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

Forest Simmons fsimmons at pcc.edu
Wed Jul 30 13:18:23 PDT 2003

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

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.


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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