[EM] Responses to some of Forest's ideas

[Richard Moore]

LAYTON Craig wrote:

>  9 A>B>C>D=E=F : 100>90>1>0=0=0     : ZI AB  : St AB 
> 38 B>D>A>C=E=F : 100>52>51>0=0=0    : ZI BDA : St B


I don't see why the second group would truncate to B in the 
informed vote. Just given the information that clones E and 
F don't have a chance, a simple informed strategy 
calculation could be made by removing E and F from the 
picture and doing a mean utility calculation (203/4 = 50.75)
and voting BDA. The information that C is a likely front 
runner would give even more reason to vote BDA.

> 40 C>B>A>D=E=F : 100>85>70>0=0=0    : ZI CBA : St C


Discounting E and F, 255/4 = 63.75, so possible strategic 
vote is CBA. Depending how confident this group is in the 
information that D will lose, they might truncate to CB or
just C, but we don't know for sure. "C only" seems unlikely 
to me unless these voters are reasonably sure of D's and A's 
defeats.


>  9 D>C>B>A=E=F : 100>10>9>0=0=0     : ZI DC  : St DC


Wrong, ZI vote here is D only (119/6 > 10). Strategic vote 
could be DC or DCB, depending on how much doubt the 
pre-election information casts on candidate D. If the 
information is believed to be unreliable, this group might 
stick with D only.


>  4 E=F>A>B>C>D : 100=100>90>12>10>0 : ZI EFA : St EFA
> 
> Undefeated (definite) Condorcet Winner is B.  Zero info approval winner
> (taking both mean and median into account) is A.  Strategic approval winner
> is C.  It seems pretty clear that I've used optimal strategy, although I
> haven't done all the math.
> 
> SU scores for the top three candidates; A = 5998, B = 8148, C = 4670.
> 
> Craig


For ZI I get A, also. For NZI, I see B as the most likely 
winner (since I don't expect to see enough CBA voters 
dropping B to make C win).

Incidentally, I've long suspected there is (in general) no 
such thing as universally available "perfect information" in 
Approval voting. It might be possible for one or a few 
voters to have such perfect information, as for instance if 
the voting is done openly by role-call and Mr. Zzyzx gets to 
hear how everyone else voted before his turn comes. But 
suppose everybody knows everybody else's sincere utilities 
before the voting takes place, and ballots are secret. The 
probability that voter X will vote for candidate Y depends 
on all the other probabilities, but each of those 
probabilities in turn depend on all the other probabilities, 
including the "X for Y" probability we are trying to 
calculate, so you can't calculate the answer if you don't 
already know it. It's a system of equations with circular 
dependencies and no unique solution. It is that lack of 
perfect information that makes it difficult to know if the 
CBA voters in the example above will truncate in the NZI case.

There are certainly special cases where there is a unique 
solution. A trivial example is one where each voter has a 
utility of 100 for one candidate and 0 for all others.

Richard