[EM] Better Than Expectation Approval Voting
Mike's exposition of basic Approval and Range strategy as variations on the theme of Better Than Expectation strategy was very interesting and valuable, including the recommendation of introducintg Approval after score or grade voting, which are much more familiar to most people. That was probably the most important part of his message, but I want to make a few more remarks about Approval strategy. 1. When there are candidates between the two front runners and you are not sure where to draw the approval line, put it adjacent to the candidate with the greatest likelihood of winning. In other words put your approval cutoff adjacent to the candidate most likely to win on the side of the candidate next most likely to win. This is what Rob LeGrand calls strategy A. 2. Suppose that order is easier than ratings for you. Joe Weinstein's idea is to approve candidate X if and only if it is more likely that the winner will be someone that you rank behind candidate X than someone that you rank ahead of candidate X. Note that when there are two obvious frontrunners Joe's strategy reduces to Rob's strategy A. 3. Suppose that on principle someone would never use approval strategy on a score/grade/range ballot, but is forced to use an approval ballot anyway. How could they vote as close as possible to their scruples? For example suppose that you would give candidate X a score of 37 percent on a high resolution score ballot, but are forced to vote approval style. In this case you can have a random number generator pick a number between zero and 100. If the random number is less than 37, then approve the candidate, otherwise do not. If all like minded voters used this same strategy, 37 percent of them would approve candidate X, and the result would be the same as if all of them had voted 37 on a scale from zero to one hundred. Now for the interesting part: if you use this strategy on your approval ballot, the expected number of candidates that you would approve is simply the sum of the probabilities of your approving the individual candiates, i.e. the total score of all the candidates on your score ballot divided by the maximum possible score (100 in the example). Suppose that there are n candidates, and that the expected number that you will approve is k. Then instead of going through the random number rigamarole, just approve your top k candidates. Forest Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Three variations of GMAT and MMT
MTGMAT: This is just GMAT with an initial search for a majority-top-ratings winner. Majority-Top-Greatest-Mutual-Approval-Top (MTGMAT): If one or more candidates have top ratings from a majority of the voters, then the winner is the one of those with the most top-ratings. Otherwise do GMAT. [end of MTGMAT definition] I don't know if this is better than GMAT. It might be. MMMT: This is just MMT with an initial search for a majority-top winner. That initial majority-top search is superfluous, because, if there are one or more candidates with top ratings from a majority, then the MMT winner will be one of those anyway. So I don't suggest MMMT. GMMT: This differs from MMT by looking for the mutual majority set whose members are rated above bottom by the most voters, and then electing the most top-rated winner in that set. It might be better than MMT. It loses most or all of MMT's brief-definition advantage over GMAT. Greatest Mutual Majority top (GMMT): A mutual majority set is a set of candidates who are all rated above bottom by the same majority of the voters--where, for each ballot in that majority of voters, the set includes at least one top-rated candidate on that ballot. The winning mutual majority set is the one with the most voters rating all of its members above bottom. The winner is the most top-rated candidate in the winning mutual majority set. [end of GMMT definition] Mike Ossipoff Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Better Than Expectation Approval Voting (2nd try readable format)
Mike's exposition of basic Approval and Range strategy as variations on the theme of Better Than Expectation strategy was very interesting and valuable, including the recommendation of introducintg Approval after score or grade voting, which are much more familiar to most people. That was probably the most important part of his message, but I want to make a few more remarks about Approval strategy. 1. When there are candidates between the two front runners and you are not sure where to draw the approval line, put it adjacent to the candidate with the greatest likelihood of winning. In other words put your approval cutoff adjacent to the candidate most likely to win on the side of the candidate next most likely to win. This is what Rob LeGrand calls strategy A. 2. Suppose that order is easier than ratings for you. Joe Weinstein's idea is to approve candidate X if and only if it is more likely that the winner will be someone that you rank behind candidate X than someone that you rank ahead of candidate X. Note that when there are two obvious frontrunners Joe's strategy reduces to Rob's strategy A. 3. Suppose that on principle someone would never use approval strategy on a score/grade/range ballot, but is forced to use an approval ballot anyway. How could they vote as close as possible to their scruples? For example suppose that you would give candidate X a score of 37 percent on a high resolution score ballot, but are forced to vote approval style. In this case you can have a random number generator pick a number between zero and 100. If the random number is less than 37, then approve the candidate, otherwise do not. If all like minded voters used this same strategy, 37 percent of them would approve candidate X, and the result would be the same as if all of them had voted 37 on a scale from zero to one hundred. Now for the interesting part: if you use this strategy on your approval ballot, the expected number of candidates that you would approve is simply the sum of the probabilities of your approving the individual candiates, i.e. the total score of all the candidates on your score ballot divided by the maximum possible score (100 in the example). Suppose that there are n candidates, and that the expected number that you will approve is k. Then instead of going through the random number rigamarole, just approve your top k candidates. Forest Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Approval strategies
Forest-- It's certainly true that the Approval strategies that I described in my previous posting on the subject aren't the only possible strategies. Of course that's because it depends on what the voter knows, or what the voter has a feel about. My posting covered a few kinds of voter perceptions or estimates. There could be different ones too. Candidates whose utility is between those of the two perceived frontrunners: There too, it depends on what the voter knows or has a feel for. If you have an estimate of the utilities of the two frontrunners, and of their win-probabilities, then you can calculate, from that, your expectation for the election, and then vote for candidates whose utility is greater than that. The expectation for the election is Pw*Uw + Pb*Uw (where those symbols have the obvious meanings given below). so vote for every candidate whose utility is greater than that. Some time ago, we showed in at least two different ways that (provided that one has the necessary information or estimates) one should vote for the better-than- expectation candidates. But it can be looked at in this way too: Pw is the probability that the worse of the 2 frontrunners has more votes than the best one. Pb is the probability better of the 2 frontrunners has more votes than the worse one. You're looking for the utility that makes equal the damage-expectation of the inbetween candidate beating the better frontrunner (in the unlikely even that s/he could), and the benefit-expectation of the inbetween candidate beating the worse frontrunner. Call that utility U. Uw is the utility of the worse frontrunner and Ub is the utility of the better frontrunner. You want: Pw(U-Uw) = Pb(Ub-U) Multiply both sides out, and collect U terms on the left: U(Pw+Pb) = PbUb + PwUw U = (PbUb + PwUw)/(Pw+Pb) But Pw + Pb is unity. So U = PbUb + PwUw Vote for the candidates whose utility is greater than that. Those are the better-than-expectation candidates. Of course, though you might have a feel for who the frontrunners will be, you might not have an estimate of their win probabilities, or numerical estimates of their utilities. For that situation, in the article that I mentioned, which described a recommended best-frontrunners strategy, the authors (who included Brams, or Fishburn, or both) suggested voting for all the candidates who seem better than the perceived midpoint between the two frontrunners' merit. That assumes an guess, for want of better information, that the two frontrunners have equal probabilities of winning. Why did Rob LeGrand say to put the Approval cutoff adjacent to one of the frontrunners' utilities? Of course if you knew that one of the frontrunners was almost certain to outpoll the other, then the election's expectation would be quite close to hir, because P for that frontrunner would be much larger than for the other frontrunner. So maybe Rob was assuming that it's a sure thing that the more winnable frontrunner will win. So when more accurate information isn't available, it's a question of which is a better guess?: 1. Both frontrunners are equally likely to outpoll eachother. or 2. The more likely front runner is sure to outpoll the other. When one doesn't know, I'd tend to go along with Brams /or Fishburn's guess, #1 above. Of course, sometimes you might feel a lot of sureness that one frontrunner or the other will outpoll the other. Yes, if you feel sure enough that a particular one of the two frontrunners will outpoll the other, then the Approval cutoff should be adjacent to hir. Otherwise, though, guess #1 is the right one. In these situations, lack of information seems more likely, and that makes #1 the typically best guess. Joe's suggestion for when you have a ranking but not ratings: Sure. If you know nothing about the ratings differences, then you'd want to at least equalize the probabilities of your pair-tie rival being above or below in your ranking. But that requires a probability estimate, and that can be difficult too. Is the probability really what you have the best feel for? The direct gut feeling? I don't think so. I think that feelings of threat, or feelings of promise are the more basic feeling, more instinctive, from our earliest ancestors. So place your approval cutoff where the feeling of threat from worse candidates feels as strong as the feeling of promise from better candidates. Or put it this way: Which feels more: The fear that your vote for x will make hir take the win from someone better, or your hope that s/he'll take the win from someone worse. It seems to me that, at the time, I called that the Threat/Promise strategy. Mike Ossipoff Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Clarification about Threat/Promise strategy
When I said that it's a matter of your fear that you'll help the inbetween candidate take the win from the better frontrunner, vs the hope that you'll help hir take the win from the worse frontrunner, of course that hope and that fear aren't just feelings about probabilities. They have to do with the product of probability and goodness or badness of the result. But we don't know them from multiplying numbers. We know them as our most basic feeling about the situation. I'm saying that that overall feeling of threat or promise is more basic than probability estimates. Or utility estimates. Likewise, when I spoke of the threat of the worse frontrunner and the promise of the better one, I was referring to what could be called the product of a probability and a badness or goodness of outcome--though it's not calculated; It's what is directly felt. Which of those candidates is the imporant one to consider in regards to that basic gut perception. Do you want to avoid taking away that hope, or do you want to try to take away that threat? Mike Ossipoff Election-Methods mailing list - see http://electorama.com/em for list info
[EM] A better between-frontrunners approval cutoff. Typo.
First the typo: I said that the expectation for the election is Pw*Uw + Pb*Uw Of course I meant Pw*Uw + Pb*Ub. A better between-frontrunners approval cutoff for when good numerical information isn't available: I feel that the worse (or better) frontrunner is a little more likely to outpoll the other frontrunner, and so I'll put the approval cutoff a little (correspondingly) closer to that slightly more win-likely frontrunner. Of course, the placement of the approval cutoff is done purely intuitively, according to how much more winnable one frontrunner seems. If one feels twice as likely to outpoll the other, then put the approval cutoff twice as close to hir. It's done intuitively, by feel, rather than numerically. Or: I don't have a feel for which frontrunner is more likely to outpoll the other, and so I'll put the approval cutoff halfway between their merits. By the way, for Score-Voting balloting, I like the slider-placement balloting, which is perceptual and intuitive instead of numerical. Mike Ossipoff Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Better Than Expectation Approval Voting (2nd try readable format)
Forest wrote: Now for the interesting part: if you use this strategy on your approval ballot, the expected number of candidates that you would approve is simply the sum of the probabilities of your approving the individual candiates, i.e. the total score of all the candidates on your score ballot divided by the maximum possible score (100 in the example). Suppose that there are n candidates, and that the expected number that you will approve is k. Then instead of going through the random number rigamarole, just approve your top k candidates. So we could justifiably call this strategy the honest approval strategy, since if preferences are sufficiently mixed and all voters use this strategy, the outcome is the same as the one with sincere Range ballots, i.e., the option with the highest total rating! Happy Holidays from Jobst to all of you! Election-Methods mailing list - see http://electorama.com/em for list info
