Dear Jobst, Thanks for your clarifications, especially the key words, "...that do not approve X at least as strongly."
Can we summarize the operational meaning of a rating R for X more simply than the following? If you rate option X at a level R, then, provided X is the option with the "greatest potential for compromise" (among those options that remain to be considered) you certify your willingness to give your vote to X as long as no more than R of the remaining ballots fail to give their vote to X. "Greatest Potential for Compromise" is to be measured by the number of (remaining?) ballots that give partial approval. Alternatively, we could say that option X has the maximum potential for compromise when R(X) (defined below) is minimal: For each option X, let R(X) be the minimum of the set of R such that at most R of the (remaining?) ballots approve X at a level of R or below. Would any of these variations (remaining vs original or R(X) vs partial approval) in defining max potential for compromise, destroy monotonicity? EC6 is good, or perhaps FC6, for Fair Chance Choice with Controlled Cooperation for Consensus or Compromise. Thanks, Forest ----- Original Message ----- From: Jobst Heitzig Date: Saturday, November 8, 2008 5:45 pm Subject: Re: [EM] Some chance for consensus (was: Buying Votes) To: [EMAIL PROTECTED] Cc: election-methods@lists.electorama.com, [EMAIL PROTECTED], Kristofer Munsterhjelm , Raph Frank > Dear Forest, > > you wrote: > > This reminds me of your two urn method based on approval ballots: > > Initialize with all ballots in the first urn. > > While any ballots are left in the first urn ... > > find the approval winner X of these remaining ballots > > circle candidate X on all of the ballots in the first urn > that > > approve candidate X, and then transfer them to the second urn. > > End While > > Elect the circled candidate on a randomly drawn ballot from > the > > second urn. > > Yes, it's inspired by that method which was due not to me but to > someone else, I think. > The important difference is, though, that in the above method > there will never be full cooperation since as long as the two > largest approval scores are more than 1 point apart, every voter > approving but not favouring the approval winner has an incentive > to remove her approval for the approval winner. In particular, > the equilibria in our 55/45 situation would look like the > following, with x+y=56: > 55-x: A > x: A+C > y: B+C > 45-x: B > So with the above method, C would win only with probability 56%. > > In the new suggestion, in contrast, the voters can make sure > that the "contract" to elect C only becomes effective when all > voters cooperate: Knowing that everybody prefers C to the Random > Ballot lottery, they can all rate C at 1. > > > It looks like your newest method is a variation where > "Approval" is > > interpreted as "positive rating," > > "partial approval" I would call it. The rating can be > interpreted not as a utility value but as a "limit on non- > cooperation". Or, the other way around, the value 100 minus the > rating can be interpreted as a "cooperation threshold". I got > the idea for this when reading the Wikipedia article on the > National Popular Vote Interstate Compact which has a very > similar provision making sure that signing the contract is > "safe" even when it is not known in advance who exactly the > other participants will be! > > > and X is circled only on those > > ballots that rate X sufficiently high* relative to the number > of ( > > remaining) ballots that do not approve X. > > ...That do not approve X *at least as strongly*. This is > important! Otherwise there would alway be an incentive to use > only the values 100 (for the favourite only), 1, and 0: Rating > an option 1 would then lead to other voters who have a higher > rating transfer their probability, without me transferring it, > too. Therefore the requirement is that a voter with rating R > only transfers her probability if more than 100-R percent of all > voters do so, too! This gives me a possibility to specify a > "safe" rating without exactly knowing who will be the other > cooperating voters. > > > If 99% of the (remaining) ballots do not approve X, then X is > circled > > only on those ballots that rate X above 99%. If less than 1% > of the ( > > remaining) ballots do not approve X, then even a ballot that > rates X > > at a mere 1% would get a circle around X. > > Right! > > > The exact relation between the required rating relative to the > lack > > of approval (on the remaining ballots) can be played with to > get > > variations of this method. > > True, but I guess as long as you use a monotonic transformation > for this, the result will be equivalent. Except that the order > in which the options are processed would vary. It is charming to > be able to simply explain it this way: "If you rate C at R, your > vote will not be transferred to C whenever R or more voters rate > C less than R". > > > In this method there is no need to rate any candidate that the > voter > > cannot conceive of as a compromise. Therefore it seems quite > natural > > to consider positive rating as some level of approval. > > Right. With this interpretation, options are considered in order > of decreasing "partial approval score". It is important not to > use the rating sum instead since then there would be a conflict > between the necessity of using a small score to be safe and > using a larger score to make sure the compromise is considered > before the polar favourite options! > > > *Some provision must be made for ties and for the case where > no > > ballot rates the current X high enough to get transfered into > the > > second urn. > > When no ballot rates X high enough, then R must be considered to > be infinity, hence no ballots get transferred and the next > option is considered. > > As for ties, I usually think of them late. However: Ties are > only relevant for the order in which the options get processes > here. The natural tiebreaker would be this: If two options have > the minimum number of zero ratings, consider the number of 1- > ratings next, then (if still equal) the number of 2-ratings and > so on. > > What do you think of the following name for this method: > > EC6 > (Equal Chances Choice with Controlled Cooperation for Consensus > or Compromise) > > Is this silly or smart? > > Yours, Jobst > ---- Election-Methods mailing list - see http://electorama.com/em for list info
