Remember the "approval stable winner" is supposed to be the candidate that stands the best chance of still being approved even when the polls and pundits raise him/her up to be the target candidate to beat during the lead up to voting night, as we saw in the lead up to Super Tuesday, for example. This is a class of methods in which the candidate X with the highest ratio of S(X) over MPO(X) is elected, where MPO(X) is the Max Pairwise Opposition against X, and S(X) is the estimated sincere zero informatinsupport for X. It is this S(X) support estimates that distinguish the different methods or versions of the method, if you will. Why zero information? Because we know from experience that the pollsters and pundits have their agendas and biases that can turn "information" into disinformation that is worse than zero information. The purpose of a Designated Strategy Voting (DSV) style method is to trust the information from the ballots themselves over the biased polls. Voters can take the polls with a grain of salt if they know that honest ballot information will be used fairly. There is no strategy free method, but if the DSV method is trustworthy it will make better strategy decisions for the voters than the pollsters and pundits will. Here are some possibilities for S(X) in no particular order: 1. Let S(X) be the number of ballots that rank or rate X above bottom. 2. Let S(X) be the number of ballots that explicitly approve X. 3. Let S(X) be the number of ballots that rank or rate X equal top plus half the number of ballots that rank X strictly between Top and Bottom. 4. Assuming Range style ballots, let S(X) be the sum of the scores of X over all of the ballots. 5. Let S(X) be the number of ballots on which X is rated at least as high as the midrange of the possible ratings. 6. Assuming the ratings are in the range from zero to one, let S(X) be the number of ballots on which the number of candidates rated strictly above X is strictly greater than the total of all ratings on that ballot. This S(X) is what I call the strategic zero information approval total, for reasons that I have explained elsewhere, and I will sketch now for easy reference: It is a consequence of a general principle (the corner point principle) In range voting the feasible decision region for each voter is an n-dimensional hypercube where n is the number of candidates. A corner of such a cube is a point where all of the candidates are voted at the extremes, i.e. approval style strategy is optimal. So here is the question we are faced with: how do we convert sincere ratings into optimal approval ballots in a zero information setting? It has often been observed that from a statistical point of view, if each voter were to approve with probability p percent every candidate rated on herr ballot p percent of the way between min range and max range, then provided the electorate were sufficiently large, the range election outcome would not be affected. The question comes up ... what is the expected number of approvals on your ballot if you were to use this method to convert your ballot from fractional score o approval. The answer from elementary probability theory is very simple: it is merely the sum of the candidate ratings on your ballot after they have been normalized between zero and one. This result allows us to determine how many candidates to approve without any need to flip coins or spin spinners. The method has its determinacy restored after our brief excursion into a Monte Carlo thought experiment. If the sum of the normalized ratings rounds to n, then approve your top n favorite candidates. This is what I call
"strategic zero information approval." I have suggested six estimates for S(X). Have I overlooked any goods ones? Any tweaks? Other comments? I should mention that the resulting methods based on any of these six possible definitions of S(X) result in monotonic, clone free methiods that satisfy Independence from Pareto Dominated Alternatives (IDPA) and the Favorite Betrayal Criterion (FBC). If you would rather trade in the FBC for the CC (Condorcet Criterion) you can use "covering enhancement" on the S(X)/MPO(X) ratio order to "upgrade" to an uncovered winner without sacrificing any of the mentioned criteria except the FBC. Thanks, Forest ---- Election-Methods mailing list - see https://electorama.com/em for list info |
Here's an important tweak: The version of MaxPairwiseOpposition that we use needs to include power truncation, otherwise we fail the Plurality Criterion. In this context power truncation means a candidate truncated on a ballot is counted as "opposed" by every other candidate on the ballot, even the other truncated candidates With this power truncation in place every method (one thru six) of my previous message is equivalent to ordinary Approval when the ballots populate only the top and bottom slots ... no in-between preferences indicated. As you know the same can be said for all Condorcet Compliant methods;when there are no intermediate rankings there is always (except when the two candidates with the most top votes have the same number of top votes) a pairwise beats-all candidate, and that candidate is the one that is ranked top or equal top on the greatest number of ballots. Another possible tweak to make the method more symmetrical is to throw in Symmetric Completion for the intermediate ranks (between top and bottom). Put this version with method number three for measuring support S(X), and we have the (IMHO) best zero information method of this type for ranked preference ballots (equal rankings and truncation allowed). To summarize this version, the ballots are ranked preference style with truncations and equal rankings allowed. The winner is the candidate X with the greatest ratio of S(X) to MPO(X), where S(X) is the number of ballots on which X is ranked equal top plus half the number of ballots on which X is ranked strictly between bottom and top, and MPO(X) is the Max Pairwise Opposition to X with Symmetric Completion below top and double that for bottom. In other words, let PO(X, Y)) be the number of ballots on which Y is ranked above X, plus half the number of ballots on which Y is ranked equal to X plus the number of ballots on which neither X nor Y is ranked at all. Then MPO(X) is the max (over Y) of PO(X,Y). As stated before this method satisfies mono-raise, mono-add, clone winner. clone loser, Independence from Pareto Dominated Alternatives, and the Favorite Betrayal Criterion. And like all good EM List methods, it reduces to Approval when the voters vote some candidates equal top and truncate the rest. Also the information needed for computing the winner is capable of efficient encoding, ballot-by-ballot in an additive form. What's not to like? As usual, in practice this version or any of the other versions can be fit into a Candidate Proxy or other VPR (Vote for Published Ranking) framework for the convenience of the voters. In my next message I will talk about the biggest obstacle to cardinal ratings... how to rate the candidates in the zero info setting, i.e. sincere ratings without any information about the preferences of other voters. What does sincere rating even mean? On Fri, Mar 6, 2020 at 2:52 PM Forest Simmons <[hidden email]> wrote:
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The power truncation fixes the Plurality failure, but at the expense of sacrificing the original objective. I can see another way of fixing the Plurality problem, but at the expense of the FBC. Perhaps we are up against another kiind of impossibility. Back to the drawing board. On Sat, Mar 7, 2020 at 11:40 AM Forest Simmons <[hidden email]> wrote:
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Here's an example that illustrates the problem: 26 A 24 C=A 25 C=B 25 B First without power truncation MPO(A) = MPO(B)= 50=W(A)=W(B), MPO(C)=26. W(C)=49. The ratio of W(X) to MPO(X) is unity when X is A or B, but is 49.26 which is much lager than unity when X is C. So without power truncation X is the winner. This violates Plurality because A for example has more first place rankings than C has any kind of ranking. With power truncation since all of the votes are at the two extremes (Top when not Bottom) the method reduces to approval and candidates A and B are tied for first place with 50 percent approval against 49 percent for C. But the original spirit of the method was to ask, for example, how much support would C have compared to the other candidates if C were the sitting duck on the hot seat, i.e. the projected winner? No doubt that W(C) would still be 49. But would A's support be only MPO(C), i.e. would A only be supported on by those voters that ranked A above C? It seems to me that those voters that voted A=C would continue to support A, so A's support would beits full approval value, and similarly for B. So in general, how can we give candidates like A full credit for continuing support when ranked equal top with the target candidate without scuttling the FBC It works out OK in this example, but in general when a lower ranked candidate is raised to equal top (like A) it might wreck C's chances for winning without replacing C as the winner, i.e. could get in the way of the FBC. That's the problem. So, as I said, back to the drawing board! On Mon, Mar 9, 2020 at 1:56 PM Forest Simmons <[hidden email]> wrote:
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