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. ---- Election-Methods mailing list - see http://electorama.com/em for list info
