[EM] Pairwise Rated Benchmarks

[Bart Ingles]

I have been advocating the use of rated sincere-voter scenarios for two
purposes:

1)  To allow setting up hypothetical situations where there is a
reasonable chance of predicting how voters will behave under various
actual methods when tactics are considered, and thus of predicting the
outcome under different methods, without necessarily saying which
outcome is the correct one.

Under Approval voting, especially, it would be difficult to predict how
voters would behave solely on the basis of information present in
rankings.  There are also situations in all of the ranked methods where
information about the strength of voters preferences is needed to be
able to predict how they will actually vote (when it would be reasonable
to use order-reversal, etc.)

For this purpose, no benchmark is needed for sincere ratings to be
useful.


2)  To allow for a determination of who "should" win.  This implies some
sort of standard or benchmark.  I doubt whether there will be much
agreement on which standard would be correct, but the prospects are no
worse than for agreement on an actual election method.  It seems to me
that if we are going to argue about which election method chooses the
"right" winner, there must be a way to define which winner is the right
one independent of the method.  Since we have the sincere ratings, there
is no need to worry about things like resistance to strategy, and we
should be able to come up with a more accurate standard than is possible
in a real election method -- shouldn't we?

I have used Average Ratings to argue that election methods using ranked
ballots are capable of bad results in some situations.  While I believe
Average Ratings was good enough for the situations where I used it, I
can understand someone questioning its use as an overall standard.  One
specific objection was that Average Ratings might give extreme voters
more power than is warranted.

I proposed limiting the maximum rating difference that would be
recognized between a voter's choices, but that doesn't do anything about
the range between highest and lowest choice.  If you limit the
difference there, what do you do about intermediate choices?

If you use a pairwise method as a benchmark, you can place a cap on the
rating difference between any two candidates, without worrying about
what to do with other candidates.  Below I show a sincere rated
scenario, along with the actual results using Condorcet, rated pairwise,
and a rated pairwise method where all pairwise votes are limited to 30
on a scale of 0-100.

The ratings assume absolute ratings, clipped to a range of 0.00-1.00
where 1.00 is fully qualified for office and 0 is no more qualified than
an average person chosen at random (I had to pick some basis for
absolute ratings).


A.  SINCERE SCENARIO:

      1.00 .90                       .10  0
45      A                              B  C
15      B   C                             A
40      C                              B  A


B.  RESULT UNDER PLAIN CONDORCET:

A:B  =  45 : 55
A:C  =  45 : 55
B:C  =  60 : 40

B > C > A;  B wins.


C.  RESULT UNDER PURE PAIRWISE RATINGS:

A:B  =  (45*0.90) : (15*1.00) + (40*0.10)  =  40.5 : 19
A:C  =  (45*1.00) : (15*0.90) + (40*1.00)  =  45 : 53.5
B:C  =  (45*0.10) + (15*0.10) : (40*0.90)  =   6 : 36

C > A > B;  C wins.


D.  RESULT WITH PAIRWISE RATINGS DIFFERENCE CAPPED AT 30% OF MAXIMUM
VALUE:

A:B  =  (45*0.30) : (15*0.30) + (40*0.10)  =  13.5 : 8.5
A:C  =  (45*0.30) : (15*0.30) + (40*0.30)  =  13.5 : 16.5
B:C  =  (45*0.10) + (15*0.10) : (40*0.30)  =     6 : 12

C > A > B;  C still wins.