There's a smooth, seamless gradation from the top-2 (or best-frontrunner) strategy to Weber's method. In this message I'd like to describe that, including some things I'd left out in previous messages.
Top-2 strategy: Vote for whichever of the 2 most likely top votegetters that you prefer to the other, and for everyone whom you like better than him/her. If you're really sure they're the frontrunners, then you definitely want to vote for the better one, but not the worse one. And if you're sure they're the only candidates who'll be in a tie for 1st, then it's harmless (if profitless) to vote for other candidates too. The next most likely kind of tie would include one of the probable top-2, instead of one. Ties involving two nonfrontrunners can be ignored for now. If a candidate's merit is between those of the putative frontrunners, then one strategy suggested is to vote for him if he's better than the average merit of the frontrunners. That assumes that he's equally likely to tie each of them. But say you have reason to believe that one frontrunner is more likely to outpoll the other than vice-versa. Say that X & Y are the expected frontrunners, and it's 70% that X will outpoll Y, and 30% that Y will outpoll X. Then, if Z is some other candidate, whose utility is between X & Y, vote for Z if (.7)(Ux-Uz) < (.3)(Uz-Uy). That also gives: Vote for Z if Uz > .7Ux+.3Uy. Since .7 & .3 are the probabilities that X & Y will win, assuming that they'll be the top votegetters, that means voting for Z if he/she is better than the election's expected utility. One could consider other possible ties between X or Y and other candidates, but, since they involve considering voting for Y or not voting for X, we shouldn't take the top-2 strategy that far. Considering those other ties means going on to Weber. Say that, additionally, it's 80% that X & Y will be the top votegetters and will be in a tie for 1st if there is one. (Isn't it reasonable to have those probabilities the same if we have no reason to expect the likelihood of a tie to depend on who the frontrunners are?). And say that you estimate a 2% probability that the top votegetters and candidates who'll tie if anyone will are neither X nor Y. Then, for Weber, Pxy = .8 Pxz = (.18)(.7)*Pfz (where Pfz is Pz divided by the sum of the Pi of the candidates other than X & Y. Pi is the probability that i will be in a tie for 1st, and is gotten from Wi, by Tideman's estimating method). Pij, where i & j are 2 candidates other than X & Y, = (.02)PfiPfj To get the Pi of the candidates other than X & Y, just call the most winnable one's winnability 1. Then assign numbers to the others accoring to how much less winnable they are. Take the square roots of those numbers to get the Pi. Pfi is i's Pi divided by the sum of the Pi of the other candidates other than X & Y. This way of estimating Weber's Pij is probably more reliable than estimating the Wi and using Tideman's estimating method. Mike Ossipoff _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp. ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
