[Election-Methods] Smith exposes our false statements

[Abd ul-Rahman Lomax]

I'll be brief. I actually do know how to do it, sometimes.

At 04:36 AM 7/26/2007, Michael Ossipoff wrote:
>So I repeat that, in public elections, Schudy's statement was 
>correct, when he said that it is never optimal to rate someone other 
>than top or bottom.

I have currently posted a counterexample. Perhaps Ossipoff addresses 
it later, but prior to the present post he has not.

The example: many voters, large enough that a three-way tie is of 
negligible effect on utilities.

Three candidates. Range 2 (CR-3).

Voter with utilities of 2, 1, 0.

Zero knowledge, so all vote patterns from the electorate minus our 
voter are equally possible.

Expected utility of Approval Votes of 220 or 200: 39/27 (improvement 
over not voting: 12/27).

Expected utility of Sincere Vote, 210: 40/27, improvement over not 
voting: 13/27.

Improvement over Approval by voting Sincerely: 1/27.

Take a look. If there is an error in the calculation, I'd like to 
know. It's not terribly difficult; I hit upon the method of 
discarding, from the outset, moot votes, that is, votes where the 
voter cannot affect the outcome, since these votes cannot affect the 
utility. Prior simulations have looked at absolute utilities, causing 
the effect to be buried in the noise, I think. Further, prior 
simulations were hi-res Range, and it is entirely possible that 
hi-res, beyond a certain point, reduces utility such that Approval 
Strategy is max.

However, in this particular case, converting the *election* to 
Approval, thus limiting all votes to 0 or 2, reduces expected utility 
*further.*

These are counter-intuitive results for me, and I expect for others, 
so they bear careful attention, I'd suggest.

Please, if you can, find the error in the proof; sufficient 
information has been given as to how to do it, and my spreadsheet has 
been posted, but you'll need Excel or some spreadsheet program that 
can read Excel files. I think googledocs would allow Ossipoff to read 
this spreadsheet on the web; I'll post a text version when I can get to it.