In my posting yesterday, about election-utility strategy for Approval, I replaced Ua(Pb+Pc+Pc) with Ua(1-Pa), because when I wrote that I must have already started interpreting the Pi as win-probabilities, though they're actually just tie-or-near-tie probabilities.
Of course, as written, the replacement isn't valid. But if kWi replaces Pi at that point in the demonstration, it works. Wi is the estimated probability that candidate i will win, and k is some constant that's the same for all the kWi. So Ua(Pb+Pc+Pd) becomes Ua(kWb+kWc+kWd), or kUa(Wb+Wc+Wd) = kUa(1-Wa) So it's: kUa(1-Wa) - [kWbUb+kWcUc+kWdUd] > 0 Ua > WaUa+WbUb+WcUc+WdUd So, when Weber's many-voter assumptions, and the assumption that Pi is proportional to Wi, are made, the utility-maximizing strategy votes for the candidates who are better than the voter's expectation for the election. One possible remaining objection is that I've assumed that the win probabilities add up to 1, even though there could be ties. But when you guess the candidates' win probabilities, can you guess them so accurately that you can say that they're decisive win probabilities, not just probabilities of winning ultimately (maybe after a tiebreaker)? So it isn't so unreasonable for the Wi to add up to 1. Even when the voter isn't explicitly using election-utility strategy, it can still be said that if the voter is maximizing his utility expection, then, by these assumptions, he's voting for the candidates whose utility is better than that of the election for that voter. So, as Richard was discussing, Approval, when people vote to maximize their utility expectation, maximizes the number of voters for whom the outcome is better than they expected, or better than the incumbant, if that's the utility that they expected, or better than average, if it's zero-info. Mike Ossipoff _________________________________________________________________ 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
