Re: [EM] Simpson-Kramer
Dear Mike, however, at least when each voter casts a complete ranking of all candidates then the Simpson-Kramer method is identical to what you call Plain Condorcet. This cannot be said about Condorcet's proposals (who, by the way, doesn't discuss partial individual rankings either). So after all, you didn't give any justification why you call this method Plain Condorcet(wv) and not Simpson-Kramer(wv). By the way: Most of the terminology (e.g. Copeland set, Smith set, Schwartz set) that is used in this mailing list comes from a paper by Fishburn (Condorcet Social Choice Functions, SIAM Journal on Applied Mathematics, vol. 33, pp. 469--489). Also Fishburn presumes that each voter casts a complete ranking of all candidates. He writes on page 470 of his paper: It will be assumed throughout that all voters have linear preference orders on the candidates or alternatives so that individual indifference between distinct candidates does not arise. So if you really wanted to be consistent you wouldn't use Fishburn's terminology either. Markus Schulze Election-methods mailing list - see http://electorama.com/em for list info
[EM] plain Condorcet debate
My two cents on the rampaging plain Condorcet vs. Simpson-Kramer vs. minimax debate: The main thing I don't like don't like about the term plain Condorcet is that it sounds too much like plain yoghurt. It sounds as if the voters are getting a half-rate cup of Condorcet that lacks strawberries or vanilla flavoring. That, and the fact that its historical accuracy is dubious at best... I call this method minimax, a term that comes from game theory, I believe. Not that it really matters though, because I don't think the method is worth using anyway. cheers, James Election-methods mailing list - see http://electorama.com/em for list info
[EM]Definite Majority Choice, AWP, AM
Ted, Russ, Forest, James,Juho and others, I think that Ted's draft public Definite Majority Choice proposal is excellent, in the sense that anything that might be slightly better would be more complicated and/or less intuitive. Two contending methods that use the same style of ballot are James G-A's Approval-Weighted Pairwise and my Approval Margins. I've found a couple of examples that illustrate differences between the three methods. The first is copied from a Sep.22,04 James G-A post. 3 candidates: Kerry, Dean, and Bush. 100 voters. Sincere preferences 19: KDB 5: KDB 4: KBD 18: DKB 5: DKB 1: DBK 25: BKD 23: BDK Kerry is a Condorcet winner. Altered preferences 19: KDB 5: KDB 4: KBD 18: DKB 5: DKB 1: DBK 21: BKD 23: BDK 4: BDK (these are sincerely BKD) There is a cycle now, KBDK On the sincere preferences ballots, the approval scores are B48, K46, D43, while on the altered preferences ballots, the approval scores are B48, D47, K46. Approval Margins uses a defeat-dropper method, measuring the strengths of the defeats by the margins between the approval scores (but like AWP,determines their directions purely by the rankings.) Approval Margins: DK 47-46 (m+1) KB 46-48 (m-2) BD 48-47 (m+1) B's defeat, with an approval margin of -2, is the weakest and so is dropped. B, the Buriers' favourite but the sincere (and voted) Approval winner,wins. DMC gets the same result by eliminating D and K. AWP differs from AM in the way that it weighs defeats. Quoting James: For a given defeat A over B, the magnitude of the defeat is defined by the number of voters who place A above their approval cutoff and B below their approval cutoff. Approval-Weighted Pairwise: DK 06 KB 46 BD 44 AWP elects the sincere CW, K! I used to think that electing the voted approval loser was absurd if we assume that the votes are sincere, but by that logic we should resolve all top cycles by electing the Approval winner. From that point of view, sometimes electing the approval loser is only a degree worse than not always electing the approval winner! Still, I don't see this example being a great advertisement for AWP versus AM because the winner is the sincere Approval winner. My next example is the one I used in my last post on AM. An old example given by Adam Tarr. Sincere rankings: 49 RCL 12 CRL 12 CLR 27 LCR C is the CW. Suppose there is pre-polling and so the L supporters decide to approve C, while the C supporters sincerely divide their approvals. Further suppose that the R supporters all decide to completely Bury C. Then we might get: 49 RLC 06 CRL 06 CRL 06 CLR 06 CLR 27 LCR Now all the candidates are in the top cycle: LCRL. The approval scores are L82, R55, C51. Approval Margins: LC 82-51 = +31 CR 51-55 = -4 RL 55-82 = -27 AM elects L, backfiring on the Buriers! Unfortunately this time DMC eliminates C, and then the Buriers' candidate R wins. Approval-Weighted Pairwise: LC 49 CR 45 RL 06 AWP gives the same good result as AM! Chris Benham Find local movie times and trailers on Yahoo! Movies. http://au.movies.yahoo.com Election-methods mailing list - see http://electorama.com/em for list info
[EM] Raynaud not monotonic
Say in a Raynaud election the ranked ballots are 7:ABC 2:BAC 5:BCA 6:CAB C is eliminated and A wins. But if the BAC voters uprank A, giving 9:ABC 5:BCA 6:CAB B is eliminated and C wins. So A went from winning to losing with extra support. -- Rob LeGrand, psephologist [EMAIL PROTECTED] Citizens for Approval Voting http://www.approvalvoting.org/ __ Do you Yahoo!? Yahoo! Small Business - Try our new resources site! http://smallbusiness.yahoo.com/resources/ Election-methods mailing list - see http://electorama.com/em for list info
[EM] Re: Definite Majority Choice, AWP, AM
Hi Chris, Nice example. But there is still a counter-strategic incentive under DMC -- see below. On 24 Mar 2005 at 08:11 UTC-0800, Chris Benham wrote: Suppose there is pre-polling and so the L supporters decide to approve C, while the C supporters sincerely divide their approvals. Further suppose that the R supporters all decide to completely Bury C. Then we might get: 49 RLC 06 CRL 06 CRL 06 CLR 06 CLR 27 LCR Now all the candidates are in the top cycle: LCRL. The approval scores are L82, R55, C51. Approval Margins: LC 82-51 = +31 CR 51-55 = -4 RL 55-82 = -27 AM elects L, backfiring on the Buriers! Unfortunately this time DMC eliminates C, and then the Buriers' candidate R wins. Approval-Weighted Pairwise: LC 49 CR 45 RL 06 AWP gives the same good result as AM! Yes, with perfect polling knowledge, the R strategy might work. But Rock/Paper/Scissors strategy like this doesn't occur in a vacuum. If R voters are coordinated enough to bury C in both approval and rank, they have to operate on the assumption that CRL voters might also suspect something and might all disapprove R instead of splitting. Without CRL's 6 approval votes, R would be eliminated by the definitive CR defeat. R's ordinal-burial of C would backfire and elect L. If I were an R voter, that would be the *last* thing I'd want! Ted -- araucaria dot araucana at gmail dot com Election-methods mailing list - see http://electorama.com/em for list info
Re: [EM]Definite Majority Choice, AWP, AM
Dear Chris! First, I'd like to emphasize that DMC, AWP, and AM can be thought of as being essentially the same method with only different definition of defeat strength, so it seems quite natural to compare them in detail as you started. Recall that the DMC winner is the unique immune candidate when defeat strength is defined as the approval of the defeating candidate, so with that definition, Beatpath, RP, and River become equivalent to DMC. Perhaps it is helpful to look at the defeat strength like this: When A defeats B, then the defeat strength is composed as a linear combination of the following three components: AM AWP DMC no. of voters approving A but not B + + + no. of voters approving A and B 0 0 + no. of voters approving B but not A - 0 0 My second point is this: Your wrote: I used to think that electing the voted approval loser was absurd if we assume that the votes are sincere, but by that logic we should resolve all top cycles by electing the Approval winner. From that point of view, sometimes electing the approval loser is only a degree worse than not always electing the approval winner! Here you state the obvious problem when looking at both approval and defeat information. Forest's ingenious argument was that we should at least not elect a candidate where both kinds of information agree that the candidate is defeated, leaving us with his set P of candidates which are not strongly defeated. But when we take both kinds of information serious, it does not seem appropriate to me to always elect a candidate from the two extremes of P like Approval and DMC do. Still, DMC has the obvious advantage of extreme simplicity. I would find it much more natural if the winner was somewhere in the middle of P! The simplest way to achieve this is to use Random Ballot among P, which adds two nice properties to the method: (i) Randomization for better long-time fairness and strategy-proofness, and (ii) taking into account the third major kind of preference information: direct support. In your second example, this would result in R with 61% probability and L with 39% probability, so that the R voters would take the risk of getting a worse outcome than before with 39% probability, which should suffice to deter them from using that strategy. The results of the first example were posted by me already. People who don't like randomization could instead use TAWS (Total Approval Winner Stays): Process the candidates in order of increasing approval, always keeping the winner of the pairwise contest between the next candidate and the candidate at hand. With a large number of candidates, I guess this will still give a winner more to the less approved end of P, but in your first example it elects B as DMC does, and in your second one L as AM and AWP do. Forest alternatively proposed to choose from P via IRV (not monotonic) or using Winner Stays as in TAWS but starting from the top candidate on a Random Ballot. Yours, Jobst Election-methods mailing list - see http://electorama.com/em for list info
[EM] Re: Approval Questions
Sorry, I was thinking in terms of equilibria that are stable under Rob's ballot-by-ballot DSV procedure. On Wed, 23 Mar 2005, Jobst Heitzig wrote: Dear Forest! You answered to me: The point is that when all ways to fill in the ballot are admissible strategies, there is never as group strategy equilibrium unless a sincere CW exists. My question here was: Is there such an equilibrium with Approval Voting when only ballots with sincere preferences are allowed! Not in general, but yes when there is a CW. The same cycle x ABC y BCA z CAB with max(x,y,z) (x+y+z)/2 illustrates the lack of such an equilibrium. If A is the winner, then the next time around, B and C supporters can collude to adjust their approval cutoffs to make C the winner (to their mutual advantage), etc. around the cycle. Well, that depends! B and C voters not always have such an incentive, since the following situation *is* an equilibrium of the desired kind: 4 ABC 2 BCA 3 CAB Here A wins and those who prefer C to A can *not* make C the winner since voting 2 BCA 3 CAB will make B the winner instead! In Rob's algorithm, once A is in the lead, the ABC voters stop approving B. Then eventually C surpasses B in approval, so the CAB voters stop approving A, so C eventually surpasses A in approval. When the approval order is CAB, then the A supporters start approving B. When the approval order reaches CBA, then the B supporters stop approving C. Etc. So your kind of equilibrium does not imply equilibrium in Rob's algorithm. So, it is *not* true that the existence of an equilibrium of the desired kind implies that there is a CW, and that was why I asked whether there might perhaps always be such equilibria. Unfortunately, they're not... Anyway, I'm still interested in Approval Strategy, because I would like to find out how much one can strategize in our Condorcet-Approval hybrids by only moving the approval cutoff around. I gave an argument a few years ago for the case of a CW, first in the context of a one dimensional issue space, and later more generally. I don't remember the date. I'm sure that Abrams, Fishburne, or Merrell already published better arguments, long ago. Hm. I'm not sure I got the point here. To prove that when a CW exists there is a group strategy equilibrium under Approval Voting seems easy: The equilibrium is when all approve of the CW and all preferred candidates, isn't it? Except those who rank the CW in last place (e.g. by truncating). It's not easy to show that Rob's algorithm (the batch version) will always converge to the CW. The ballot by ballot version doesn't always converge to the CW, though for most ballot orders it will. Election-methods mailing list - see http://electorama.com/em for list info
[EM] Re: Approval Questions
Forest, you wrote: Sorry, I was thinking in terms of equilibria that are stable under Rob's ballot-by-ballot DSV procedure. And: In Rob's algorithm, once A is in the lead, the ABC voters stop approving B. But why should they do so when A wins already? They have no incentive whatsoever to change their behaviour and thus provoke reactions which can only lead to a worsening of their outcome, have they? Jobst Election-methods mailing list - see http://electorama.com/em for list info
[EM] Re: public acceptability
From: James Green-Armytage [EMAIL PROTECTED] Subject: [EM] I forgot something important... James wrote... I forgot to mention something important before I sent my last post, CWO may be worth fighting for. I wrote: Here is one possible progression for single winner elections (to decide on representatives): 1. plurality and runoffs 2. IRV 3. CWO-IRV 4. ranked pairs(wv), with CWO 5. cardinal pairwise (with CWO?) James went on to say that he hoped that Direct Democracy could furnish a shortcut to Condorcet methods. Forest replies ... I have a shortcut to Direct Democracy that meets all of Russ's simplicity criteria: [Start Method Description] Voters distinguish favorite and also approved candidates, as in Majority Choice Approval. While two or more candidates remain in competition, all candidates (acting as proxy electors for their direct supporters) eliminate (by majority vicarious vote) one of the two remaining candidates with lowest approval. [End Method Description] Explanation: This is TAWS (total approval winner stays) with the voters directly supplying the approval information, and the ordinal information coming from the candidates that they directly support. In the USA we already use electors. But because of the severe discretization error inherent in the Electoral College, these electors represent the voters in only crude proportion at best. [Best means those states that do not require their electors to vote as a block.] Furthermore, since the voters usually don't even know their electors, they don't really have any reason for trusting them as representatives. Wouldn't you rather have your favorite candidate as an elector? The proxy weight of each candidate is the number of ballots on which s/he was designated as favorite. To minimize the possibility of spoiled ballots, we allow voters to designate more than one favorite. If a ballot has k candidates designated as favorite, then that ballot contributes only 1/k to each of their proxy weights, i.e. favorites are counted cumulatively for proxy purposes. There are lots of possible variations. For example, instead of basing the method on TAWS, we could sort the approval seeded list using sink sort, bubble sort, or some other way. My favorite sort in this context: While some candidate (pairwise) beats an adjacent candidate with more approval, swap the members of the adjacent pair with the least approval difference. This sort has a nice reverse symmetry to it: reverse the approval order and the pairwise beats, and the final order is reversed, as well. Forest Election-methods mailing list - see http://electorama.com/em for list info
[EM] Re: Sincere Methods
Date: Wed, 23 Mar 2005 18:50:26 +0100 From: Jobst Heitzig [EMAIL PROTECTED] There is no sincere way to specify utilities. Prove me wrong and tell me what a sincere utility could possibly be! How about the probability that the candidate in question would represent me accurately in his vote on a random vote in congress. Then the total CR for that candidate would be an estimate of the expected number of voters that would be accurately represented by that candidate on a random vote in congress. Forest Election-methods mailing list - see http://electorama.com/em for list info
[EM] Re: Approval Questions
On Fri, 25 Mar 2005, Jobst Heitzig wrote: Forest, you wrote: Sorry, I was thinking in terms of equilibria that are stable under Rob's ballot-by-ballot DSV procedure. And: In Rob's algorithm, once A is in the lead, the ABC voters stop approving B. But why should they do so when A wins already? They have no incentive whatsoever to change their behaviour and thus provoke reactions which can only lead to a worsening of their outcome, have they? True, in this case greed doesn't pay. But suppose that all of the votes are not in, and all you know is that the approval order (so far) is ABC, which happens to match your preference order. Where would you put your approval cutoff? For me it would depend on whether B was closer to A or to C in my estimation. If closer to A, then I would say to myself, Approving A and B will keep A in the lead, and put a cushion between B and C. If closer to B, then I would say to myself, I better put the cushion between A and B. At any rate, Rob's strategy A, which says, Put the approval cutoff next to the current approval leader on the side of the current approval runner up is the simplest DSV strategy (for ordinal ballots) that almost always converges to the CW when there is one. As you have demonstrated, in general it is not always possible to determine from ordinal information alone where rational voters should put their cutoffs, even given accurate polling information or information about partial election results. The reason? Because there truly is information in approval ballots that cannot be derived even from a set of completed ordinal ballots. It seems to me that this supports our idea that voter supplied approval cutoffs do indeed add valuable information. Finally, here's another (half-baked) idea that I have been thinking about: Suppose that L is a lottery with maximum support S(L) such that for each positive epsilon and each candidate C in S(L), there is a lottery L' within epsilon of L such that C wins (or ties for first) when voters approve according to L'. In other words, by announcing winning probabilities close to L, we can make any candidate in S(L) win the approval contest, IF the approval voters believe our predictions (with enough faith to place their approval cutoffs so as to maximize their expected payoff under L). It seems to me that each member of S(L) should have some probability of winning. What should that probability be? The lottery L is kind of like the basketball referee setting up the ball between the two centers to start play. So perhaps L is a fair way of assigning probabilities. On the other hand, perhaps, having used L to identify the set S(L), we can now throw away L, and pick the winner by random ballot from S(L), giving direct support a chance. One reason I like this lottery L is that we can mimic a constructive (piecewise linear) path following proof of Sperner's Lemma to find such a lottery. But, under what conditions is it unique? Forest Election-methods mailing list - see http://electorama.com/em for list info
[EM] A precise abstract definition of preference
Though I've told why, for the purpose of my criteria (and Steve's criteria) it doesn't matter what prefer means, and, in fact, though I've told why it doesn't matter if prefer doesn't mean anything, some may be uncomfortable about the fact that I haven't posted a precise definition of prefer that doesn't refer to everyday life. Here is such a definition: Definition of prefer: A preference is an information-record consisting solely of a designation of a set of voters and a designation of an ordered pair of candidates. A sentence or clause that says [some set of voters (referred to in this paragraph as S)] prefer X to Y means A preference consisting of a designation of S and a designation of (X,Y) is recorded in reference to the example being discussed. [end of precise abstract definition of prefer] This definition of prefer means something only if an example is being discussed, but criteria are only used during the discussion of an example. Mike Ossipoff _ Express yourself instantly with MSN Messenger! Download today - it's FREE! http://messenger.msn.click-url.com/go/onm00200471ave/direct/01/ Election-methods mailing list - see http://electorama.com/em for list info
[EM] Simpson-Kramer
Dear Markus-- You say: however, at least when each voter casts a complete ranking of all candidates then the Simpson-Kramer method is identical to what you call Plain Condorcet. I reply: Not good enough. The fact that Simpson-Kramer can sometimes give the same result as PC doesn't mean that Simpson-Kramer is PC. You continue: This cannot be said about Condorcet's proposals (who, by the way, doesn't discuss partial individual rankings either). I reply: Condorcet proposed that when there's no CW (consistent opinion, or words to that effect, in some translations) then we should ignore the proposition (pairwise defeat) that has the least support. And do so till there is a consistent opinion. That sounds like PC. Norm posted a quote that showed that Condorcet discussed partial rankings. But thanks for showing that you can't deny the things that I said in my previous Simpson-Kramer posting, meaning that you admit that Simpson-Kramer is not PC, and that PC is not Simpson-Kramer. Since we agree that MinMax is Simpson-Kramer (to deny it would be to deny the authority of your authors), then we agree that MinMax must not be PC. That's because if Simpson-Kramer isn't PC, and if MinMax can only be one thing, and if MinMax is Simpson-Kramer, then MinMax can't also be PC. You continue: So after all, you didn't give any justification why you call this method Plain Condorcet(wv) and not Simpson-Kramer(wv). I reply: But why would I call PC Simpson-Kramer(wv) when Simpson-Kramer elects the candidate whose greatest pairwise vote against him in a pairwise-comparison (not necessarily a pairwise defeat of his) is less than anyone else's greatest pairwise vote against them in a pairwise comparison? You continue: By the way: Most of the terminology (e.g. Copeland set, I reply: I don't often mention the Copeland set. You continue: ...Smith set, Schwartz set) that is used in this mailing list comes from a paper by Fishburn (Condorcet Social Choice Functions, SIAM Journal on Applied Mathematics, vol. 33, pp. 469--489). Also Fishburn presumes that each voter casts a complete ranking of all candidates. He writes on page 470 of his paper: It will be assumed throughout that all voters have linear preference orders on the candidates or alternatives so that individual indifference between distinct candidates does not arise. So if you really wanted to be consistent you wouldn't use Fishburn's terminology either. I reply: I've posted definitions of the Smith set and the Schwartz set. Is it really necessary for me to post them again for you? I don't depend on Fishburn's definitions. Let me know if there's a term that I use that I haven't defined and which needs a definition. Fishburn refers, by the name Condorcet, to the method which I call PC. If Fishburn's assumption that you quoted means that Fishburn's method definitions, including his Condorcet definition, don't apply unless everyone ranks all the candidates--no problem. I've defined PC without any such assumption. You're welcome, then, to refer to PC if you want to name a method defined without the assumption that everyone votes a complete ranking. Condorcet also proposed a method that Tideman reasonably interpreted as what we call Ranked-Pairs, except that Tideman measures defeats by margins. Condorcet's RP and PC methods have in common the fact that circular ties are solved by successively dropping weakest defeats or keeping (locking-in) strongest defeats. Other Condorcet versions describable in that way, such as Smith//PC, SD, SSD, or CSSD are less literal interpretations of Condorcet's PC wording. It may well be that if Condorcet actually conducted a defeat-dropping Condorcet election, he'd use SD, SSD or Smith//PC instead of PC. Anyway, I justify calling SD, SSD, and Smith//PC Condorcet for that reason, and because they solve circular ties by successively dropping weakest defeats. I call CSSD, and therefore BeatpathWinner, Condorcet by extensiion from SSD. Mike Ossipoff Markus Schulze _ On the road to retirement? Check out MSN Life Events for advice on how to get there! http://lifeevents.msn.com/category.aspx?cid=Retirement Election-methods mailing list - see http://electorama.com/em for list info
[EM] James, about PC and Simpson-Kramer
James-- You wrote: My two cents on the rampaging plain Condorcet vs. Simpson-Kramer vs. minimax debate: The main thing I don't like don't like about the term plain Condorcet is that it sounds too much like plain yoghurt. It sounds as if the voters are getting a half-rate cup of Condorcet that lacks strawberries or vanilla flavoring. I reply: Aren't they? They're getting the basic Condorcet, the original version, with SFC and WDSC, but which hasn't been demonstrated to meet GSFC and SDSC, as have SD, SSD, CSSD, BeatpathWinner and RP. You continue: That, and the fact that its historical accuracy is dubious at best... I reply: Fact? Are you saying that's a fact because Markus said it? Look up the translations of Condorcet's proposals. You'll find PC. One place to look would be _Theory of Committees and Elections_, by Duncan Black. The title is probably as I wrote it, unless it's ...Elections and Committees. You continue: I call this method minimax, a term that comes from game theory, I believe. I reply: Which method? Simpson-Kramer or PC? You said that you're replying to the discussion of whether or not Simpson-Kramer is PC. I told why Simipson-Kramer is not PC. If you believe that Simpson-Kramer is PC, then tell me which statement in my two most recent Simpson-Kramer postings you disagree with, and why. Since Simpson-Kramer is not PC, and since, according to Markus, Levin, and Nalebuff, MinMax is Simpson-Kramer, do you see the problem in calling PC MinMax? You continue: Not that it really matters though, because I don't think the method is worth using anyway. I reply: The Libertarian organization, Free State Project, used, and may still use, PC, and has been quite satisfied with it. It's certainly possible that you and I don't agree on what is important. I consider SFC and WDSC worth getting. Maybe you don't. There's no reason to expect everyone to value the same guarantees. You've got to talk to some people who aren't yet familiar with voting systems. Then you might be more tolerant of siimpler and more modest methods such as PC and CR, which, while modest proposals, would be powerful improvements. Mike Ossipoff _ Dont just search. Find. Check out the new MSN Search! http://search.msn.click-url.com/go/onm00200636ave/direct/01/ Election-methods mailing list - see http://electorama.com/em for list info
[EM] Clarification about pork-chops meeting criteria
Chris-- I mentioned two separate reasons why Plurality meets Non-Drastic Defense. Let me quote what I got from a website: Non-Drastic Defense: Each voter must be allowed to vote as many alternatives as s/he wishes tied for top, and if more than half the voters vote some alternative y (tied for) top, then no alternative voted below y by more than half of the voters may be chosen. [end of NDD definition at the website] I should have checked your posting before starting this posting, but it seems to me that, in your posting, you worded NDD differently from that, in such a way that Plurality passes because NDD's premise makes a requirement that can't be met in a Plurality example, meaning that there can be no Plurality failure example, because there can be no Plurality example that the criterion applies to, and because your NDD didn't have the requirement for allowing equal top ranking. It was in regards to that that you said that a pork chop passes NDD for the same reason. I agree that something's wrong when a method passes for that reason. My criteria are universally and uniformly applicable. Well, all of them but SARC, which I rarely if ever use, and will drop if I can't make it universally applicable without changing it too much. You stated a few criteria of your own, and they all stipulated, in their premise, something that could only be possible if the ballot has at least 3 rank positions. That's another example of a criterion which Plurality passes for the same reason that a pork chop passes. No one can vote 3 rank positions in Plurality, and so there's no Plurality example that your criterion applies to. Therefore there's no plurality failure example for your criterion . Now, if you had added a requirement that the method must allow at least 3 voting levels, then Pluralilty would fail, by fiat, in the way that it fails the actual NDD quoted above. But your criterion didn't have that requirement. So Plurality passes it for the same reason as pork chop does. I don't remember if there was another reason why Plurality would pass, even if that weren't so. But, returning to NDD, the version that I quoted above, from the website: Plurality fails that version of NDD, which is more likely to be the official version, since it's at the website. And it fails for a similar reason: Pluralitiy fails because it doesn't allow the voter to vote as many alternatives as s/he wishes tied for top, and NDD requires that a method allow that. It's a rules criterion, because it says that a method fails if it doesn't have a certain type of rule--in this case it's about balloting rules. That's something that I avoid, because, as I said, I want my criteria to be universally and uniformly applicable. If some methods can pass simply because their examples can't meet the requirements in the criterion's premise, that isn't very uniform applicabililty. If some methods fail simply because their rules aren't of a certain kind explicitly required by the criterion, then that isn't very uniform applicability either. Without that rules-requirement, Plurality would pass the criterion, because if more than half of the voters vote Y at top, then no one voted below y by most of the voters can win. That's why the criterion needs the rules-requirement. Ok, this posting is a mess, but I hope it clarifies my previous reply. If there's a point to be found in all this, it's that it would be better if other criteria were universally and uniformly applicable, as the defensive strategy criteria are. Mike Ossipoff _ On the road to retirement? Check out MSN Life Events for advice on how to get there! http://lifeevents.msn.com/category.aspx?cid=Retirement Election-methods mailing list - see http://electorama.com/em for list info
