Dyadic Rated Pairs

[Forest Simmons]

On Mon, 24 Sep 2001 DEMOREP1 at aol.com wrote:

> fsimmons at pcc.edu wrote in part-
> 
> Not really a margin. Take the following example:
> 
> 45 A > B >> C
> 35 C > A >> B
> 20 B > C >> A
> 
> The pair strengths are ...
> 
> S({B,C})= 2*45+2*35+20 = 180
> S({C,A})= 2*45+35+2*20 = 165
> S({A,B})= 45+2*35+2*20 = 155
> 
> [No margins, because no subtraction]
> 
> So according to Dyadic Rated Pairs the over all ranking is
> 
> B > C > A
> 
> Despite the fact that A is the Ranked Pairs winner, the SSD winner, the
> Approval Winner, the Universal Approval winner, the Dyadic Approval
> winnner, the ACMA winner, and even the IRV winner.
> 
> This example is enough to get me to abandon Dyadic Rated Pairs.
> -----
> D- Beating the dead horse some more ----
> 
> A vote is 100 percent for or 100 percent against (with ranges in between).
> 
> Thus for one voter
> 
> A > B >> C  may mean 
> 
> 100 A > 99 B >> -100 C
> 
> to another voter it may mean
> 
> 100 A > 51 B >> 20 C
> 
> and yet to another voter it may mean
> 
> 40 A > 20 B >> -50 C
> 
> and to another voter it may mean
> 
> -5 A > -10 B >> -95 C
> 
> That is -- *relative* votes are only just that --- *relative*.
> 

True enough. That's one reason why we don't trust Borda. Yet Borda beats
most methods in social utility expectation. So the main reasons we don't
like Borda is that it gives a lot of incentive for reversing preference
orders and it has clone problems.

The numerical weights that I use in Dyadic Rated Pairs do not introduce
either of these problems.

But for the pure relativists I will repeat the example without using
weights. (The weights are only an artifice to get an ordering of the
pairs, and as I mentioned in a recent posting, Ranked Pairs on Pairs will
get us an ordering of pairs, with just a little more effort.) 

Here's the example again:

45 A > B >> C
35 C > A >> B
20 B > C >> A

{B,C} is stronger than {A,B} in the A faction, and ties in the C faction,
so the margin is 45 minus 20, which is 25. 

{B,C} is stronger than {C,A} in the C faction, and ties in the A faction,
so the margin is 35 minus 20, which is 15. 

{C,A} is stronger than {A,B} in the A faction, and ties in the B faction,
so the margin is 45 minus 35, which is 10. 

So {B,C} is the CW  and {A,B} is the Condorcet Loser in the Strength of
Pairs contest.

The order of pair strengths is {B,C} > {C,A} > {A,B}
 
Locking in the strongest pair gives B > C.

Locking in the next strongest pair yields C > A, which doesn't contradict
the first pair.

Trying to lock in the last pair A > B contradicts the partial order
induced by the previously locked in pairs, so we throw it out.

In summary, B > C > A .  We got this result without assigning numerical
weights to the inequalities of various strengths. Only their strength
order was used. 

Forest