EM list-- As I was saying before, the utility expectation gain that you get by voting between a particular pair of candidates can be assumed equal to Pij(Vi-Vj)(Ui-Uj). That assumption is used by academic authors on the subject. Though, for a particular method, Pij isn't really the same as the probability that 2 same-voting voters could change the win from j to i, if they each vote a vote-difference of 1 between i & j, and if everyone votes a vote-difference of 1 between each candidate pair, it's still convenient to use that standard factor, and to consider each term to be multiplied by that unwritten factor. Then, when making the determination for a particular method, each term of the determination has to be multiplied by an adjustment factor, as I described in my previous posting. Of course only with Borda is Vi-Vj anything other than 1. *** Plurality: You're voting between A & B, and between A & C. (Va-Vb) is 1, as it is in every method other than Borda, and Pij is something that I'm not including, because, at first, I'm just considering that standard factor that I mentioned to be in front of each term. So that just leaves (Ua-Ub). Ua-Uc is 1. Ua-Ub is 1-B. So, adding those, the whole amount, for the two pair votes, is 1+1-B = 2-B. For the reason that I described in the previous posting, that has to be multiplied by 3/2. So it's 3/2(2-B). When B varies from 1 to 0, then, Plurality's table entries vary from 3/2 to 3. *** Approval: If B is greater than .5, then it's above average, and you vote A over C and B over C. Adding up the utility differences, it's (Ua-Uc) + (Ub-Uc). That's 1 + B. [Sometimes I use A & B to mean Ua & Ub] If B is less than .5, then it's below average, and you vote A over B and A over C. Adding up the utility differences, it's (Ua-Ub) + (Ua-Uc). That's 1 + (1-B) = 2-B. If B = .5, then both of those 2 expressions equal 1.5 If B is 0, then the 2nd expression applies, and it equals 2. If B is 1, then the 1st expression applies, and it equals 2. So, before adjustment, Approval varies from 1.5 to 2. But multiplying by 3/2, it varies from 2.25 to 3. *** I'll send this now, and send separate postings for Borda & IRV, immediately after this one. *** Mike Ossipoff ______________________________________________________ Get Your Private, Free Email at http://www.hotmail.com