[EM] Reversal Software output 6/9/1999

[Blake Cretney]

Steve Eppley wrote:

> Hi,
> 
> I added Blake Cretney's "Path Voting" method (which is a 
> variation of Schulze's method) and Condorcet(Margins) into
> my Reversal software.

Thanks.
 
> "Cret" is Blake's Path Voting method.  When calculating the
> size of a pairwin, it uses the pairwise margin of victory.
> (Schulze uses the number of voters who ranked the pairwinner
> ahead of the pairloser.)  Blake made a philosophical argument
> for preferring pairwise margins of victory, but I don't find 
> it more compelling than the philosophical argument for 
> preferring pairwise support (or call it opposition, depending 
> on which side you're on).  More important than philosophical
> criteria, in my opinion, are other criteria we usually use
> for comparing voting methods.
> 
> "Marg" is the Condorcet(Margins) variation.  Hugh Tobin

Calling this method Condorcet seems to be an error started on this
list.  I'm going to refer to Minmax to refer to the same method since
this term is used in academic journals.  I use Minmax(Margins) to
refer to the Marginal variant.

> wrote (long ago) that he preferred it more than CondorcetVA,
> for the same philosophical reason that Blake prefers 
> Path Voting to Schulze.
> 

Dumais is equivalent to Minmax(Margins) and Path Voting for 3 or
fewer candidates.  The following is my proof.

If a candidate is the Condorcet winner for the ballots (as opposed to
sincere preferences) then this candidate will win in all 3 methods. 
So, they behave identically when there is a CW.

When there is not, we have a cycle.  With no loss of generality
A>B>C>A, with C>A being the win of lowest margin.  A is therefore the
winner of Minmax(Margins).  

I will define x to be the margin C>A, y A>C, z B>C.  Since C>A is the
lowest margin, x<y x<z.  

If A>B is the highest margin, then y>x y>z z>x
The path B>C>A has strength x.  This is lower than A>B strength y.
The path C>A has strength x.  This is lower than A>B>C strength z.
So, A is the PV winner.

Counting the Dumais scores (margins of victory - margins of defeat)
A: y-x>0
B: z-y<0
C: x-z<0
So, A is the winner.

So, the three methods are equivalent if A>B is the highest margin.

If B>C is the highest margin, then z>y z>x y>x
The path B>C>A has strength x.  This is lower than A>B strength y.
The path C>A has strength x.  This is lower than A>B>C strength z.
So, A is the PV winner.

Counting Dumais scores
A: y-x>0
B: z-y>0
C: x-z<0

Since A pair-wise beats B, A will win on the second round.
So, all methods are equivalent if B>C is the highest margin.

I have exhausted all possibilities.  My conclusion is that the three
methods are equivalent for three candidates (and no ties).  Any
difference must be due to programming error.

> Note that the only difference between Path Voting and 
> Schulze, in this software's tests, is whether they comply 
> with Mike Ossipoff's Strong Defensive Strategy criterion: 
>    "A majority who prefer X more than Y should have a way 
>    to defeat Y without any of them having to explicitly 
>    rank any alternative i equal to or ahead of any 
>    alternative preferred more than i."  

I guess the point here is that you don't consider leaving candidates
unranked as explicitly ranking them equal.  It is only implied.

> (In this 3-alternative software, this means that the 
> supporters of the condorcet winner B can ensure that A 
> doesn't win merely by truncating B>A>C to B.)  Schulze 
> and VA comply with SDSC, but Path Voting does not.

Using Path Voting, Dumais, or Minmax(Margins) the supporters of B can
ensure that A doesn't win by ranking A last.  I don't see why any more
defensive strategy is needed than this.

I think that it can be shown that not submitting one's full ranking
in a winning-votes-only method will tend to reduce the likelihood that
a favourite candidate will win.  This being the case, I ask two
questions:

1.  Why is it useful to have truncation resistance, if anyone who
would leave candidates unranked can avoid the punishment by simply
randomly ranking those candidates?  This seems like something intended
to trap the gullible.

2.  Why would anyone use truncation as a defensive strategy if it
tends to hurt their chances of winning, and the defensive strategy of
voting a reverser last is available?

VA seems like a very strange method to me, an attempt by Mike
Ossipoff to combine Approval and Concorcet.  In other methods, a
sincere vote is the best (or at least equal to the best) vote when no
strategic information is known, and then as knowledge is picked up,
strategic possibilities result.  The winning-votes-only methods are
unique in that a sincere vote isn't always the best vote even when NO
information is known.

It makes no sense to vote
A>B=C
unless you have some strategic goal in mind (and even that is far
fetched).  This is I think what you are referring to when you talk of
a "philosophical reason" for avoiding VA.

> I intend to write software that tests for resistance to
> truncation.  It will assume all voters' sincere preferences
> are "strict" (no pairwise indifference) but that some of
> those with an ABC preference order may vote just A, and
> will calculate when this can change the outcome from 
> sincere condorcet winner B to A.  It is expected that
> Path Voting and Condorcet(Margins) will prove less 
> resistant to truncation than Schulze and CondorcetVA.

I certainly don't doubt this.  If you punish voters for leaving
candidates unranked, it is easy to make a method where they can't do
so as a strategy.

---
Blake Cretney
See the EM Resource:  http://www.fortunecity.com/meltingpot/harrow/124
My Path voting Site:  http://www.fortunecity.com/meltingpot/harrow/124/path