EM list-- I've found the mistaken assumption that made Borda look improbably good in the comparison of utility expectations for sincere voters. When I first posted the table, I said that, to get the improvement in utility expectation for a sincere voter, the table entry should be multiplied by the probability that some arbitrarily- chosen pair of candidates are the 2 frontrunners, and are so close that 2 same-voting voters could reverse their comparison with eachother, changing the winner from one to the other. There were other assumptions that I didn't state because it didn't occur to me that I was making them: Each voter votes between each pair, and votes a vote-difference of one between each pair. I said to multiply the table entry by that probability because I considered that to be the probability, for each candidate-pair, that, in any of those voting systems, 2 same-voting voters could change the winner between those 2. That isn't true, because the 2 assumptions in the previous paragraph aren't true. But that assumption is still ok as a standard assumption, as long as the table entry for each method is adjusted in accordance with the way that assumption differs from the truth with that method. Making those adjustments, Approval is better than Borda all the time. Approval is better than IRV all the time. Plurality is better than IRV slightly more than half the time. *** How do the different methods winner-changing probabilities differ from the standard one that I defined? In Borda, it isn't true that the voters are only voting a vote difference of 1 between each pair of candidates. Each voter votes, among the 3 candidate pairs, vote differences of 1, 1, & 2. So each voter's average vote-difference is 4/3. So there are 4/3 times as many votes voted among each pair in Borda than I'd assumed. That makes it 3/4 as likely that the pivotal voters (the 2 voters whose expectation change we're estimating, and whose probability of changing the outcome we're interested in) will be able to change the outcome. So Borda's 3 should be multiplied by 3/4 to get 9/4 = 2.25 The other methods have to be adjusted similarly. In Approval and Plurality, each voter is only voting between 2 candidate pairs, not all 3. So the assumption that everyone votes between each candidate pair isn't true. There are only 2/3 as many votes between each pair as I'd assumed, and so Approval's & Plurality's table entries should be multiplied by 3/2. With IRV, part of its determination needs to be adjusted in that way. An immediately subsequent posting will describe how I got the results that I got, and will include a new table. Mike Ossipoff ______________________________________________________ Get Your Private, Free Email at http://www.hotmail.com