[EM] Comparative Effectiveness of Approval and Condorcet in the case of a three candidate cycle.

[Forest Simmons]

Russ brought up the issue of effectiveness of Approval.

I think that we are mostly in agreement now that Approval locks on to the 
CW fairly quickly when there is a CW.  "Quickly" can even mean during the 
first election if DSV is used, or if partial results are made available to 
the voters before most of them cast their approval ballots.

Suppose that we have a three candidate cycle.  How effective is Approval 
compared to Condorcet in this setting?

In this setting, Approval voters may have a hard time applying Strategy A, 
especially if all of the candidates appear to have nearly equal support at 
all ranks.

In this case Approval voters should ask themselves if their middle 
candidate is better or worse than half way between the other two 
candidates.  If better, then approve, otherwise, not.

In the borderline case, go with the decision of a friend, or wait for 
partial results to come out (if possible).

If none of these possibilities are available, flip a coin.  If the coin 
flip result gives you a bad feeling, go the other way.  Your subconscious 
is wiser than you think.

But let's consider the worst possible case: you have absolutely nothing to 
help you decide.  Then just approve your favorite only.  As we showed in a 
recent posting this is exactly as likely (in this zero info three 
candidate case) to work in your favor as approving both favorite and 
middle.

In fact, we showed that as long as you approved your favorite and did not 
approve the candidate you considered worst, then given that your ballot is 
pivotal, there is a two thirds probability that your approval ballot will 
tip the election outcomein a direction that you consider favorable. 
[Satisfaction of the Participation criterion guarantees that it cannot 
make the outcome worse.]

[If you make your decision on the basis of any information at all, this 
2/3 probability is improved drastically.]


So, by way of comparison, let's see if Condorcet can match this:


Suppose that your sincere preference ballot is A>B>C, and that there is a 
cycle among these candidates.  There are two possible directions for the 
cycle:

Case I.  A beats B beats C beats A.
Case II. (the reverse of case I): A beats C beats B beats A.


What is the setup that would put two of these candidates in a Condorcet 
near tie?

The two weakest defeat strengths would have to be within one of each 
other.

Case I.i  The strong defeat is A>B.

     Subcase I.i.a  The B>C defeat is equal to the C>A defeat.
                    In this subcase Condorcet gives the win to A.
                    Your ballot neither helps nor harms.

     Subcase I.i.b  The B>C defeat is stronger than the C>A defeat by one.
                    (Same result as previous case)

     Subcase I.i.c  The B>C defeat is one weaker than the C>A defeat.
                    In this subcase your ballot changes the winner from
                    candidate C to A, definitely in your favor.


Case I.ii  The strong defeat is B>C.

     Subcase I.ii.a  The A>B defeat is equal to the C>A defeat.
                     In this subcase your ballot changes the winner from
                     candidate B to A, in your favor.

     Subcase I.ii.b  The A>B defeat is one less than the C>A defeat.
                     After your ballot is taken into account B is still
                     the winner: no help, no harm.

     Subcase I.ii.c  The A>B defeat is one greater than the C>A defeat.
                     A is the winner before and after your ballot is
                     counted. No help, no harm.

Case I.iii   The strong defeat is C>A.
              In all three subcases of this case the two weak defeats are
              both increased by the same amount (one) so the winner C is
              not changed (no help, no harm).

Case II.i    The cycle is A>C>B>A and A>C is the strong defeat.

      Your ballot does not affect the result in any of the three subcases
      of Case II.i, because it does not change either of the two weak
      defeats ( C>B and B>A ) since they are both contrary to your ballot
      (still A>B>C).

Case II.ii   Cycle A>C>B>A and C>B is the strong defeat.

      Subcase II.ii.a  The A>C and B>A defeats are equal in strength.
                       Your ballot changes the winner from C to A.

      Subcase II.ii.b  The A>C defeat is one less than the B>A defeat.
                       The winner remains C.

      Subcase II.ii.c  The A>C defeat is one greater than the B>A defeat.
                       The winner remains A.

  Case II.iii   Cycle A>C>B>A and B>A is the strong defeat.

      Of the three subcases, the only one that your ballot improves is
      the one in which A>C is one weaker than C>B.  Your ballot improves
      the winner from C to B.


Of the eighteen cases, your ballot only improves the result in four cases. 
Of course, your favorite was already the winner in six of those cases, so 
no improvement was possible.  So taking that into account, we can say that 
your ballot improved the result in four of the twelve possible cases, 
about half as effective as Approval.

Of course we didn't consider the use of truncation in Condorcet.  But 
that's only fair, since the advantage of Condorcet over Approval is 
supposed to be that you can vote your sincere preferences without loss of 
voting power.

This little case by case study seems to show that this supposition is not 
true, at least in three candidate cycle case that we are considering here;
use of fully ranked ballots is less powerful than ballots that rank two of 
the three candidates equally (i.e. approval ballots).

Forest