[EM] yet MORE massive confusion by Venzke:

[Warren Smith]

>WDS:
> Proof sketch:
> Because that winner X will be Condorcet winner.
> For each Y not in {A,B},  X is ranked above Y one-half of the time,

>Venzke:
One-half isn't a reasonable guess for this. You're not considering the
equal ranking at all.

--WDS: wrong.
one-half was not a "guess." It was an "obvious exact result"
and it took into full account the equalities.
The typical situation is U_A=0.666, U_B=0.333, or vice versa.
Then if X is A in former case:
prob = 1/3:    U_Y>U_A so A=Y in the vote
prob = 2/3:    U_Y<U_A so A>Y in the vote
or in the vice versa case (X is B):
prob = 2/3:    U_Y>U_A so A=Y in the vote
prob = 1/3:    U_Y<U_A so A>Y in the vote
so that on average (since the above two happen asymptotically 50-50)
prob = 1/2:    U_Y>U_A so A=Y in the vote
prob = 1/2:    U_Y<U_A so A>Y in the vote

>Venzke:
>Not sure how you get these numbers...

--WDS: hopefully you are figuring it out now....

>Venzke:
Two problems:
1. You didn't show burial here. Burying involves order reversal.
2 . How did you show that the second scenario is superior?


--WDS: 
(1) Aha.  Well in that case, ranking B (say) coequal bottom is not a "burial"
by your definition.  It seems like burila to me, but if that is how you define it,
then fine, it isn't. Anyhow, what I'm saying is, this form of
"semi burial" is "superior" in the sense that somebidy other than A or B can win
with probability bounded above 0.

>Venzke:
>Isn't this identical to Thm 7?

--WDS: no.  5/12 is not 1/2.

>WD:
> Now for IRV:
> X is ranked top by V/2+o(V) voters before vote-alteration 
> and after alteration is
> top-ranked by 3V/(2C)  voters.   Meanwhile Y is top-ranked 1/3+o(1) 
> of the time 

>Venzke:
>Where does this come from? Y could be top-ranked on *all* the ballots.

--WDS: No. Remember.  EVERYTHING is about: we are in a random in [0,1] utilities scenario,
in the  V=# of voters = large  limit.
Y therefore could NOT be top ranked on all the ballots (except with probability=0).

>WDS:
> before vote-alteration and after alteration is
> top-ranked by V/C  voters.   So each Y will get eliminated (and since the
> vote-transfers
> are independent random, this will happen in subsequent IRV rounds also)
> Q.E.D.

>Venzke:
>It looks like you forgot that the votes have to be split up symmetrically.

>Counter-example:

51 X=Z
49 Y=Z

Z wins even if we say that A and B are X and Y.

--WDS:
sigh.  It appears my advice to "put these theorems in your pipe and smoke them"
was throughout taken to mean "bhong".  

This counter example is failing to use full rankings (forbidden).
And it is probability=0 in the whole model.
So it is just irrelevant garbage.

-wds