[Election-Methods] Smith exposes our false statements

[Abd ul-Rahman Lomax]

At 02:22 PM 7/26/2007, Chris Benham wrote:

>>Take a look. If there is an error in the calculation, I'd like to
>>know.
>
>What "calculation"?  Look at what?  With "many voters" your 
>"improvement over not voting"
>figures look too high.

I've posted the spreadsheet as such and a web page that shows the 
spreadsheet information. However, earlier I found an error in the 
calculations. What had happened was that I had what I thought was 27 
equal probability vote patterns where the voter can affect the 
result. It appears that there was a problem with that. I had missed 
some patterns. But, then it turns out that the patterns come in pairs 
(except for the ties), and there are five patterns only for each 
pairwise election, for 15 total. It's actually simpler than I thought.

When the error was corrected, the utility for both Approval and 
Sincere Range strategy become 40% over not voting. I've suggested 
that writers here, if they care, predict what would happen if this 
Range election became an Approval one. I wrote something before about 
it, but my numbers were off. Hopefully, they have been corrected. 
What is the effect on the expected utility of the Approval-style 
voter if the election is changed from Range 2 to Range 1, i.e., Approval?

I've described the method: list all possible votes, excluding the 
ballot of the voter, where the voter can then affect the outcome. 
With zero knowledge and proper care, all these possible votes should 
be equal in probability (it's easy to get this wrong, which is why I 
wrote "care"). One can then calculate the utilities, which are either 
the utility of the candidate elected, or the average of those tied. 
And then one can determine the expected utility.

3 candidates, zero knowledge, our voter has utilities of 2, 1, 0, and 
it is a Range 2 election.

Now that I've polished the spreadsheet a bit, it automatically 
calculates the winners and utilities now, I'd been putting them in 
manually. And so I can essentially turn a dial on the candidate 
utilities and see what happens. It's quite interesting. And, of 
course, I have the Approval election pulled out: it's a subset of the 
vote patterns, all those with even votes only (aside from the 
condorcet loser, which is expressed with a vote total of -1 now).

In any case, if anyone else has done this calculation or has a tool 
which can do it, it would be useful to verify the result. 40%, 
exactly, over not voting.

By the way, that is the *relative* utility assuming that the vote 
makes a difference. I've stated this many times, perhaps Benham was 
referring to this in saying my numbers were too high. With many 
voters, of course, the absolute improvement of voting is quite small.

However, people can and do assume that how they behave, so will 
others, so we do look at the utilities assuming that many others will 
behave as we do. Otherwise we really would not bother to vote unless 
we were bored to tears and had nothing else to do....

>
>
>>
>>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.
>Right.  "Posted" where?

Well, a little later, after I send this, the old spreadsheets -- 
which are wrong -- are at 
http://beyondpolitics.org/OptimalRangeVote.xls, and there is an HTML 
document at the http://beyondpolitics.org/OptimalRangeVote.htm.

I will put up the new, improved spreadsheet at
http://beyondpolitics.org/Range2Utility.xls and
http://beyondpolitics.org/Range2Utility.htm

I find it extremely interesting. I can play with the utility numbers 
and see what happens. The case that is shown in the spreadsheet is a 
special one, exactly balanced utility, and it's clear that utility 
gets worse, in comparison with Approval, when the utilities aren't 
spot on in the sincere vote. However, there are other aspects that 
make this not as bad as it might seem.... In particular, it is 
interesting to look at what happens to utility if the election 
becomes Approval. It's not surprising, really.