interval  [0,100], and each lottery increases the probability of no alternative except the one that has a 
rating increase on one or more ballots.

As for Clone Independence.  This method has the same marginal clone independence that Range voting 
has.  If all clones are rated at the same level, then no problem.  Otherwise, the closer they are rated to 
the same level, the less the effect of cloning.

My Best,

Forest