# [EM] Approval Stable Winner: Back to the Drawing Board

Forest Simmons fsimmons at pcc.edu
Mon Mar 9 14:35:30 PDT 2020

```Here's an example that illustrates the problem:

26 A
24 C=A
25 C=B
25 B

First without power truncation MPO(A) = MPO(B)= 50=W(A)=W(B), MPO(C)=26.
W(C)=49.

The ratio of W(X) to MPO(X) is unity when X is A or B, but is 49.26 which
is much lager than unity when X is C.

So without power truncation X is the winner.

This violates Plurality because A for example has more first place rankings
than C has any kind of ranking.

With power truncation since all of the votes are at the two extremes (Top
when not Bottom) the method reduces to approval and candidates A and B are
tied for first place with 50 percent approval against 49 percent for C.

But the original spirit of the method was to ask, for example, how much
support would C have compared to the other candidates if C were the sitting
duck on the hot seat, i.e. the projected winner?

No doubt that W(C) would still be 49.  But would A's support be only
MPO(C), i.e. would A only be supported on by those voters that ranked A
above C?

It seems to me that those voters that voted A=C would continue to support
A, so A's support would beits full approval value, and similarly for B.

So in general, how can we give candidates like A full credit for continuing
support when ranked equal top with the target candidate without scuttling
the FBC  It works out OK in this example, but in general when a lower
ranked candidate is raised to equal top (like A) it might wreck C's chances
for winning without replacing C as the winner, i.e. could get in the way of
the FBC.

That's the problem. So, as I said, back to the drawing board!

On Mon, Mar 9, 2020 at 1:56 PM Forest Simmons <fsimmons at pcc.edu> wrote:

> The power truncation fixes the Plurality failure, but at the expense of
> sacrificing the original objective. I can see another way of fixing the
> Plurality problem, but at the expense of the FBC.  Perhaps we are up
> against another kiind of impossibility.
>
> Back to the drawing board.
>
>
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