Warren Smith reminds us from time to time that range voting minimizes in some sense something called "Bayesian Regret" which is the difference between the social utility of the "best candidate" and the one elected by sincere ballots. A related but different concept is what I call "ballot disappointment," which tries to quantify the disappointment for one voter in one step of a sequential elimination method. Suppose that an elimination step replaces candidate X with candidate Y, as the new "champion." How much disappointment does that incur for the sincere voter of a ranked preference ballot B? For example if the method must eliminate candidate X in favor of some candidate Y that covers X, it would be nice if Y were the candidate (among those covering X) that incurred the least total disappointment for this transition. Here's my proposal. For each candidate Z let f(Z) be the first place (that is random favorite) probability for candidate Z over the entire set of ballots. Then on ballot B the disappointment in going from candidate X to candidate Y is ... ....zero if Y is ranked ahead of or equal to X else the sum (over all candidates Z ranked ahead of Y) of f(Z). So if Y is ranked higher than X on ballot B, then the voter of ballot B has little cause for complaint, otherwise the disappointment is the probability that a better Y would have been chosen by random ballot. This is the foundation of my new Landau method based on ranked preference ballots with out the need for approval cutoffs. Ordinarily my proposal for the initial candidate in the sequence would be a candidate chosen by random ballot or else the approval winner, but I'm studiously avoiding requiring the voters to make approval judgments, and I want to have a deterministic version of the method, as well. So in the next message I have a deterministicsolution that does not require voter to make approval judgments.. ---- Election-Methods mailing list - see https://electorama.com/em for list info |
Here's a method that I consider to be good in its own right, not only as a starting point for "Minimum Disappointment Covering Enhancement." Assume ranked preference ballots with equal ranking and truncation allowed. Also assume access to a "random favorite" probability distribution, whether from a separate poll or by inference from the ballot set itself. A ballot B is said to "like" candidate X if a random favorite is less likely to be ranked ahead of (i.e. above) X than not on ballot B. The method elects the candidate liked by the greatest number of ballots. This method is monotone whether or not the random favorite distribution is computed on the fly. It also satisfies clone winner and clone loser the same way that range voting does, i.e. as long as the clone sets are ranked (or truncated) together. On Tue, Feb 18, 2020 at 1:51 PM Forest Simmons <[hidden email]> wrote:
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Now to review the "Minimum Disappointment Covering Enhancement" method: Ranked ballots with the possibility of equal ranking and truncation are required, along with access to or the ability to compute (from the ballots) a random favorite distribution. An initial candidate x0 is obtained by some election method based on ranked preference ballots of the type mentioned above. It could be the method described in my previous message, or it could be River, Random Ballot, Greatest Implicit Approval, or any other monotonic, clone-free method. 1. If x0 is uncovered, then elect x0. 2. Otherwise, among all of the candidates that cover x0 replace x0 with the one that minimizes the sum of squares (over all ballots) of the ballot disappointments for the transition from the previous value to the updated value of x0. 3. Return to step 1. Remember that the ballot disappointment in the transition from candidate x to candidate y is zero if y is ranked ahead of x, else it is the probability that random favorite would pick a candidate ranked ahead of y. On Tue, Feb 18, 2020 at 1:51 PM Forest Simmons <[hidden email]> wrote:
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Hi Forest, I like this method, though it seems to contain a huge gotcha. If you truncate candidates who collectively have more than half of the random favorite win odds, your ballot will approve everyone. I imagine this is deliberate and a big reason why a pre-election poll is envisioned to help inform the voters. I have been sitting on a similar idea for a long time and maybe I should just write it out now. I had the thought that if you have an approval election where voters are required to approve a majority of the options, the "median" one of those options could be expected to get 100% approval. (Exceptions for strategic voting and the possibility that there is no underlying issue space to explain the preferences.) Whichever option gets the most approval would be your best guess for the median. On reflection this seems not actually right, since the options could all be located far from all the voters so that multiple options get 100% and none of these are the median option. But no matter, this issue is quickly fixed. If we want to apply the idea to find the median VOTER (and his preference for the outcome), the options need to correspond to or be weighted by the voters themselves. Using first preference weight (identical to random favorite win odds) seems like a reasonable way to weight the options, since in issue space the voters should be closest to their favorite candidate. You could explain this requirement to approve a specific minimum value of options, as an analogy to proposing a coalition to form a government. A viable proposal for a ruling coalition (usually) has to cover a majority of the voting power. What is the expected effect? If the electorate is split 51-49 between two factions, but there are actually candidates near the median (which we might expect, due to the increased viability of such candidates), then basically the winner will come from the 51, but the 49 will choose who it is. This is in contrast to the 51 and 49 factions "privately" selecting one best nominee each, allowing the median voter to choose which nominee wins. In that case, as we see, the median voter is part of the winning faction but is normally an extremist within that faction. I do think the concept has a serious risk of electing a candidate who doesn't really have any support. You could have a separate round of voting in advance to determine reasonable finalists, or you could do the entire thing in one go, deduced from a single set of fully ranked ballots. A two-round approach probably gives the voters a greater sense of agency. If they know the candidate weights, they know how their second round ballot is going to get interpreted. If they want to try a strategy, they at least can tell what they're doing. A downside is trying to figure out how to present the math to the voter. In terms of ballot format, the simplest possible presentation of the idea is to have two rounds: One round narrows the field somehow to three finalists, and the second round is effectively Majority Favorite//Antiplurality. I.e. in the second round, you have to rank all three candidates. A majority favorite wins. Otherwise everyone approves two of the three options. (Half-approval for the middle option would also be recognizable as following the principle.) I don't have a name for the idea yet. Something about the mandatory nature of suggesting a coalition seems appropriate. But one might point out that the method doesn't actually identify any coalition, and the whole thing is just a means to guess where the median is. Kevin
Le jeudi 20 février 2020 à 17:15:26 UTC−6, Forest Simmons <[hidden email]> a écrit :
Here's a method that I consider to be good in its own right, not only as a starting point for "Minimum Disappointment Covering Enhancement." Assume ranked preference ballots with equal ranking and truncation allowed. Also assume access to a "random favorite" probability distribution, whether from a separate poll or by inference from the ballot set itself. A ballot B is said to "like" candidate X if a random favorite is less likely to be ranked ahead of (i.e. above) X than not on ballot B. The method elects the candidate liked by the greatest number of ballots. This method is monotone whether or not the random favorite distribution is computed on the fly. It also satisfies clone winner and clone loser the same way that range voting does, i.e. as long as the clone sets are ranked (or truncated) together. ---- Election-Methods mailing list - see https://electorama.com/em for list info |
Kevin, Thanks for your comments and catching that snafu. It turns out that if you just impose that truncated candidates get zero approval, the method is still monotonic (and clone free to the same extent as range is provided equal ranking and truncations are allowed). I like the idea of two rounds ... one to establish the firstplace/favorite preferences, and the other for determining the approvals informed by the information garnered in the first round. It seems like the less sophisticated voters might be happy to delegate their second round votes to their favorite or to some other published preference order. In the case of the published order the approvals would be filled in automatically according to the rules we have been contemplating. Perhaps the delegated favorites could have a little more flexibility??? You probably remember Joe Weinstein. A long time ago he suggested that instead of basing your approval cutoff on expected utility of the winner, you could use the 'median probability" idea; if your subjective probability sense tells you that the winner is less likely to be someone that you deem to be better than X than otherwise (someone you like no better than X), then approve X. When I first thought about using that heuristic in a DSV setting, I ended up with non-monotonic methods based on multiple rounds. Eventually all of the probability got concentrated in the top cycle, and further rounds continued around the cycle. At that point it was the same as Rob LeGrand's approval strategy A (foreshadowed by Weintein's remarks when he first introduced his idea). Another way of looking at the problem is that the winning probabilities (based on previous rounds or on sets of sampled ballots) never stabilized, so how could you define "THE winning probability." But when we replace the nebulous subjective probabilities with the well defined first place preference distribution, then we have something more definite to go on, and we get a monotonic method! Changing topic slightly back to the idea of disappointment; In the definition of disappointment "felt by" ballot B when the champion changes from candidate X to candidate Y, I wrote... If candidate Y is ranked above X, then the disappointment is zero. It should have said "above or equal to X," since equal is no worse, hence no disappointment in this sense. Nevertheless, if candidate Y is truncated on ballot B, then B should be counted as having 100 percent disappointment even if X was also truncated, and even if the random ballot probability of a candidate being ranked (i.e. above truncation) on ballot B is less than 100 percent. Thanks Again, Forest On Thu, Feb 20, 2020 at 8:59 PM Kevin Venzke <[hidden email]> wrote:
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Giving the favorite candidates a little more flexibility might help them to thwart "chicken" gamers. On Fri, Feb 21, 2020 at 4:02 PM Forest Simmons <[hidden email]> wrote:
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I don't hate the idea of giving the favorite candidate more flexibility. Candidates would either indicate their approvals in advance, or wait for the second round and then figure it out (simultaneously?). I guess I don't see how to avoid the scenario where candidates themselves play chicken with this. I do like that if the second round is presented as delegating a decision to a single candidate, the math can be kept off the ballot. You could do everything in a single round expressing a single vote, and task the candidates with approving each other according to the weighting requirement. On your actual topic, I have to say, I am skeptical of defining disappointment for individual steps of a multi-step process. I'm wary that the process could do something to respect that some % of voters are disappointed to see a tentative winner status switch from candidate A to candidate B, when an honest assessment will show that A could never be the winner of the whole election. In that case the "disappointment" measured for these voters is a little abstract. Kevin
Le vendredi 21 février 2020 à 18:15:31 UTC−6, Forest Simmons <[hidden email]> a écrit :
Giving the favorite candidates a little more flexibility might help them to thwart "chicken" gamers. On Fri, Feb 21, 2020 at 4:02 PM Forest Simmons <[hidden email]> wrote:
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