Re: [EM] Recursive Elimination Supervisor
Tony, I am a little worried that this simplification gives room for a "Independence of Irrelevant Alternatives" problem to creep in. Many methods suffer from this IIA problem (which says that the Winner shouldn't change when some other candidate sits out) and it may be too much to expect that we can make IIA hold in our recursion, yet in a way a weak variant of IIA is the whole basis of my idea: Weak version of IIA: The winner shouldn't change if the worst candidate is thrown out. This seems like a reasonable requirement for a decent method. The unsimplified version implicitly uses a slightly stronger version (but still close to the weak version): The winner shouldn't change if the worst candidate or next to the worst candidate is thrown out. In this simplified version we're using a much stronger version of IIA: The winner shouldn't change if the Seed Loser is thrown out (unless the winner is the seed loser). Forest On Tue, 27 Feb 2001, Forest Simmons wrote: > Tony, > > here's a simpler version of the Recursive Elimination Supervisor, > based on a suggestion of yours. > > Step 1. Use the seed method in reverse to find the "Seed Loser" SL, from > among the N candidates. > > Step 2. While the SL sits out, recursively supervise the seed method to > find an N-1 stage recursive winner RW from among the N-1 remaining > candidates. > > Step 3. Compare SL and RW directly. Whichever is better is the N stage > recursive winner. In case of a tie between these two, choose RW. > > > Forest > >
Re: [EM] Recursive Elimination Supervisor
Tony, here's a simpler version of the Recursive Elimination Supervisor, based on a suggestion of yours. Step 1. Use the seed method in reverse to find the "Seed Loser" SL, from among the N candidates. Step 2. While the SL sits out, recursively supervise the seed method to find an N-1 stage recursive winner RW from among the N-1 remaining candidates. Step 3. Compare SL and RW directly. Whichever is better is the N stage recursive winner. In case of a tie between these two, choose RW. Forest
Re: [EM] Recursive Elimination Supervisor
Interesting. This might turn out to be the best IRV mitigation. Not that IRV advocates have ever accepted any mitigation. As I always say, they seem determined to impose all the worst of IRV on the voting public. But mitigations such as this should be offered to them, so that we can later show that not only did they know about IRV's problems, but they were also offered a number of mitigation compromises, none of which they accepted. Then there's always the possibility that a new method can turn out to be one of the best, or maybe the best, in some important regard. It would be a mistake to take it as certain that the best methods known so far will always be the best known methods, and so sometimes we've checked out new methods to find out if they challenge Condorcet for the title of the best. So far none of them have, but when a new method looks good, it should of course be checked out, and so let's find out if this particular new method is up there with the best. Mike Ossipoff _ Get your FREE download of MSN Explorer at http://explorer.msn.com
[EM] Recursive Elimination Supervisor
Forest, I found your scheme fascinating, and it worked with an example or two I tried it with. For three candidates, we first select one who wins conventional IRV, and who we can figure is unlikely to be the CL. We're not sure, but that isn't necessary, since the principal purpose of this step is just to cut the number of candidates in the next step to two. Then we compare the two "lower" candidates. The one who loses this comparison is definitely not the CW, and is eliminated from counting. Next, we compare the two remaining candidates. Since the candidate already completely eliminated is not CW, one of the remaining two must be the CW, if there is one, and that is the one that wins. You have invented a Ranked Pairs method! What's interesting is the way the method deals with a situation in which there is no CW. I haven't thought through the complications sufficiently to know, but with three candidates, is there a likelihood that your method would pick a different winner than straight IRV? Some other interesting things: The first step is to find a temporary IRV winner in order to find the first loser. How about just using IRV inverted to elect a first loser directly? Then eliminate him/her/it, transfer votes and proceed as usual. Or suppose: 1. Use IRV to select a temporary winner, say, A. 2. Select a loser from between B and C, say it's C. That leaves A and B. 3. Select a winner from between A and B, say it's B. That leaves A and C. 4. Select a loser from between A and C. Suppose it's A. That leaves B and C. 5. Select a winner from B and C. Suppose it's C. Stop here and call C the winner. You could keep iterating. Does it converge to a single winner every iteration? If so, is it a better winner in some sense than you'd get straight from IRV? How much of a difference does it make if you use something other than IRV? Just some interesting possibilities. I think. Certainly an interesting scheme. Tony
[EM] Recursive Elimination Supervisor
Any trivial method can correctly distinguish winner and loser in a two way contest. How about a three way contest? The answer is the same, yes, any old trivial method can correctly distinguish winner and loser in a three way contest, IF (this is one of those big IF's) the method is not just left to its own devices, like some forlorn latchkey child. For example, let's use IRV as a "seed method" for our recursion. IRV does just fine in a two way single winner contest. And for definiteness, let's assume that our version of IRV breaks two way ties by drawing a ballot at random. Now, let's go to work on the three way contest: Step 1. Have IRV tell us which of the three candidates he thinks should win, temporarily label this candidate SW (for Seed Winner), and set it aside for later use. Step 2. Have IRV point out from among the remaining two candidates the one that is the worst (or the tie breaker if they are tied.) IRV can handle this because it is just a two way contest. Label this candidate RL (for Recursive Loser). Step 3. Trade in SW for RL, so that now, of the three, RL is sitting out. Step 4. While RL sits out, have IRV pick the winner from the other pair. (IRV can do this because he can handle a two way contest.) This time label IRV's pick W for Winner and give IRV a star on his forehead. Here's why W deserves to be labeled as Winner: The candidate RL found in step 3 could not be a deserving winner (or at least not uniquely so) because IRV distinguished him to be a loser from among two candidates, and IRV doesn't make mistakes on two way contests. Therefore, a true winner was to be found among the other two. In step 4 IRV picks a deserving winner from among the two. Notice that although IRV is prone to get into trouble when unsupervised, a little guidance from the older children helps him to earn a gold star. We have just shown that IRV can pick a three way winner when correctly supervised. Similarly (by switching the directions of all the preferences) a well supervised IRV can pick a three way loser competently. Now let's help IRV handle a single winner four way contest: Step 1. Have IRV pick a seed winner SW from the four candidates and set this seed winner aside as before. Step 2. Supervise IRV to pick a Recursive Loser RL from the other three candidates. IRV does fine on this task with supervision as we just finished showing. (This is a recursive step in the general case.) Note that this RL is not a deserving winner, or at least not uniquely so. Step 3. Trade in SW for RL, and note that with RL sitting out, there must still be at least one deserving winner among the remaining three. Step 4. While RL sits out, supervise IRV to find the Winner W from the remaining three candidates. We showed above that IRV can handle this task under supervision (This is also a recursive step in the general case.) It is easy to see that the Winner in step 4 is truly a deserving winner, since she was competently chosen from a group of three that had to contain a deserving winner. So under appropriate supervision IRV can handle four way contests. Now let's do the general case. To avoid the above cumbersome circumlocutions that result from constant care to include the possibility of multiple deserving winners, I give a version that doesn't apply when there are ties or cycles of deserving winners: Step 1. Let IRV pick the Seed Winner SW from among the N candidates, and set this SW aside temporarily. Step 2. Recursively supervise IRV to pick the Recursive Loser RL from the remaining N-1 candidates. (Our induction hypothesis is that supervised IRV is competent for finding best and worst among N-1 candidates.) Step 3. Trade in SW for RL, and note that with RL sitting out, the remaining N-1 candidates must contain the true winner. Step 4. While RL sits out, recursively supervise IRV to pick the Winner from among the other N-1 candidates. Comments: In the first step we let IRV pick the seed winner just to make him feel important. The seed winner could be chosen at random, but we don't want to hurt IRV's feelings, so we let IRV choose the one to set aside, and give it a fancy name, "the seed winner." In practice it would only make a difference if there were multiple deserving winners. In that case we can think of IRV's influence as providing flavor to the winner. Using another seed method besides IRV would give a different flavored winner. I'm too tired to write in the circumlocutions for the case of non-unique deserving winners into this general recursion statement. Let someone else copy them from the first two sample induction steps above. I haven't done the details on paper, but I believe it can be easily proven in an induction parallel to the recursion that all (big W) Winners and "Recursive Losers" (RL's) in the recursion are Condorcet Winners of the direct and reverse preference rankings, respectively, in the case where deserving winners are unique. This super
