Definition of "Pairwise Method"

[Bruce Anderson]

On Apr 16, 12:08am, Steve Eppley wrote:
> Subject: Re: Definition of "Pairwise Method"
> Bruce Anderson wrote:
> [snip]
> >To address Steve's objection concerning ambiguity in my
> >definition-1 above, let me make it even more explicit as follows.
> >
> >DEFINITION 1:  Let p(x,y) be the sum number of voters who
> >explicitly rank x over y plus the number of voters who explicitly
> >rank x [[BUT]] do not explicitly rank y, and let q(x,y) be the sum of the
> >number of voters who explicitly rank X as tied with y plus the
> >number of voters who do not explicitly rank either x or y, and let
> >p and q be the corresponding arrays of values of p(x,y) and q(x,y).
> >Then a ranked-ballot single-winner voting method is a "pairwise
> >method" if and only if its set of winners can be calculated using
> >only the number of candidates, the number of voters, and the values
> >in p and q.
> >
> >Steve:  Is this precise enough?
> 
> The "either..or" is still ambiguous.  How about this, nearly the same:
> 
>   DEFINITION 1:  Let p(x,y) be the sum of the number of voters who 
>   explicitly rank x over y plus the number of voters who rank x and 
>   leave y unranked.  Let q(x,y) be the sum of the number of voters who
>   explicitly rank x as tied with y plus the number of voters who leave
>   both x and y unranked.  Let p and q be the corresponding arrays of
>   values of p(x,y) and q(x,y).  Then a ranked-ballot single-winner
>   voting method is a "pairwise method" if and only if its set of
>   winners can be calculated using only the number of candidates, the
>   number of voters, and the values in p and q.
> 
> Why include the number of voters there at the end?  Why disallow
> other info? 
> 
> I'd like to be able to refer to pairwise//nonpairwise methods as 
> pairwise.  Would that be tabooed by a "strict constructionist" 
> interpretation of Definition 1?
> 
> --Steve
> 
>-- End of excerpt from Steve Eppley

First, there was a typo in my original definition -- I left out the conjunction 
"but", which I inserted inside double brackets above.  With this correction, the 
two definitions above seem identical to me, and so I'd be happy with either one 
(or even both).

Second, none of this makes any sense to me if using the number of candidates is 
not allowed.  Thus, since it is always implicitly allowed, I didn't think it 
would hurt (and, for clarity, I thought it might help) to explicitly allow the 
use of the number of candidates here.

Third, I saw no harm in allowing the use of the number of voters.  Certainly, no 
one could object to a voting method because it explicitly made use of the number 
of voters in an election!  Conversely, allowing the use of a number does not 
mean that any method is required to use that number in order to satisfy the 
definition -- it only means that using the number does not disqualify the method 
from satisfying that definition.  But, with one exception, it's not important to 
me either way.  The one exception is that I sure would not want to have to 
explain to anyone why using the number of voters in order to determine the 
winners disqualifies a voting method from being a "pairwise method."

Finally, it seems to me that the whole point of defining a type-x method to 
disallow, in some sense, nontype-x methods.  For example, let S be a set of 
pairwise voting methods.  Define All-S to be the voting method that chooses as 
its winner the candidate that is the unique winner according to every voting 
method in S, if such a candidate exists, and chooses all the candidates to be 
tied as winners otherwise.  If pairwise//nonpairwise methods are called 
pairwise, then wouldn't "All-S//Plurality" and "All-S//Hare" also be pairwise?  
Is this what you want?  Again, with one exception, it's not important to me 
either way.  The one exception is that I sure would not want to have to explain 
to anyone why, for every such set S, All-S//Plurality and All-S//Hare are 
considered as being "pairwise methods."

Bruce