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).
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