I've been meaning to post this message for several days, but didn't have the opportunity to spend enough time on the computer, till today. Joe & Richard have just posted on that topic, and so it might seem as if that's why I'm posting on it today. But I've been meaning to post this for days. I mention that because their messages might make this one seem a little on the elementary side, and so I don't want it to seem as if this message is supposed to be some sort of rebuttal or enhancement to their messages.
Most likely someone here as already checked this matter out, and has maybe posted about it. I like the election-utility strategy for Approval, which is why I asked Richard about what's been demonstrated about it. It's one of the useful, practical Approval strategies, and is especially easy to use. Though election-utility depends on an assumption that might be more approximate than the assumptions that Weber-Tideman uses, all the inputs for these methods are approximate anyway, and so it's questionable whether any serious loss of accuracy results when using approximate methods. Say there are 4 candidates: a,b,c & d. Using Weber's many-voters simplifying assumptions, and his method for maximizing one's utility expectation, we should vote for candidate a if & only if: Pab(Ua-Ub)+Pac(Ua-Uc)+Pad(Ua-Ud) > 0 One assumption of Tideman's estimating method is that we can replace that with: PaPb(Ua-Ub)+PaPc(Ua-Ub)+PaPd(Ua-Ud) > 0 , where Pa is the probability, if there's a tie or near-tie for 1st place, a will be in it. That can be written: Pa[Pb(Ua-Ub)+Pc(Ua-Uc)+Pd(Ua-Ud)] > 0 We can drop the Pa, and have: Pb(Ua-Ub)+Pc(Ua-Uc)+Pd(Ua-Ud) > 0 That can be written: Ua(Pb+Pc+Pc) - [PbUb+PcUc+PdUd] > 0 Ua(1-Pa) - [PbUb+PcUc+PdUd] > 0 Ua > UaPa+UbPb+UcPc+UdPd If Pa were the probability that a would win, then this would be a statement of the election-utility strategy. Under the assumptions of this discussion, the election-utility strategy maximizes utility expectation if the probability that candidate a will be in a tie or near tie for 1st place if there is one is the same as, or proportional to, the probability that candidate a will win. Well, it's reasonable to assume that the candidate a's tie-or-near-tie probability is proportional to his probability of winning. If a is more likely to win than b, it's because a is a stronger candidate, and s/he is also more likely to be in a tie or near tie for 1st place. Now, Tideman's approximate assumption that a candidate's tie-or-near-tie probability is proportional to the square root of his win-probabiliity is probably better. But, as I said, the inputs of these methods are so approximate anyway, that we needn't quibble about accepting an approximation, and the election-utility strategy is a very useful one. In fact, I wouldn't hesitate to use election-utility, Weber-Tideman, top-2-contenders, etc, even in committee elections, for the reason stated in the previous paragraph, and because though 3-way ties aren't vanishingly unlikely in committees, they're still significantly less likely than 2-way ties. Mike Ossipoff _________________________________________________________________ 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
