[EM] Some FWD letters from Paul Dumais

[Bart Ingles]

Paul Dumais wrote:
>
> Bart Ingles wrote:
[snip]
> > In the first example, 45% of the voters prefer A over all other
> > candidates.  15% prefer B, and 40% prefer C.  The full ratings are as
> > follows:
> >
> > EXAMPLE 1:  Voters' private suitability ratings
> >
> >       Rating:
> >       100    80    60    40    20    0
> >       ----------------------------------
> > 45     A                          B  C
> > 15     B  C                          A
> > 40     C                          B  A
> > ---
> > 100 votes total
> >
> > Average sincere ratings:
> >
> > Candidate A = (45% x 100)              = 45.0 points
> > Candidate B = (15% x 100) + (85% x 10) = 23.5 points
> > Candidate C = (15% x 90) + (40% x 100) = 53.5 points
> >
> > Borda Count:
> >
> > Candidate A = 90 points
> > Candidate B = 115 points
> > Candidate C = 95 points
> >
> > Borda picks B as the winner based on rankings, although B has only half
> > the rating of the other two candidates.
> >
> > Note that under Approval Voting, the voters could vote as follows:
> >
> > 45   A
> > 15   BC
> > 40   C
> >
> > C would probably win, with around 55 votes, versus 45 for A and 15 for
> > B.  The A voters could prevent C from winning by voting for both A and
> > B, but are unlikely to for two reasons  First, A has enough votes that
> > they may think that A has a chance of winning.  Second, since they think
> > that B is a poor candidate, it will be unlikely that many of them will
> > be willing to vote for B the same as for their favorite.
> 
> Under my proposed borda count (Paul's borda count) unranked candidates
> split the available points (this is equivalent to giving 0.5 points for
> all tied pairings. It is similar to approval voting in that a candidate
> would vote A or BC or C. It is implied that A means A gets 2 points and
> B and C get 0.5 each. BC implies 2 for B and 1 for C. You will see that
> this will give a fair result in the example above. A: 110 B:72.5
> C:117.5. To ensure extra fairness, I've included a second step, which
> compares the top candidates only. For this example A and C are compared
> and C stil wins: 55:45.

Under either Borda or Nanson there is strong incentive for voters to
fully rank all candidates (even if the rankings don't always reflect the
voter's honest assessment of the candidates).  In the example above, if
the C voters refused to rank B in second place, A would pick up another
20 points.  I doubt that the C voters  would choose to do so.

As a result, B is still the likely winner under either Borda or Nanson,
with an average rating of 23.5 and a median rating of 10 points.  Note
that C has an average rating of 53.5 and a median of 90.

> 
> >
> > ----------------------------------------------------
> >
> > As bad as the Borda results were in the last example, it gets even
> > worse.  In the following example, B is considered totally unsuitable by
> > all of the A and C supporters.
> >
> > EXAMPLE 2:
> >
> >       Rating:
> >       100    80    60    40    20    0
> >       ----------------------------------
> > 45     A              C              B
> > 15     B                        C    A
> > 40     C              A              B
> > ---
> > 100 votes total
> >
> > Average sincere ratings:
> >
> > Candidate A = (45% x 100) + (40% x 50)              = 65.0 points
> > Candidate B = (15% x 100)                           = 15.0 points
> > Candidate C = (45% x 50) + (15% x 20) + (40% x 100) = 65.5 points


Note:  Median ratings:   A = 50, B = 0, C = 50


> >
> > Under Borda C should get 140, A should get 130, and B should get 30,
> > right?  The only problem is, since B is perceived as a weak candidate,
> > the A supporters are more concerned with defeating their major opponent,
> > C.  The A voters rank insincerely as:
> >
> > 45   A B C
> > 15   B C A
> > 40   C A B
> > ---
> > 100 votes total
> >
> > Now A has 130, C has only 95, and B has 75.  What should the C voters
> > do?  If they adopt the same strategy by voting CBA, they can knock out
> > A, but will be worse off with last choice B winning (as in the Borda
> > results for Example 1).  They may have to do so, though, in order to
> > discourage the other side from attempting the strategy again in the
> > future.
> >
> > Granted that such severe Mutual Assured Destruction tactics involving
> > every voter are unlikely.  However, it is possible for smaller numbers
> > of the A and C supporters to engage in the same tactic.  The sensible
> > strategy would be to try to defeat the major opponent, while being
> > careful not to give the election away to a "dummy" opponent.  The result
> > would be a tendency to equalize the scores of all three candidates,
> > resulting in a nearly random outcome.
> >
> > Note again that under Approval voting, voters could vote:
> >
> > 45  A
> > 15  BC
> > 40  C
> >
> > Here, the A and C voters have no incentive at all to vote for a second
> > candidate.  The B voters know that they have no chance of winning, but
> > can get a moderate improvement in outcome by including a vote for C
> > along with B.
> 
> Paul's Borda Count gives:
> A: 110 B:72.5 C:117.5

In this example, the A and C voters actually do have an incentive to
leave both opponents unranked, with the results you have shown.  But see
how much closer the results are in this case.  In fact, since there
appears to be some room to spare, I would expect a few of the A and C
voters to give reverse rankings, like:

25  AB
20  A
15  BC
15  C
25  CB

Now A = 97.5,  B = 97.5, C = 105

Here, the A and C voters are engaged in a game of "Chicken", with B
playing the part of the locomotive.  In doing so, they erase most of the
margin of error that should be present in this election.  Remember that
B had an average rating of 15, and a median of zero!

The likely Approval outcome was A=45, B=15, C=55.

Example 2 was not necessarily intended to show that Borda produces a
different winner, but that the incentives for tactical voting under
Borda can work in the wrong direction -- in this case obscuring what
should have been a convincing defeat for B.  Actually, I think Condorcet
has this problem as well.


> 
> Or if no strategic voting was done:
> 45 AB
> 15 B
> 40 CA
> 
> This gives: A:137.5 B:75 C:87.5
> after eliminating b: A:52.5 C:47.5
> 
> This means Borda is not immune to strategic voting, but so is approval
> voting. In any case approval voting (AV) is similar to borda count
> except AV gives one point to it's highest choices and 0 to all others.
> Borda count is the simplest and more accurate way to match sincere
> ratings.

Approval actually thrives on strategic voting.  In the examples above,
the Approval voters were all using best strategy.

Bart