When someone pointed out to Borda that his method led to strategic order
reversals, he replied that he
only intended it for honest voters. Unfortunately, that's only half the
problem; Borda is highly sensitive to
cloning:
Assume honest votes:
80 A>B
20 B>A
Candidate A wins by Borda and any other decent method.
Now clone B:
80 A>B1>B2>B3>B4>B5>B6
20 B1>B2>B3>B4>B5>B6>A
B1 wins with a Borda score of 5*80+6*20=520 compared with A's score of 6*80=480
.
Range, which awards the winner to the candidate with the highest average rating
instead of the highest
average ranking, doesn't suffer from this problem, since ratings are not
constrained to spread out like
rankings.
In short, Range is the cardinal ratings analog of Borda, without the drastic
clone problem. There is still
an incentive to exagerate sincere ratings to the extent of collapsing to the
extremes, but not to the
extent of order reversals. Honest voting with Range would give perfectly
satisfactory results, unlike the
case with Borda.
But can we find a "Borda Done Right" method based on Rankings instead of
ratings?
Yes. We just need a natural way of converting rankings to ratings that
automatically takes clone sets
into account, rating their members near each other.
One way to do that is (for each candidate X) let p(X) be the percentage of
ballots that rank X in first
place. If X is replaced with a clone set {X1, X2, ...} then the sum
p(X1)+p(X2)+ ... will be the same as p
(X) was before the replacement. Furthermore, if X is moved up in the rankings
relative to Y (but no other
relative move) then p(X) will not decrease, and p(Z) will not increase for any
other candidate Z.
These two properties (clone consistency and monotonicity) of the "ballot
favorite lottery" p are the only
ones needed for the following construction and discussion. So the result will
apply for any other lottery
distribution p that is both clone consistent and monotone.
We do the transformation from rankings to ratings in two steps: first a
conversion to raw ratings, and
then a normalization. Since the normalization will preserve the monotonicity
and clone consistency, we
will concentrate our attention mostly on the raw ratings.
But just for the record, to normalize a raw ratings ballot, subtract the lowest
rating from each of the other
ratings and then divide them all by the highest resulting rating. For example
if (on some ballot) the raw
ratings for the respective candidates are 1, .8, .5, .3, and .2, first
subtract the lowest rating .2 fromo
each of the other numbers to get .8, .6, .3, .1, and 0, and then divide by
the largest of these, namely .8
to get
1, .75, .375, .125, and 0. This is the affine transformation that normalizes
the ratings to a scale of zero
to one.
The more interesting part is the conversion of rankings to raw range scores by
use of the lottery
distribution p. For a given ballot b and an arbitrary candidate X, the raw
score of X is the sum over all Z
ranked (on ballot b) equal to or behind (i.e. lower than) X, of the values
p(Z). In other words the raw
score of X is
p(X)+p(Z1)+p(Z2)+ ... where the sum is over all Z ranked below or equal to X on
ballot b.
The way to visualize this is the candidates (or their names) stacked up on top
of each other with the
highest ranked candidate at the top of the stack, where the spacing between the
candidates Z1 and Z2
is given by the value of p(Z1) where Z1 is the higher of the two candidates.
The total height of the
candidate X in this stack of names is the raw score of X. Since the
probabilities add up to unity, the
candidates ranked equal top will all have raw scores of unity.
Now suppose that X is replaced with a clone set {X1, X2, ...}, then in the new
"stack" of candidates the
clone set will precisely fill up the space p(X)=p(X1)+p(X2)+... that separated
X from the candidate ranked
immediately below X. This is what we mean when we say that the conversion is
clone consistent.
Now suppose that X moves up in the ranking one place by moving X up relative to
the other candidates
on some of the ballots. If the distribution p changes, then p(X) is the only
value that increases.
First let's consider the effect on the ballots where no swap was made: If all
of the candidates that lost
probability are ranked below X, then the raw score of X stays the same, because
whatever is subtracted
from the ones under X is added to the space immediately below X. In this
subcase some of the other
candidates' raw scores decrease, but none increase.
On the other hand if some of the candidates above X lose probability, then X
may well push some of the
other candidates upward in raw score, but only by the same amount that X's raw
score increases at
most. In either of these subcases, no other candidate's total raw range score
(over all such ballots) will
increase more than X's range score increases.
On the ballots where X moves up in the rankings, this change itself can only
increase X's raw score, and
then from there on the considerations are the same as in the previous case.
In summary, raising X in the rankings cannot increase any other candidate's
total raw score more than
the increase of X's total raw score. Therefore the conversion is monotone.
This conversion followed by the normaliztion described above is the complete
setup for "Borda Done
Right". The Range winner based on the normalized ratings after both steps of
the conversion is the
winner according to Borda Done Right.
I suggest that for the purest form, where complete rankings are required for
input, the distribution p
should be based on the ballot favorite lottery. On the other hand, when
truncations and equal rankings
are allowed, I suggest the use of the random approval lottery based on implicit
approval.
I emphasize the seemingly subtle point that the purpose of these lotteries is
only to define the values of
p, not to introduce any randomness into the outcome of this deterministic
method. For example, in the
case of the random ballot favorite lottery, p(X) is the number of ballots on
which X is ranked first divided
by the total number of ballots. No random drawings are necessary to determine
this number.
I would also like to point out that any use of range ballots that is resistant
to the "ratings inflation" that
makes range strategically equivalent ot approval ... any such use of range
ballots can also be applied to
these rankings that we have coverted to normalized range ballots.
Andy's chiastic approval is one such approach. Range based Bucklin fits into
the same general
scheme. It seems to me that finding other valuable uses of range style ballots
is a worthwhile endeavor.
DSV methods for conversion of Range ballots into approval ballots fall into the
same category of using
range ballots as inputs. It is exciting to me that we now know some monotone
ways of doing this. Any
such method could be adapted to ranked ballots via the conversion specified
above.
And don't forget that PR methods, like RRV, based on range style ballots, can
now be done with
rankings, thanks to the above conversion process.
That's about all I have time for right now, but I want to continue this thread
in the future.
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