Mike, I think there may be far less of a difference here than meets the eye. Consider the equations for the two methods:
delta-Pij = -sum_over_X(Fi(X-Ai)*Gj(X)*product_over_k(Bik(X-Ak))) pij = -sum_over_X(Fi(X-Ai)*Fj(X)*product_over_k(Wik(X-Ak))/pt) 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. For large populations, more-than-two-way ties are much less likely than two-way ties, and so can be ignored; this means Wik is very nearly equal to Bik for large populations. Of course, you could accommodate multiway ties by simply replacing the Wik with Bik. 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). pt can vary slightly with changes in the Ai/Ak values. However, pt will be constant for each pij for a given j, so all the pij for that j will scale by the same amount, so the resulting approval decision for candidate j is not affected. Therefore, we can ignore pt altogether. 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. So the differences between the two methods all become insignificant for large populations. The only area where I expect delta-Pij to perform significantly better is in the small-population cases. There are two effects that are significant for small elections: multi-way ties, and the effect of Gj vs. Fj. More comments below. MIKE OSSIPOFF wrote: > > When I said that tie probabilities are more fundamental than delta-p, > I just meant that calculating the delta-p values depends on the > probabilities of the revelevant ties & near-ties, whereas you don't > need the delta-p values to calculate estimate the tie probabilities. My equations don't use the tie probabilities. They use the score probabilities, Fi(X); all other variables are derived from the Fi(X). (In the case of correlated score probabilities you would need additional information not found in the Fi(X).) But at any rate you would need Fi(X) to calculate pij, as well as to calculate delta-Pij. > But, just as the delta-p approach could take n-way ties into account, > for committee voting, or could assume that all ties are 2-way, for > public voting, that's also true of Weber's method. Weber can take > into account n-way ties & near-ties. 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. > 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. No, it's less accurate because it uses Fj rather than Gj. > For public elections, where only 2-way ties are considered, can > the delta-p method match the great simplicity of Weber's method? The added complexity is actually pretty trivial, though I wouldn't want to calculate either one by hand. 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. > If delta-p is more complicated, more calculation-intensive, then > , more likely than not, it has more roundoff error than Weber would, > and is slightly (maybe not significantly) less accurate. I'm not saying > that delta-p isn't accurate enough, only that it's probably less > accurate if it's considerably more calculation-intensive. 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? > Until Weber & delta-p are completely worked-out for taking into account > n-way ties & near-ties, for small committees, I don't know which is > simpler. Of course the 2-way tie experience suggests that Weber will be > simpler, but we won't know about that till both methods are worked out > for small committees. So I'm not now claiming that Weber will be > simpler for small committees. Weber's method would be *slightly* simpler, since we can always leave pt out of the equation. 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 don't want to seem too partisan about Approval strategy methods, > or Condorcet(wv) strategies. If I've seemed so, I'm not someone who'd > fight about those issues. I wanted to just state the case for > truncation in Condorcet(wv), and, in the case of the Approval strategy > methods, I objected to Richard's implication that delta-p is more > accurate than Weber, because it's more accurate when we look at > n-way ties & near-ties with delta-p, but not with Weber. It's not just n-way ties. I don't see that Weber takes into account the other small-population effect. If you put the small-population adjustments (Bik and Gj) into Weber then it becomes the exact equivalent of the delta method (pt has no effect on the outcome). Which is why I say there's less of a difference than meets the eye; both methods are formulaically very similar, and ultimately rely on the same data: the F(X) values, which are inferred from polls or other available data through a statistical model. I'm not out to politicize this issue either; I just hope to set the record straight where I think some inaccuracies have crept in. And if I've made any erroneous statements then corrections are welcome, because I certainly don't want to contribute any new inaccuracies. -- Richard ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
