EM list: With IRV, you have 2 chances of changing the outcome: In the 1st elimination, and in the pair comparison of the 2 candidates left after that elimination. Again, I'm leaving out Pij, because I'm considering each term to have the unwritten standard factor in front of it. Vi-Vj is always 1. So we just have to write Ui-Uj In the 1st elimination, though, we don't change from one outcome to another, but rather from one lottery to another. For instance, if we make B be eliminated instead of A, then we get the AC lottery instead of the BC lottery. So let me write the sum of the utility differences that we could achieve: The utility of the AB lottery is (A+B)/2. Let me write the utility difference if we make B get eliminated instead of A: (A+C) - (B+C) Of course C cancels out, and we just have A-B. I'm going to multiply the whole sum by 1/2 once, rather than each term. So we get, for our utility expectation change in the 1st elimination: 1/2( (A-B) + (A-C) + (B-C) ). Collecting terms: 1/2(2A-2C) = A-C Since A (which I sometimes use for Ua) is 1 & C is 0, that expression equals 1. *** But say we don't change the outcome in the 1st elimination. There's still one more chance. The 2nd elimination will be just one contest between one pair, and so we just have one chance to influence a pair-comparison, rather than 3. But it could be any of the 3. So the utility expectation change in the 2nd elimination is: 1/3( (A-B) + (A-C) + (B-C) ). Collecting terms: 1/3(2A-2C). Since A = 1 & C = 0, that comes to 2/3. *** It remains to adjust the 1 & the 2/3: In the 1st elimination, it's like Plurality. Each voter is only voting beteen 2 pairs, not 3. Also, even though we're interested in frontrunners for elimination, it's still as if we were interested in frontrunners. So we multiply the 1 by 3/2, and so the utility expectation change for the 1st elimination is 3/2. In the 2nd elimination, it's different in 2 ways. For one thing, each voter is voting between all the pairs, and so we don't need to make the 3/2 adjustment. For another thing, we don't need for the 2 candidates to be frontrunners in some field of candidates. They need only be within 1 vote of eachother. The standard assumption was that the 2 candidates need to be frontrunners and within one vote of eachother. With 3 candidates, there are 2 ways 2 candidates could be within one vote of eachother: They could be the top 2 or the bottom 2. One of those lets us change the outcome and one doesn't, so changing the outcome is 1/2 as probable as it would be if we didn't need for them to be frontrunners among 3. With just 2 candidates, in the 2nd elimination, we don't have that problem, and all we need is for the 2 candidates to be within 1 vote of eachother, and so we double the value for the 2nd elimination, to 4/3. Adding the adjusted values for the 2 eliminations: 3/2 + 4/3 = 9/6 + 8/6 = 17/6 = 2 & 5/6 = 2.8333... *** That isn't the 2.25 that I said in the earlier message. I must have made an error before. Maybe I divided the 2nd elimination result by 2, mistakeningly copying the 1/2 that the 1st elimination result had to be multiplied by. Anyway, from this, the table entry for IRV is 2.8333... That makes IRV usually better than Approval by this standard, unless there's still a mistake in this IRV determination. Neeless to say, that doesn't make me start advocating IRV :-) There are too many other differences that make IRV look really bad. FBC, SARC, WDSC, Corruption-encouragement, non-median candidates winning at Myerson-Weber equilibrium, Social utility, to name some differences. Social utility has a more concrete meaning than this utility expectation change for a sincere voter. It tells us how we can expect to do in some future election after the new method is adopted. It tells us our utility expectation, at this time, for that future election. But I hope someone can tell me that there's an error in this posting. Mike Ossipoff ______________________________________________________ Get Your Private, Free Email at http://www.hotmail.com