You wrote:
In the pij equation, Wik represents the probability that i will defeat k strictly on votes (i.e, they will not tie), compared to Bik which includes the possibility that i beats k in a tiebreaker. Wik is used here since the definition of pij as I understand it only includes two-way tie possibilities. I reply: But I was saying that Weber, like delta-p, can deal with n-member ties and near-ties. So I'm not saying that Weber has to only deal with 2-member ties. And when I say "...and near ties", that means situations where your ballot can change a decisive result into a tie. Change a decisive result that j wins into a tie that i has some probability of winning, and which has some utility to you different from j's utility, depending on who is in the tie. 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. And I emphasize that, when Weber takes n-member ties & near-ties into account, that doesn't mean that Weber is achieving delta-p's accuracy by borrowing something from delta-p. n-member ties & near-ties are part of the committee landscape, and are not the property of delta-p. Weber and delta-p are 2 approaches, either of which could take all the possibilities into account, for committee votes, or could just consider 2-member ties, for public elections. You continued: Also, pt is the probability that there is a tie for first place. Dividing by pt converts pij to a conditional probability. pij is the probability, given a tie for first place, that it is between i and j (i and j both score some value X, and all other candidates score less than X). I reply: Yes, that's what I've meant by Pij. You continued: The distinction between Gj(X) and Fj(X) needs a little more consideration. If we only consider the probability of j being involved in a first-place tie (2-way or multi-way), so that we want to calculate the probability that our vote will convert j from a possible tie-break winner to a decisive (by one vote) winner, we can use Fj(X). But we should also consider the probability that our vote converts j from a loser by one vote to a tie-breaker participant. Gj(X) accounts for both possibilities. Gj(X) is different from Fj(X) only if Fj(X-1) is different from Fj(X). When the population is large, this is the case within a very small margin of error. When the population is small, the more accurate Gj(X) should be used. I reply: That's what I meant when I said n-way ties _& near-ties_. The near-ties are situations where your ballot can change a decisive win into a tie, not necessarily 2-member. You continued: So the differences between the two methods all become insignificant for large populations. I reply: But the methods don't have that difference, when I'm referring to Weber that takes into account n-way ties & near-ties. You continued: My equations don't use the tie probabilities. They use the score probabilities, Fi(X); all other variables are derived from the Fi(X). I reply: I noticed that when I reread the posting. It's a very different approach from Weber's. If Weber's method is ignoring something, that that would make it less accurate, but, when talking of committee Weber, I didn't intend for it to ignore n-way ties or near-ties. You continued: But at any rate you would need Fi(X) to calculate pij, as well as to calculate delta-Pij. I reply: 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 continued: Obviously you could take the pij equation above and replace all the Wik with Bik, and that would account for n-way ties. That would make Weber nearly as accurate as delta-Pij. I reply: Exactly. If Weber doesn't make the simplifying assumptions that it uses for public elections, it's as accurate as delta-p. But that doesn't mean that Weber is then borrowing from delta-p. They're 2 different approaches, either of which could make or not make the many-voter simplifying assumptions. I'd said: >So, let's not say that Weber's method is less accurate because, in >that article, it doesn't take into account n-way ties & near-ties. You replied: No, it's less accurate because it uses Fj rather than Gj. I reply: If that means ignoring the possibility that your ballot could change a decisive win into a tie, Weber needn't make that assumption. That's why I've been referring to Weber with n-member ties & near-ties. You continued: The added complexity is actually pretty trivial, though I wouldn't want to calculate either one by hand. I reply: Maybe, but it just seemed to me that Weber is more direct, and that delta-p is more computationally roundabout, calculating various intermediate quantities. But I only have experience with Weber with the many-voter assumptions, and so I can't say anything for sure about one being simpler than the other till I've used both in simplified & not-simplied form. You continued: Also, if I'm just going to estimate the pij (as with the geometric mean or some other approximation), I could make a valid claim that this is also an estimate of the delta-Pij strategy, because, at least for large-scale elections, pij and delta-Pij are practically the same strategy. I reply: If they're both designed to maximize utility expectation, and they both are valid and don't make simplifying assumptions that the other doesn't make, and they both use the same inputs, then they're effectively the same method. Experience with both, with & without the many-voters simplifying assumptions, will tell which approach is easier. You continued: The only added calculations are in the Bik and Gj terms. Are you suggesting that if I use calculated Bik or Gj values then it will be less exact than if I just throw in the Wik and Fj terms in their place? I reply: Not at all. But if delta-p gets its results in a more computationally roundabout way, and the methods are otherwise doing the same thing, then that could make delta-p very slightly less accurate, due to roundoff error. Again, I can't really say that delta-p is more computationally roundabout, till you post an example that makes the many-voters simiplifying assumptions that Weber made in his article, and until , for both methods, there are examples written in which the same kind of probability-info is available to both methods, and they both don't make the many-voters simplifying assumptions. But it _appears_, at this point, that delta-p is likely to be more computationally roundabout. I haven't said exactly what I mean by Weber without the many-voters simplifying assumptions. Here's all I mean. This isn't intended as an instruction, just a rough definition: 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. That's what I mean by Webster's method. Easier said than done, of course, for committees. For public elections, Weber does that in a very simple way, with the simplifying assumptions that are possible under those conditions. You continued: But if it's simplicity you're after, then the grosser approximation of the geometric mean could be used even for small groups. Just include an adjustment to the winning probability for each candidate you've already determined you will vote for. I reply: Sure, the assumption that ties are 2-way isn't really so bad even in committees. 3-way ties & near-ties aren't vanishingly unlikely in committees, but they're still significantly less likely than 2-member ties & near-ties. Most likely, one's guesses about candidates' utilities, and whatever probability-input is used, aren't going to be reliable enough so that there'd be enough precision to be spoiled by the 2-way assumption. So, right now, since I haven't worked out n-way strategy, if I had to do that calculation in a committee, I'd just use many-voter Weber. In our polls last year, my Approval strategy was the one based on unacceptable candidates. But otherwise, my Approval strategy would have been the one that votes down to the likely CW, and no farther, or to the better of the 2 strongest contenders. Approval was the expected CW (and of course that estimate turned out to be right). I'd have probably just voted down to the CW & no farther, rather than try to guess the other strongest contender. In any case, though, I'm sure that I like Approval better than whatever would have been the other strongest contender, and so that strategy too would have led me to vote down to Approval and no farther. Mike _________________________________________________________________ Join the world�s largest e-mail service with MSN Hotmail. http://www.hotmail.com ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
