[EM] RE: FBC comparison: WV, margins, MMPO, DMC

Simmons, Forest simmonfo at up.edu
Fri Sep 9 17:41:40 PDT 2005

James has asked for evidence that "winning votes" (as a measure of defeat strength) gives more incentive for insincere rankings than "winning approval" does. 
Adam and Kevin have argued that Winning Approval tends to encourage Favorite Betrayal more than Winning Votes does.
But a thorough examination of the possibilities does not seem to support that view.
When winning votes is used as a measure of defeat strength, you know that if your compromise does beat your favorite, then you can still help your compromise by strengthening that defeat.  In order to do this, you have to rank your compromise strictly ahead of your favorite.
But when winning approval is used, the only way you can strengthen the defeat of Compromise over Favorite is by approving Compromise, which is no skin off anybody's nose.  Neither order reversal nor order collapse will help Compromise at all when Compromise already beats Favorite without the help of your ballot.
For the other case (where Favorite beats Compromise pairwise) imagine the following conversation between two voters from the A>C>>B faction in a DMC election:
John:  I'm afraid that our favorite candidate A doesn't quite have the viability that our compromise C has, so just to be safe, we'd better throw our support behind C.  Let's rank C ahead of A.
Jane: But I'm already approving C.  Isn't that enough support for our compromise?
John:  Perhaps.  In fact it would be the maximum effective support that you could give in the event that C beats our favorite candidate A, but I think that we need to help ensure that C does beat A.
Jane:  Why do you say that? I thought we wanted A to beat C.
John:  Well yes, but I think that C has a better chance of beating B head-to-head than A does.
Jane:  More power to C.  If C can beat B pairwise, and A cannot, then the race will be between A and C, and in this race we prefer A, though we won't move to Canada if C wins.
John:  Well, there's just one little problem.  If A beats C pairwise, AND also has more approval than C, then C gets eliminated before she can beat B.
Jane: Let's see if I understand your fear:  You think that (more likely than not) without our order reversal A will doubly defeat C, so that it is likely that C won't survive to eliminate B.
John: That's right.
Jane: But I thought that C didn't have to survive the "double defeat" phase in order to defeat B doubly.
John:  That's true, but although we are confident that C can defeat B, we don't know that C can defeat B doubly.
Jane: That's why I'm approving C ... to make C's defeat over B as strong as possible.
John:  But don't you see, the polls seem to show that all of the following are true:
        (1)  B has more approval than C.
        (2)  A has more approval than C
        (3)  A beats C pairwise.
        (4)  C beats B pairwise.
        (5)  B beats A pairwise.
Jane:  I see the possibility, but (5) doesn't seem too likely given conditions (2), (3), and (4).  And (1) doesn't seem too likely, given (4).  What if even one of these conditions turned out to be false?
John:  In that case it would be better to vote our sincere order.
Jane: If one of the five conditions fails, would it be better for us to vote A=C or to vote A>C ?
John: A>C.
Jane: Then do what you want, I'm going to stick with A>C.
Summary: This conversation shows that it would take a lot of confidence in a set of unlikely (but possible) conditions to entice a rational voter into Favorite Betrayal under DMC even if condition (3) is thought to be likely.  Furthermore, if condition (3) is known to be false, then there is no Favorite Betrayal incentive at all. 
Does that help?
Below is a case by case study of DMC's resistance to "compromising" from the EM archives of Feb 2002:
What if we used the Demorep style ballots (candidates ranked and approved
or disapproved) to carry out Approval Seeded Bubble Sort?

Bubble Sort (like Single Elimination) always puts a member of the Smith
set at the top.

Would this method be vulnerable to compromising?  Not much, if at all.

Case 1a.  Compromise is seeded above favorite and is never challenged by
favorite. No problem.

Case 1b.  Compromise is seeded above favorite and is already defeated by
at least one other candidate before being challenged by favorite. No

Case 1c.  Compromise is at the top when challenged but not defeated by
favorite. No problem.

Case 1d.  Compromise is at the top when challenged and defeated by
favorite. Then favorite will remain at the top until defeated by a
candidate that also defeated compromise.

Case 2a.  Favorite is seeded above Compromise and is never challenged by
Compromise. Disapproving Favorite wouldn't help, because Compromise's
order among the remaining candidates would be unchanged.

Case 2b.  Favorite is seeded above Compromise and is challenged and
defeated by Compromise. No problem.

Case 2c.  Favorite is at the top when challenged (but not defeated) by
Compromise.  Favorite has dashed the hopes of Compromise. Anybody who
deposes favorite will have to beat Compromise first. No problem.

Case 2d. Favorite has already been defeated before being challenged (but
not defeated) by Compromise. At this point we would have some regret that
Compromise was blocked from further chance of proceeding to the top.
However, given that Compromise had less approval than Favorite and that
Favorite beat Compromise, we shouldn't expect Compromise to have great
chances of beating all the guys ahead of Favorite, notwithstanding
pre-race polls to the contrary.

This last case (2d) would be a pretty flimsy excuse for voting Compromise
over Favorite. I suppose it could happen if your preference of Favorite
over Compromise were extremely weak, or if you had extremely precise,
detailed information about voter preferences that gave Compromise a good
chance of beating every candidate that could beat Favorite despite
Compromise having less approval than Favorite and being unable to beat

Note that this Approval Seeded Bubble Sort has less vulnerability to
compromising than Random Candidate Single Elimination, because in this
latter method, Favorite can get into a position to block Compromise's
progress by random (as opposed to merit based) means.


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