Richard:
I'd said: What I'm saying is that Weber and delta-p could both use the assumption that all ties are 2-way, or they could take into account n-member ties & near-ties. You replied: If you take those effects into account for both methods, or ignore those effects for both methods, then the only difference is that pij requires calculating the probability that there is any first-place tie. If you remove that requirement (which as I pointed out is an unnecessary complication since it is merely a scaling factor), then you end up with exactly the same calculations: the same level of accuracy, and the same difficulty. They are in fact identical methods. I reply: Ok, I didn't know that. Though I figured they'd give the same results, the wording is so different that they sound like different approaches. If the actual calculations are the same, so both approaches are equally difficult, that's good, because it simplifies the choice of strategy method. I'd said: I noticed that when I reread the posting. It's a very different approach from Weber's. You replied: Does this mean there's a different way to calculate pij than the one I gave? I'm not aware of one. If there is one then that would be a different approach, but so far I'm not convinced that they are different. I reply: When I said that, I assumed that they were completely different approaches, completely different calculations that get the same result, because their wording is so different. That patticular statement of mine, copied above, was in reply to when you pointed out that the delta-p approach doesn't use tie or near-tie probabilities. What I was saying, then, was that I hadn't realized that the approaches were that different. But of course it's possible for the same calculations to be done, under 2 very different wordings, one of which speaks of tie probabilities and one of which doesn't, and which can appear to be completely different approaches. I'd said: That's the way that sounds best to me, for Pij, or for the n-way ties & near-ties. But Hoffman is available too, for public elections or commmittees. Hoffman could sum the probability density in regions where n candidates have the same vote-total, & in regions where to vote totals differ only by one. The individual vote-total probability distribution approach sounds like it wouldn't get complicated in the way that Hoffman would, when more candidates are added. But I don't know how the probabilities would be gotten from the individual candidates' vote total probability distributions. If I remember correctly, that's what Crannor's method does. You replied: >From my reading of your description of Hoffman's method, it sounds like the method of calculation is to integrate probabilities over the boundaries between the winning regions of candidates. That's like taking the formula I gave and replacing the summmations with integrals, and I presume the binomial distribution is replaced with a continuous distribution (Gaussian?). I reply: Yes. The probability density at a point in outcome space is related to that point's distance from the most likely outcome-point position, by the Gaussian distribution. You continued: So Hoffman's method doesn't seem to be different in the calculation aspect. What is distinct about it is the method of inference of the winning probabilities; did you say that it was based on the outcome of a previous election? I reply: Yes. Or the previous iteration in Crannor's DSV. You continued: Again, how are they different? And if they aren't, how could one be borrowing from the other? I reply: I'd just assumed that they're different because they sound so different, in their wording. I'm not still claiming that they're different now. I'd said: Maybe, but it just seemed to me that Weber is more direct, and that delta-p is more computationally roundabout, calculating various intermediate quantities. You replied: Unless you can present a simpler way of calculating Pij or Ptij, they are computationally equivalent. For M candidates and N voters, calculating a single Ptij or delta-Pij value requires M*N inferences to determine the F(X), M*N calculations (of 1 or 2 additions, and 0 or 2 divide-by-two operations) to determine the cumulative probabilities (Ck(X), which stand in for the Bik(X)), and N subtractions to determine the Gj values. This is followed by N*(M-1) multiplications, and the summation of the resulting N values. If there's a simpler way (that isn't an estimate), I'd like to know. I reply: I haven't brought the Weber method, without the simplifying assumptions, any farther than the rough definition that I gave. If I'd actually pursued both approaches, then I'd have noticed that they're the same. The reason why I said that Weber seemed more direct was because it only looks at tie & near-tie probabilities, and the difference in utilities depending on whether you change those ties & near-ties. The delta-p approach _appeared_ more computionally roundabout because of the intermediate quantities. But, not having actually worked-out either, I of course wasn't in a position to say for sure that one approach is roundabout. And, when I hear that the calculations are the same, despite the very different sound of the wording, I have no reason to doubt that, since I haven't gone into those calculations. Appearances can be deceptive, and, though the methods seemed different, I'm not now claiming that they're different. I'd said: Find, somehow, the probability of all the possible n-member ties & near-ties, before your vote is cast (last). For each of the 2^N ways that you could vote, a sum can be written, each of whose terms multiplies the probability of a tie or near-tie by the difference in the utility of what would happen if you didn't change that tie or near-tie and what would happen if you did. Find the way of voting that maximizes that sum. You replied: If you calculate this based on your 2^N (or 2^M, since I use M to represent number of candidates) ways of voting, you will have far more calculations to do. Fortunately, this is approval, so you only have to determine whether to vote for your second favorite assuming you've already approved your favorite, and you third favorite based on whether you approved your first two choices... I reply: ...and isn't it also assumed that you're not voting for any candidate whom you like less than the one currently being considered? At first my concern was that if you later decide to vote for someone lower in your sincere ranking, that could reduce the utility of a tie that had made it worthwhile to vote for your 2nd choice, making it no longer desirable to vote for the 2nd choice. But then I realized that that's skipping, something that's been said to be called for only situations so implausible that they can be ignored. So one could safely use the stepwise procedure that you describe, if it can be assumed that skipping doesn't pay. You continued: , and so on...that's M possible ways of voting. I reply: Yes, much easier. Considering all 2^M ways of voting would be good to avoid if possible. But if it were felt possible that skipping conditions could obtain, wouldn't it then be necessary to look at all 2^M ways of voting? But I'm not saying that skipping conditions are likely enough to warrant considering them. Mike _________________________________________________________________ Send and receive Hotmail on your mobile device: http://mobile.msn.com ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
