Tideman suggested estimating the proportion relationship of the tie probabilities by saying that the probability of i being in a tie for 1st place is proportional to the square root of his probability of winning. He gave a justification for that rough estimate. I described his geometrical justification, and suggested another justification too, at http://www.barnsdle.demon.co.uk/vote/sing.html , in the Approval & Plurality Strategy article listed in the main menu. It's just an estimate, for when there's nothing else to go on. Unlike tie probabilities, win probabilities are something that people have a natural feel for. If someone declares candidacy, we have an impression of how realistic their candidacy is. Of course we don't need the actual tie probabilities, just a set of numbers that are in the same proportion relation to eachother as the tie probabilities are. That's what the square roots of the win probabilities can be used to estimate. Of course one might rate merit or winnability by giving the best or most winnable candidate 1, 10, or 100, and then rating the others as fractions of that. Or by giving the worst or least winnable one 1, and rating the others as multiples of that. When writing the winnability ratings below, I forgot to do it that way, but I'm using the example as-is, instead of starting over. 4 candidates: A, B, C, & D. Rate how good they are: A 10, B 8, C 6, D 4 Rate how realistic their candidacy seems: A 3, B 5, C 7, D 9 Since the square root of win probability is estimated to be the probability of being in a tie, then we can multiply those tie probabilities of i & j to get an estimate of the probability that i & j will be in a tie for 1st place. Only an estimate. It seemed more convenient to do the square root of the product instead of square-rooting before multiplying. Strategic values of B & C: Sb = sqr(3X5)(-2) + sqr(5X7)(2) + sqr(5X9)(4) = 30.9 Positive strategic value. Vote for B. Sc = sqr(7X3)(-4) + sqr(7X5)(-2) + (7X9)(2) = -14.3 Negative strategic value. Don't vote for C. So you vote for A & B. For Plurality strategy, you need the strategic value of A: Sa = sqr(3X5)(2) + sqr(3X7)(4) + sqr(3X9)(6) = 57.25 A has by far the largest strategic value, so we vote for A in Plurality. Based on the winnability estimates and our utilities, our best strategy in this example is to vote for the least winnable candidate, in Plurality. p.s. There was a typo in a recent letter from me about my wager probability definition. I'd said "The probabilities of the wagered-for outcomes of the Wi are numbers such that..." I meant: "The probabilities of the wagered-for outcomes of the Wi are Pi values such that..." As I said, if "if the Pi are all equal" is added, then the definition specifies a certain particular set of Pi, and the definition is a complete definition, whereas if the Pi needn't be equal then it only describes a necessary condition for the Pi to be the probabilities. For one isolated unrepeatable event, I'd say that its probability is what it would be if lots of other events were added to it, to make the series in the wager definition or the frequency definition. No doubt the probability of an unrepeatable event has already been defined, and could be looked up somewhere. I posted my definitions not because I believe that no other definition is available, but to show that I haven't been using the word "probability" without a meaning all this time. My wager definition & frequency definition are efforts to write what probability of an unrepeatable event has always meant to me when I've used the word. Mike Ossipoff _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com
