[EM] WMA

[Kevin Venzke]

Hi Forest,

--- En date de : Dim 2.5.10, fsimmons at pcc.edu <fsimmons at pcc.edu> a écrit :
> 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.

This concerns me or at least isn't completely clear to me. If my vote
was A>B and I change it to A=B, this increases B's random ballot
probability. Surely it necessarily decreases A's random ballot probability.
Or not? Do we not insist that it adds to 100%?

> 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.

Yes, and other candidates can have decreased 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.

But suppose the probability of an alternative rated strictly above W
decreases? Then other candidates could increase in approval. Assume for
example that on this ballot, W was not and will not be approved. Some
candidate previously unapproved, but preferred to W, could benefit.

> 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.

It seems to me that you are neglecting ballots that won't vote for W.

Kevin