At 09:28 AM 7/27/2007, Chris Benham wrote: Looking at these six situations and for each one measuring the equity gain from voting 200, >210 or 220 versus not voting we see that the total gain for each is >the same (+2). > >So assuming that without our voter's vote there is never an exact >3-way tie, it doesn't make any difference >in this election what rating our (sincere A2,B1,C0) voter gives B. >But including that very small chance means >that 200 and 210 are slightly better than 220.
With the correction of the missing votes, the spreadsheet now contains all the possible vote combinations where the voter has a possible gain (or loss) in utility from how the voter acts. Benham's analysis seems correct to me, based on the new results, which are, indeed, that with the voter utilities of 2, 1, 0, the expected return is almost exactly equal for the two possible approval votes. I'm not sure about one thing he says, which I have not verified, yet, which is what happens with three-way ties. Setting that aside, the two possible approval votes also have the same expected utility as the sincere range vote. However, there is something very interesting question that remains. Take my spreadsheet, which covers all possible vote combinations where the voter's actino has a say -- unless there is some other error, which is getting less likely, and restrict the votes to only Approval votes. I.e., take only the vote totals which are divisible by two. What happens to the outcome utility? Instead of publishing it now, I'll ask for predictions. Doe it increase, stay the same, or decrease. In other words, for the *voter's* expected utility with optimal strategy, but balanced utilities, which method should the voter rationally prefer, Range 2 or Approval? I think I may have written something about this before, but the foundation data, the list of votes, was in error. Having corrected that, what results can be seen for changing the election from Range 2 to Approval? >>This election is a counterexample to the claim that optimal voting >>in Range is never the sincere vote. > >The "claim" you refer to was made by no-one. What was pointed out >was that the "sincere" vote is never >better than some approval-style vote (giving no intermediate ratings). That claim may be true. However, I ask a very pertinent question above. Which voting method, Approval or Range 2, provides the best expected outcome for the voter, whether the voter votes sincerely or approval style? I'm now in a position to answer that definitively, if I haven't made any more errors. No, the table did not line up, and it is presenting the data in a different way and does not state the conclusions, the actual expected utilities. Since different ways of presenting the data should still produce the same result, here are my results: Both the sincere vote and the approval vote have a utility exactly 40% above that of not voting, given the initial conditions. And the remaining question is .... what if we exclude all votes that are not divisible by 2, leaving us with only Approval style votes, which is exactly equivalent to not allowing any intermediate votes? Does the utility improve, stay the same, or decrease? And by how much? We can now answer exactly and easily, if my list of votes and consequences is correct. And it is a far shorter list now, only 15 possible votes. (I had missed combinations before, but many of the combinations were equivalent, my selection of preceding vote patterns was far more complex than it needed to be, and simplifying it appears to have removed the error.) And we can then look at Range 3, and higher. It's not as difficult as I thought it might be. If there are P pairwise elections and the election is Range N, the number of combinations is P*(2*N+1), I think. It's not exponential, that was an error. I'll put up the revised spreadsheet and post the URL when I can. And it does now have an explanation in it. If there are shortcomings in the explanation, I'd like to know. It should be extremely easy to understand and review, it's very straightforward now. ---- Election-Methods mailing list - see http://electorama.com/em for list info