[EM] How to convert a set of CR ballots to a set of Approval ballots

[Forest Simmons]

Both amalgamation rules give the same result in the case of n=3, and
perhaps that is the only case that is really needed:

Each voter rates each candidate with a three, one, or two to
express definite approval, definite disapproval, or somewhere in between.

In this case the two slots not containing candidate X are the slots that
are amalgamated to yield the approval ballot.



On Wed, 30 Jul 2003, Forest Simmons wrote:

> I promised I would give the rationale, so here it is:
>
> If we were to go from resolution n to resolution two in one step, we would
> approve (i.e. amalgamate at the top) all of the candidates rated on the
> ballot above the front runner, and disapprove (i.e. amalgamate at the
> bottom) all of the candidates rated below the front runner. Whether or not
> we approved the front runner would depend on other considerations such as
> its rating relative to the second placer.
>
> In this method we conservatively amalgamate only two slots at a time
> instead of the sweeping amalgamation described above.
>
> Which two slots should be amalgamated?
>
> Any two strictly below or strictly above the front runner would be
> consistent with the above one step strategy, but we want to do better than
> that.
>
> In this method we amalgamate the two slots furthest from the front runner,
> and if the front runner is in the exact center, we use the highest scoring
> candidate X that is not in the center to break the tie, i.e. we amalgamate
> the two slots furthest from X.
>
> A refinement is possible.
>
> What we really want to do is amalgamate the adjacent slots that have the
> least chance of making us regret their amalgamation.  We would regret
> their amalgamation if at any subsequent stage the two front runners ended
> up in the same slot (but wouldn't have if we had not amalgamated).
>
> Suppose that we amalgamate the two adjacent slots whose leading candidates
> have the lowest sum of CR scores. In other words suppose that the leading
> candidates in slots k and k+1 have scores of x and y, respectively, and
> that x+y is the minimum in this regard.
>
> Since this sum is so small, at least one of the two slots is very unlikely
> to produce a front runner at a later stage, so there is little chance that
> this amalgamation will be regretted later.
>
> It may well happen that two pairs of adjacent slots will produce the same
> sum x+y, especially if this sum is zero due to empty slots on the ballot.
>
> In that case the tie can be resolved by going with the minimal sum pair
> closest to the end of the range that is furthest from candidate X, where X
> is the highest scoring candidate not in the exact center.
>
> It's hard to imagine how one could get any benefit from voting an
> insincere order without it backfiring.
>
> If one thinks he can get an advantage by voting only at the extremes of
> the original CR ballot, let him do so.  The final approval ballot will be
> equivalent to the original CR ballot, with no chance of an adjustment to
> take advantage of the information in the other ballots.
>
> Personally, I would feel very confident in rating the candidates in
> proportion to my respective utilities for them, and would trust the method
> to sort out my optimal approval strategy more than I would trust the
> information from any earthly pre-election polls.
>
> Forest
>
>
> On Tue, 29 Jul 2003, Forest Simmons wrote:
>
> > This method recursively converts a set of CR ballots with resolution n to
> > a set of CR ballots of resolution n-1, and stops at resolution two, i.e.
> > approval ballots.
> >
> >
> > We start with a set of Cardinal Ratings (CR) ballots on which every
> > candidate is rated by a whole number in the range one to n.
> >
> > Provisional CR scores are calculated for the candidates by summing or
> > averaging their ballot ratings.
> >
> > Then each ballot is modified to a CR ballot with resolution n-1 as
> > follows:
> >
> > Let X be the candidate with the highest provisional score whose rating on
> > this ballot is not precisely at the midrange value.
> >
> > If X is above midrange, then move all of the candidates except the lowest
> > level candidates down one level, i.e. decrement their ratings in such a
> > way that the lowest two levels are amalgamated.
> >
> > Otherwise lower only the top level, i.e. amalgamate the top two levels.
> >
> >
> > Now recursively convert these n-1 resolution CR ballots to resolution n-2,
> > etc. until they are completely converted to approval ballots.
> >
> > The approval winner is the method winner.
> >
> >
> > That's it.
> >
> > I'll explain the rationale behind this in another posting.
> >
> >
> > I like it best with Five Slot grade ballots.  Each voter gives an A to the
> > candidates that she is sure that she wants to approve in any case, and an
> > F to each candidate that she would approve in no case.  The undecided are
> > sorted into three piles ... leaning towards approval, leaning away from
> > approval, and not knowing which way to lean.
> >
> >
> > Forest
> >
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