[EM] Winning-votes intuitive?

[atarr at purdue.edu]

I apologize for ataking a while to get back to this thread; I have been out of e-mail contact.

Blake wrote, I responded, and Blake wrote again,

>>> The point is that if my first choice is A, the method penalizes me 
>>> for not choosing between B and C, by strengthening one or both 
>>> candidates, and therefore weakening A.
>>
>> Certainly not both candidates!  In the zero-information election, you 
>> don't know which one you should weaken... you may swing the wrong 
>> 3-way tie by casting an insincere later vote.  It does not seem 
>> intuitively obvious to me that casting random later votes will 
>> generally help you more than it hurts you. 
> 
> If you agree that it generally is more helpful to a candidate to have a 
> loss decreased then a victory increased, it follows that random ranking 
> is better than leaving candidates unranked at the end of the ballot. 
> Winning-votes ensures that a vote in either direction will either have 
> no effect or ratchet up the size of the victory between two candidates.
> The only exception being the discontinuity at a tie, where the value 
> goes back to 0 (except in Markus's system).

I beleive your contention is false.  In the zero-information 3-candidate case, I could just as easily cause a victory to become a defeat as I could cause a victory to have more winning votes than another.  Both cases just involve tripping a running count over a border.  One is not inherently more likely than the other, unless you have some knowledge of the polls.

(Both cases do cause a brief tie on that border.  We shouldn't concern ourselves with the tie too much, since in a public election you can expect that many other like-minded people will vote similarly to you, and this discontinuity will be vanishingly rare.)

Let me provide a simple pair of examples that demonstrate what I am trying to say.  Here's the first one:

A beats B, 70% winning votes (25% losing)
B beats C, 50% winning votes (45% losing)
C beats A, 52% winning votes (40% losing)

Say I have no poll information coming in, and I prefer A, with no B/C preference.  If I merely vote A, C wins the election.  Your argument is that, if I (any many others like me) randomly vote C over B, we change nothing, while if I (any many others like me) randomly vote B over C, we can push B's margin of victory past C's, and give the election to A, our favorite.  At any rate, the losing votes do not factor in.

This is of course correct, and follows your premise exactly.  But what about this example:

A beats B, 70% winning votes (25% losing)
B beats C, 52% winning votes (45% losing)
C beats A, 50% winning votes (40% losing)

By virtue of a slight perturbation (the sort that would fall within polling error margins in the real-world, nonzero-information case) candidate A now wins the election.  In this case, if I (any many others like me) randomly vote B over C, we change nothing, while if I (any many others like me) randomly vote C over B, we may turn B's victory over C into a defeat, and we turn C into a Condorcet winner.  This causes the defeat of A, our favorite.

The moral of the story?  Losing votes do matter just as much as winning votes in winning votes methods.  They just don't matter until they become winning votes.  I've heard the argument here that this is too sudden and sharp a change, since we suddenly switch from considereing ONLY the votes on one side to considering ONLY the votes on the other.  This is sort of a silly argument, since every election method has some boderline where all of a sudden one vote causes a completely different result.  How you count the votes (winning votes vs. margins, for example) only decides where this border falls; it does not make this border any less stark.

> But is my premise true?  It certainly is for three candidates.  With 
> three candidates, both methods are equivalent to Minmax.  Minmax can be 
> phrased as giving the victory to the candidate whose defeat is least 
> strong.  Having a stronger victory only helps in so far as it increases 
> another candidates defeat.

Right... but having a weaker victory may turn a victory into a defeat.  This is the missing link in you analysis.  More votes on the side of a victory might make the victory stronger than another victory, but more votes against a victory might turn that victory into a defeat.  There is just as much inherent symmetry here as there is in margins methods.

-Adam