[EM] Census re-districting instead of PR for allocating seats to districts.
I haven't been following this discussion closely, but I've long thought that the best way of allocating seats to multi-member districts is to just say that subject to every district having at least one seat we do the allocation after the votes have been cast, based on the numbers of people who actually vote. (Then within each district I favour STV-PR rather than any list system..) Competition between districts should help motivate an overall high turnout. But maybe there would be added incentives for skulduggery. :( Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Approval-Runoff
Mike Ossipoff wrote (9 March 2012): Kevin: You wrote: I don't think Approval-Runoff can get off the ground since it's too apparent that a party could nominate two candidates (signaling that one is just a pawn to aid the other) and try to win by grabbing both of the finalist positions. If this happened regularly it would be just an expensive version of FPP. [endquote] I'd believed that it would just be seen as a minimal change from Runoff. You mean that, because of Approval in the 1st election, it would be too easy for a faction to put two identical candidates in the runoff? Yes, now that you mention it, that's probably so. Approval-Runoff suggestion withdrawn. Some years ago I suggested a 2-round system which uses approval in the first round, and then (if the most approved candidate is not approved on a majority of ballots) has a run-off between the most approved candidate A and the candidate that is most approved on ballots that don't approve A. That removes the problem (compared to normal Approval-Runoff) of the same subset of voters choosing both finalists, and also greatly reduces the Push-over (aka turkey raising) incentive. I also consider this to be some improvement on normal (vote-for-one ) TTR. Of course it loses plain Approval's compliance with the FBC, because voting for your favourite could cause the runoff to be between Favourite and Worst leading to win for Worst instead of between Compromise and Worst leading to a win for Compromise. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] an extra step for IRV (and some other methods?)
I have an idea for adding an extra step to IRV which has the effect of throwing out its compliance with Later-no-Harm in exchange for Minimal Defense, while trying to hang on to Later-no-Help. *Voters strictly rank from the top however many or few candidates they wish. Until one candidate remains, provisionally eliminate the candidate that is highest ranked (among candidates not provisionally eliminated) on the fewest ballots. The single candidate left not provisionally eliminated is the provisional winner P. [So far this is IRV, used to find a provisional winner. Now comes the extra step.] Interpreting candidates ranked above P as approved and also P as approved if ranked, elect the most approved candidate.* This method might be called IRV-pegged Approval (IRVpA). It is more Condorcet-consistent than IRV, because when IRVpA produces a different winner that candidate must pairwise beat the IRV winner (so it keeps IRV's compliance with Mutual Dominant Third). Also the IRVpA winner must be more approved than the IRV winner. I'd be interested if anyone can show that this fails Later-no-Help. Some other methods might gain from adding the same extra step, for example Schulze(Margins), MinMax(Margins) and Descending Solid Coalitions. It will fix any failures of Minimal Defense (and my Strong Minimal Defense criterion) and Plurality. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] suggested improvement on Mutual Majority criterion/set (add-plump proofed MM)
I've decided to bin (i.e.I now withdraw) my suggested Add-Top Proofed
Mutual Majority as something highly desirable or a real improvement on
plain Mutual Majority.
I defined it thus:
*If the number of ballots on which some set S of candidates is voted
strictly above all the candidates outside S is greater than the number
of ballots on which all the members of S are voted below equal-top
(i.e. strictly below some/any outside-S candidate), then the winner must
come from S.*
I've discovered that it's incompatible with the Condorcet criterion.
2: AB=C=D=E
1: AX
1: BX
1: CX
1: DX
1: EX
The criterion says that the winner must come from {A B C D E}, but X is
the CW (pairwise beating all the other candidates 4-3).
Following a suggestion from Kevin Venzke, I now instead propose
Add-Plump Proofed Mutual Majority (APPMM):
*If the number of ballots on which some set S of candidates is voted
strictly above all the candidates outside S is greater than the number
of ballots on which any outside-S candidate is voted strictly above any
member of S, then the winner must come from S.*
Kevin gives this demonstration of how it differs from regular Mutual
Majority:
28: AB
27: BA
45: CD
Both MM and APPMM say that the winner must come from {A B}, but if we
add 12A ballots APPMM says the same thing while MM now says nothing.
I haven't found or thought of any method that meets both of
Mono-add-Plump and regular Mutual Majority (aka Majority for Solid
Coalitions) but fails APPMM.
Chris Benham
I wrote (13 Jan 2012):
On 21 Dec 2011 I proposed this criterion:
*The winner must come from the smallest set S of candidates about which
the following is true: the number of ballots on which all the members
(or sole member) is voted strictly above all the non-member candidates
is greater than the number of ballots on which a (any) non-member
candidate is voted strictly above all the members of S.*
That is fairly clear, but the wording could perhaps be improved, say:
*If the number of ballots on which some set S of candidates is voted
strictly above all the candidates outside S is greater than the number
of ballots on which all the members of S are voted below equal-top
(i.e. strictly below some/any outside-S candidate), then the winner must
come from S.*
I tentatively suggested the name Add-Top Proofed Solid Coalition
Majority. A bit less clumsy would be Add-Top Proofed Mutual
Majority. Maybe there is a better name that either does without the
word Majority or includes another word that qualifies it. For the
time being I'll stick with Add-Top Proofed Mutual Majority (ATPMM)
I gave this example:
45: AB
20: A=B
32: B
03: D
My criterion says that the winner must be A, but Mike Ossipoff's MTA
method elects B.
I did endorse MTA as an improvement on MCA, but since it (and not MCA)
fails this (what I consider to be very important) criterion (and is also
a bit more complicated than MCA) I now withdraw
that endorsement. I still acknowledge that MTA may be a bit more
strategically comfortable for voters, but I can't give that factor
enough weight to make MTA acceptable or win its comparison with MCA.
Chris Benham
Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Two more 3-slot FBC/ABE solutions
Following on from my recent definition of the APPMM criterion/set, I'd like to propose two not bad 3-slot methods that meet the FBC.. Recall that I defined the APPMM criterion thus: *If the number of ballots on which some set S of candidates is voted strictly above all the candidates outside S is greater than the number of ballots on which any outside-S candidate is voted strictly above any member of S, then the winner must come from S.* The APPMM set is the set of candidates not disqualified by the APPMM criterion. APMM//TR: * Voters fill out 3-slot ratings ballots. Default rating is Bottom (signifying least preferred and not approved.) The other slots are Top (signifying most preferred) and Middle. From the set of candidates not disqualified by the APPMM criterion, elect the one with the most Top ratings.* APMM//CR: * Voters fill out 3-slot ratings ballots. Default rating is Bottom (signifying least preferred and not approved.) The other slots are Top (signifying most preferred) and Middle. From the set of candidates not disqualified by the APPMM criterion, elect the one with the highest Top minus Bottom ratings score.* So far I can't see that these are technically any better than my earlier suggestion of TTPBA//TR, and unlike that method they fail the Tied at the Top Pairwise Beats All criterion. But like that method they meet the Plurality and Mono-add-Plump criteria, and also have no problem with Kevin's bad MMPO example. I'm happy for APMM//CR to be also called APMM//Range. This method is more Condorcetish than APMM//TR, for example: 49: CB 27: AB 24: BA BA 73-27, BC 51-49, AC 51-49. APMM//TR elects A, while APMM//CR elects B (like TTPBA//TR). I am sure that APMM//TR has no defection incentive in the Approval Bad Example, and the other method also does in the example normally given. Of course some other points-score scheme (perhaps giving greater weight to to Top Ratings) is possible. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Kristofer: MMPO objections
Forest, I am a bit confused by the first of your two interesting suggestions: 1. Put 50 percent in each of the diagonal positions. (A candidate would beat a clone of itself half of the time.) Err..50% of what? Chris Benham Forest Simmons wrote (5 Jan 2012): Kristopher, I agree that Plurality failure is bad in a public proposal and hard to defend in any case. In the case of MMPO the question is moot because Plurality failure is so easily fixed by either of the following natural tweaks: 1. Put 50 percent in each of the diagonal positions. (A candidate would beat a clone of itself half of the time.) 2. Put the respective truncation totals down the diagonal positions. (These totals are the pairwise oppositions of the Minimum Acceptable Candidate.) With this second fix, you can also create a list of oppositions against MAC, and if MAC's max opposition is smaller than any other candidate's max opposition, then various possible courses of action exist: (a) throw out these candidates and start over. (b) elect the approval winner (i.e. the one with min opposition from MAC, which is the same as the one with most opposition against MAC). (c) use the fall back lottery to elect the winner. Election-Methods mailing list - see http://electorama.com/em for list info
[EM] TTPBA,TR
Mike, One thing that I like about the tied-at-top methods is that they elect A in the ABE, meaning that one-sided coalition support is sufficient to defeat C, but without giving the election away to B. By the ABE, do you mean this? 27: AB 24: B (sincere is BA) 49: C Of course the election of A violates the Plurality Criterion, but that's fine with me. I wrote in the post suggesting this method that TTBA//TR meets the Plurality criterion. So does Kevin's ICA method. In the above example no ballots have any any candidates tied in top position (i.e. more than one candidate top-rated), so in that case TTBA//TR is the same as Condorcet//TR (and ICA is the same as Condorcet//Approval). http://wiki.electorama.com/wiki/index.php?title=Tied_at_the_top x: A 1: C=A 1: C=B x: B x is any number bigger than 1. MMPO elects C. As currently defined, ICT elects C in Kevin's MMPO bad-example. No it doesn't. 27: AB 24: BC (sincere is BA) 49: C ICT has burial strategy. In the ABE, the B voters can make B win by burying A, by middle- rating C but not A. I assume that you are talking about the above example. A candidate that is not the most top-rated can't win unless its the sole TTBA winner. In the above example there are no TTBA winners so the TTBA//TR winner is C Chris Benham Mike Ossipoff wrote (14 Jan 2012): Tied-at-Top-Pairwise-Beats-All, Top Ratings. In keeping with Kevin's naming, and reflecting its relation to ICA, it could be called Improved Condorcet-Top (ICT). I'll use that because it's shorter. One thing that I like about the tied-at-top methods is that they elect A in the ABE, meaning that one-sided coalition support is sufficient to defeat C, but without giving the election away to B. Of course the election of A violates the Plurality Criterion, but that's fine with me. To me, the _practical_ advantage described in the previous paragraph is worth more than the non-practical, aesthetic, Plurality Criterion. ICT has burial strategy. In the ABE, the B voters can make B win by burying A, by middle- rating C but not A. Then A doesn't have any indifference on his side, in hir comparison with C. But B still beats C, because BC is still greater than CB. For the same reason, C still doesn't beat everyone. And B still beats A, because BA + B=A is greater than AB. So B is now the only beats-all candidate. B wins. As currently defined, ICT elects C in Kevin's MMPO bad-example. No one is indifferent between A and B. So, since A=B is zero, then AB + A=B is no greater than BA. Likewise vice-versa, of course, since A B are symmetrically-related. Therefore, neither beats the other. Maybe that can be fixed, by defining beat in the opposite way, so that x beats y if xy is greater than yx + x=y, and then saying that the winning set is the set of unbeaten candidates. In summary, ICT does three things that some find unacceptable: 1. Plurality Criterion violation 2. Successful burial strategy 3. Noncompliance in Kevin's MMPO bad-example. #1 and #2 aren't a problem to me. #2 could be, but I don't know what burial-deterrence ICT has. With the sole exception of MMT, the conditional methods meet Mono-Add-Plump. They probably meet the Plurality Criterion too, because of their close relation to Approval. If B defects, those methods elect C, in compliance with the Plurality Criterion. Burial strategy has no meaning in the conditional methods. As I've been saying, they're a completely new kind of method, with a new kind of strategy, a milder strategy. Mike Ossipoff Election-Methods mailing list - see http://electorama.com/em for list info
[EM] suggested improvement on Mutual Majority criterion/set (and MTA reviewd)
On 21 Dec 2011 I proposed this criterion: *The winner must come from the smallest set S of candidates about which the following is true: the number of ballots on which all the members (or sole member) is voted strictly above all the non-member candidates is greater than the number of ballots on which a (any) non-member candidate is voted strictly above all the members of S.* That is fairly clear, but the wording could perhaps be improved, say: *If the number of ballots on which some set S of candidates is voted strictly above all the candidates outside S is greater than the number of ballots on which all the members of S are voted below equal-top (i.e. strictly below some/any outside-S candidate), then the winner must come from S.* I tentatively suggested the name Add-Top Proofed Solid Coalition Majority. A bit less clumsy would be Add-Top Proofed Mutual Majority. Maybe there is a better name that either does without the word Majority or includes another word that qualifies it. For the time being I'll stick with Add-Top Proofed Mutual Majority (ATPMM) I gave this example: 45: AB 20: A=B 32: B 03: D My criterion says that the winner must be A, but Mike Ossipoff's MTA method elects B. I did endorse MTA as an improvement on MCA, but since it (and not MCA) fails this (what I consider to be very important) criterion (and is also a bit more complicated than MCA) I now withdraw that endorsement. I still acknowledge that MTA may be a bit more strategically comfortable for voters, but I can't give that factor enough weight to make MTA acceptable or win its comparison with MCA. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] TTPBA//TR (a 3-slot ABE solution)
I have conferred off-list with Kevin Venzke, and now agree with him that the Tied at Top Pairwise Disqualification, Top Ratings method I suggested (20 Nov 2011) almost certainly does fail the FBC, so I withdraw that proposal and instead suggest this simpler method: *Voters submit 3-slot ratings ballots, default rating is Bottom signifying least preferred, Top rating signifies most preferred, the other ratings slot is Middle. According to the Tied-at-the-Top pairwise rule (TTP), candidate X beats candidate Y if the number of ballots on which X is given a higher rating than Y *plus the number of ballots on which X and Y are both rated Top* is greater than the number of ballots on which Y is given a higher rating than X. If any candidates (or candidate) TTP beats all other candidates, elect the one of these with the highest Top-Ratings (TR) score. Otherwise elect the candidate with the highest TR score.* I call this Tied at Top rule Pairwise Beats-All// Top Ratings (TTPBA//TR). It is similar to Kevin Venke's Improved Condorcet//Approval (ICA) method, the only difference being that it uses Top-Ratings instead of Approval. It was Kevin who invented the special tied at the top pairwise rule. http://wiki.electorama.com/wiki/index.php?title=Tied_at_the_top http://wiki.electorama.com/wiki/Improved_Condorcet_Approval http://nodesiege.tripod.com/elections/#methica TTPBA//TR (or TTBA,TR) meets the Plurality and Mono-add-Plump criteria. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] ACF grade voting
Forest, Why are your suggested grade options labelled A, C, F and not A, C, E? You can make the same wonderful argument that 2-slot ballots can work just as well as 3-slot ballots. And why limit the voters to one coin-toss each per candidate? A voter who wishes to give candidate x a grade of B on the scale A-B-C-D-E can first toss a coin to decide between A and C on an imaginary A-C-E ballot and if that comes up A then approve x on the actual 2-slot ballot but if it comes up C then toss the coin again to decide between approving x or not. Chris Benham Forest Simmons wrote (30 Dec 2011): Suppose the ballot limits grade options to A, C, and F, but a sizeable faction would like to award a grade of B to a particular candidate. If half of them voted a grade of A and the other half a grde of C, the resulting grade points would be the same. So in elections with large electorates there is no need to have grade ballots with all five grade options. Those who want to award a B grade can flip a coin to decide between A and C. Those who would like to award a grade of D can decide between C and F with a coin toss. The grade averages will come out the same as if the higher resolution grade ballots were used. If two or more candidates are statistically tied, the tied candidate with the greatest number of A's and C's should be elected. Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Who wronged the A-plumpers
Mike, A voting method algorithm stands or falls by its properties, i.e. its criterion compliances and failures. Another school of thought is less concerned about strict pass/fails of criteria and stresses how well the method does in computer simulations at maximising social utility and/or minimising negative emotions like regret. Before continuing, though, I should clarify that middle isn't the best descriptive name for what a middle rating means or is used for. Instead of calling it a middle rating, let's call it (on the ballot) an accept coalition rating. Let's call the ratings on the ballot top and accept coalition. If you don't give a rating to a candidate, we can call that a bottom rating. A voter accepts a candidate if s/he votes hir top or merely gives to hir an accept coalition rating. Ballots are simply for voters to register their (hopefully and officially presumably sincere) preferences among the candidates, not to formally issue open invitations to the voters to play sordid strategy games. For the method to wrong someone, it has to act wrongly or wrongfully. That is nicely circular...granted. But let's consider what MMT does, to judge whether what it does is wrong: You then go on to more-or-less explain the details of MMT's algorithm, putting a big positive spin on each of its components. Now, please note that I'm not using MMT's rule to justify MMT's rule. I'd like to note that, but I can't see what else you are doing. Among the candidates in those mutual majority sets, MMT elects the most popular one (the one with the most top-ratings). Defining popular that way instead of say fewest bottom-ratings or best top-ratings minus bottom-ratings score (or perhaps even by some non-positional measure) looks quite arbitrary and uncompelling How wrong is it to elect the most popular candidate among those among whom majority has determined that the winner must be chosen? It's wrong because it's an algorithm that needlessly fails some desirable criteria. There are other methods (and there need only be one) that don't fail those criteria while sharing all of MMT's desirable criterion compliances. MMT's rules were chosen to achieve FBC compliance, avoid the co-operation/defection problem, and enforce majority rule in some way. That is fine and admirable, but there are much better methods that do the same thing. Sincere preferences: 49: C 27: AB 24: BA 20: A The AB and BA voters should obviously coalition-accept each other's candidates. What should the 20 A voters do, who are indifferent between B and C? Well, they should know that A probably doesn't have top ratings from a majority of the voters. And they should know that A probably is not even the Plurality winner. Err... why should they know that? A's top-ratings score is 47, not far behind C's top-ratings score of 49. If all the voters have very good information about each others' preferences and are aware of and happy to use the best strategy (no matter how insincere or weird) then all deterministic methods are about as good as each other at picking the right winner. And how does Chris justify saying that the result is wrong? He says that it shouldn't be possible for voters to foul up their voted-favorite's chance of winning. No, that would refer to Mono-add-Top. I said that methods must meet Mono-add-Plump (compared to which Mono-add-Top is a bit arbitrary and expensive). Also the result gives bad failures of other criteria, like the tied-at-the-top rule modified pairwise beats-all criterion (compatible with FBC etc.) and my new Add-Top Proofed Solid-Coalition Majority criterion (doubtless also compatible with FBC, ABE etc.) and Condorcet Loser. Could there be a method that would protect the A-plumpers from their own stupidity? Sure. Is the voting system obligated to do that? No. The voters are adults, responsible for their own actions. If the stupidity is just honest voting, then the voting system is obligated to protect those voters if it can (without giving up some other desirable property). If the honest voting is just plumping, I say that the voting system can do that easily. And notice that though Chris is affronted by noncompliance with Mono-Add-Plump, by an FBC/ABE method, Chris isn't bothered by IRV's particularly flagrant form of nonmonotonicity. Why the inconsistency and self-contradiction, Chris? You might have got a clue from what I wrote in an earlier post: ...if failure of Mono-add-Plump isn't self-evidently *completely ridiculous* (and so much so that anything not compatible with Mono-add-Plump compliance is thereby made a complete nonsense of), then I have no idea what is. The only way this view of mine could be dented (and I made a bit wiser and sadder) is if it was proved to me that compliance with Mono-add-Plump isn't compatible with some other clearly desirable (IMO) property or set of
[EM] Chris: Regarding the criteriion failures you mentioned for MMT
Mike, In an earlier message of yours (the last one I responded to) you wrote: MAMT is an addition to the list of FBC/ABE methods to choose from. People should be looking into its properties. Tell me what you know, so far, about its properties, ... That is almost the only thing I did. You didn't ask me to confine myself to properties that I personally think are *important* or to explain why I think they are important. You said that MMT fails Later-No-Help: With MMT, you can help your favorite by entering into a mutually-chosen, mutually-supported, majority coalition. Everyone supporting that coalition does so because they consider it beneficial to their interest. How is that a failure?? I assume you know what the criterion specifies and are asking me why meeting Later-no-Help is a good thing. Failing LNHelp while meeting LHHarm creates a random-fill incentive. One of the problems with that is that is unfair to sincere truncators. Why should they be penalised for declining to play silly games with candidates they don't care about? Another is that all methods that fail LNHelp are vulnerable to Burial strategy. You said that MMT fails Mutual Dominant 3rd: I don't know what that criterion is. It is a weakened version of Smith that is compatible with LNHelp compliance (and so Burial Invulnerability) and also compliance with LNHarm. It says that that if there is a subset S of candidates that on more than a third of the ballots are voted strictly above all the outside-S candidates and all the S candidates pairwise-beat all the outside-S candidates then the winner must come from S. From your recent past statements I know I don't have to sell the desirability of compliance with this to you. I gave this example: 49: A 48: B 03: CB I can't take seriously any method that doesn't elect B here. Can you? Isn't this just the sort of small (probably wing) spoiler scenario that motivates many to support electoral reform? You said that MMT fails Mono-Add-Plump: I've already commented on that a few times. Yes, and I obliquely responded to your comment. But to be blunt, if failure of Mono-add-Plump isn't self-evidently *completely ridiculous* (and so much so that anything not compatible with Mono-add-Plump compliance is thereby made a complete nonsense of), then I have no idea what is. The only way this view of mine could be dented (and I made a bit wiser and sadder) is if it was proved to me that compliance with Mono-add-Plump isn't compatible with some other clearly desirable (IMO) property or set of properties. This doesn't come anywhere near cutting it: Your favorite initially won only because of mutual majority support. The plumpers declined that mutual support, as is their right. Having declined mutual support, should it be surprising or unfair if they no longer have it? Is it surprising or unfair that some new voters should in effect have their ballots given negative weight because they refused to play silly games with some candidates they weren't interested in and maybe knew nothing about? Err*yes*. As Jameson said, the chicken dilemma, also called the co-operation/defection problem, or the ABE problem, is the most difficult strategy problem to get rid of. However, there are a number of methods that do get rid of it, while complying with FBC and furnishing majority-rule protection: You (Chris) proposed one some time ago. Does it meet the criteria that you require, in addition to FBC and avoidance of the co-operation/defection problem? Can it be worded in a brief and simple, and naturally and obviously motivated way, for public propsal? I've been distracted and thinking about other things. I'll get around to addressing those questions, along with my closer look at Forest's MMMPO method. Chris Benham Mike Ossipoff wrote (15 Dec 2011): Chris: You said that MMT fails Mono-Add-Plump: I've already commented on that a few times. You said that MMT fails Condorcet's Criterion: But, as you know, CC is incompatible with FBC. You said that MMT fails Mutual Dominant 3rd: I don't know what that criterion is. But, in any case, to say that a failure of it is important, you'd have to justify the criterion in terms of something of (preferably) practical importance. You said that MMT fails Minimal Defense: Plurality meets Minimal Defense. So my answer will refer to the universally-applicable counterpart to Minimal Defense: 1CM. Of course MMT fails 1CM. MMT doesn't recognize one-sided coalitions. Rather than being an accidental failure, that is the point of MMT. To justify using 1CM against MMT, you'd need to tell why it's necessary to recognize one-sided coalitions. You'd need to justify it other than in terms of a criterion requiring that recognition. You said that MMT fails Later-No-Help: With MMT, you can help your favorite by entering into a mutually-chosen, mutually-supported, majority coalition.
[EM] Forest: MAMT
In my last post (13 Dec 2011) I wrote: A better method would (instead of acquiescing majorities) use the set I just defined in my last post. *If there is a solid coalition of candidates S (as measured by the number of ballots on which those candidates are strictly voted above all others) that is bigger than the sum of all its rival solid coalitions (i.e. those that contain some candidate not in S), then those candidates not in the smallest such S are disqualified. Elect the most top-rated qualified candidate.* That method I suggested wouldn't meet the FBC (it has now occurred to me), so I suspend my ..better method.. claim. In my other EM post the same day, I wrote: I propose a replacement for Mutual Majority which addresses this problem and also unites it with Majority Favourite. Preliminary definitions: A solid coalition of candidates of size N is a set S of (one or more) candidates that on N number of ballots have all been voted strictly above all outside-S candidates. Any given solid coalition A's rival solid coalitions are only those that contain a candidate not in A. Statement of criterion: *If one exists, the winner must come from the smallest solid coalition of candidates that is bigger than the sum of all its rivals.* [end criterion definition] This wording could perhaps be polished, and I haven't yet thought of a name for this criterion and resulting set. (Any suggestions?) It might be possible to use the set as part of an ok voting method. Thinking about it a bit more I now doubt that the last sentence is true, but still I think it wouldn't be as bad for that purpose as the usual Mutual Majority set. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] suggested improvement on Mutual Majority criterion/set
Back in December 2008 I criticised Marcus Schulze's beatpath Generalized Majority Criterion (which says in effect that if any candidate X has a majority-strength beatpath to candidate Y then Y can't win unless Y has a majority-strength beatpath back to X) in part because the concept is vulnerable to Mon-add-Plump. That is, extra ballots that plump for candidate A can cause A to fall out of the set of candidates that the criterion specifies are qualified to win. Then it was pointed out to me that to some extent the Mutual Majority (aka Majority for Solid Coalitions) criterion has the same problem. A candidate X can be in the set of candidates that are qualified to win and then some extra ballots that plump for X are added and then the set of candidates the criterion specifies are qualified to win expands to include one or more new candidates. X doesn't actually fall out of the set (as with beatpath GMC), but nonetheless according to the criterion X's case has been weakened by the new ballots that plump for X. I propose a replacement for Mutual Majority which addresses this problem and also unites it with Majority Favourite. Preliminary definitions: A solid coalition of candidates of size N is a set S of (one or more) candidates that on N number of ballots have all been voted strictly above all outside-S candidates. Any given solid coalition A's rival solid coalitions are only those that contain a candidate not in A. Statement of criterion: *If one exists, the winner must come from the smallest solid coalition of candidates that is bigger than the sum of all its rivals.* [end criterion definition] This wording could perhaps be polished, and I haven't yet thought of a name for this criterion and resulting set. (Any suggestions?) It might be possible to use the set as part of an ok voting method. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Oops! Forgot to include Chris's text. Chris MMT reply, complete this time.
Mike,
As I pointed out in my last message, I made a mistake with the example I
gave. There should have been only 10 BA votes.
45: C
06: DA
39: AB
10: BA
So there are a hundred voters and no what you call mutual-majority
candidate set.
But if it weren’t big enough, and if the D voters wanted to add
themselves to it, then they’d have only to vote D=AB. By MMT2’s
definition of a mutual majority candidate set.
I see. It seems that contrary to what I claimed, this method does meet
the FBC as you say.
But overall IMO it pays far too high a price for no defection
incentive and FBC compliance. It has random-fill and Burial incentives
and fails Mono-add-Plump.
Chris Benham
Mike Ossipoff wrote (9 Dec 2011):
Chris said:
As far as I can see the examples I gave apply equally well to MMT2.
I've pasted them in at the bottom.
He was referring to his posting copied and replied to below:
I think this (MMT2) fails the FBC. Say sincere is:
45: C
06: DA
39: AB
20: BA
There is no mutual majority set (by your latest definition)
My latest MMT version is still MMT2. It’s my latest, final,
and best MMT version.
By its definition of a mutual-majority candidate set, in
your example, {A,B} is a mutual-majority candidate set.
But if it weren’t big enough, and if the D voters wanted to
add themselves to it, then they’d have only to vote
D=AB. By MMT2’s
definition of a mutual majority candidate set.
Therefore, there would be no violation of FBC in your
example.
Your example illustrates a general fact: It’s possible to be
counted in support of any mutual majority candidate set without voting
anyone
over your favorite. MMT2 meets FBC.
Election-Methods mailing list - see http://electorama.com/em for list info
[EM] MMT2 meets FBC, fails Mono-Add-Plump, as it should.
Mike,
Sorry, there was a typo (20 BA voters instead of 10) in my
demonstration of MMT2's failure of FBC in my last post. So I'll go
through it again.
MMT2 defines mutual-majority candidate set as:
A set of candidates who are each voted above bottom by each member of the
same majority of voters--where that set includes at least one
top-rated candidate
on the ballot of every member of that majority.
45: C
06: DA
39: AB
10: BA
So in this example {A,B,D} isn't a mutual-majority candidate set
because D isn't voted above bottom by each member of the same majority
of voters, right? And because there is no such set the MMT2 winner is
C, right?
Say those votes were all sincere. If the 6 DA voters change to AB (or
A=B or BA) the winner changes to A, a candidate those voters prefer to C.
45: C
06: AB (sincere is DA)
39: AB
10: BA
Now {A,B} is a mutual-majority candidate set and MMT2 elects A.If the
method meets the FBC, those 6 voters must have some way of voting D not
below equal-top and get a result they like as much. What is it?
Mono-Add-Plump makes even less sense for MMT than for MDDTR.
The failure scenario is:
Your favorite wins by having the most top ratings among a
mutual-majority candidate
set. Now some new voters arrive and plump for hir. As plumpers, they
aren't counted
in the mutual majority. But they are counted in the total number of
voters, thereby
increasing the majority requirement. No longer is there a
mutual-majority candidate
set. No longer is your favorite the winner.
Is anyone claiming that that result is wrong?
Err..yes, I claim that at least one of the results must be wrong. Even
if we ignore the mono-add-plump failure and look at the two elections
independently (of each other), it is highly likely that at least one of
them will be a failure of some other desirable criterion compliance.
And, by the way, with MMT, the Mono-Add-Plump failure, and the LNHa
compliance
and LNHe failure don't create a random-fill incentive.
Logically, I don't see how it couldn't.
49: C
21: A (new voters, whose ballots switch the MMT2 winner from A to C)
27: AB
24: BA
(121 ballots, majority threshold = 61)
If the 21 A truncators randomly choose between middle-rating B or C then
A's chance of winning changes from zero to more than 50% (more than
11/21 have to middle-rate C for A to not win).
Chris Benham
Mike Ossipoff wrote (8 Dec 2011):
FBC:
In MMT2, if you top-rate a compromise, along with your favorite, then you'll
be counted in the majority supporting a mutual-majority candidate set that
s/he is in.
That's because MMT2 defines mutual-majority candidate set as:
A set of candidates who are each voted above bottom by each member of the
same majority of voters--where that set includes at least one top-rated
candidate
on the ballot of every member of that majority.
Mono-Add-Plump:
Mono-Add-Plump makes even less sense for MMT than for MDDTR.
The failure scenario is:
Your favorite wins by having the most top ratings among a
mutual-majority candidate
set. Now some new voters arrive and plump for hir. As plumpers, they
aren't counted
in the mutual majority. But they are counted in the total number of
voters, thereby
increasing the majority requirement. No longer is there a
mutual-majority candidate
set. No longer is your favorite the winner.
Is anyone claiming that that result is wrong?
Your favorite initially won only because of mutual majority support. The
plumpers
declined that mutual support, as is their right. Having declined mutual
support,
should it be surprising or unfair if they no longer have it?
And, by the way, with MMT, the Mono-Add-Plump failure, and the LNHa
compliance
and LNHe failure don't create a random-fill incentive.
The LNHe failure consists only of perhaps being able to benefit from
mutual majority
support.
I should say again that, henceforth, when I say MMT, without a
distinguishing number,
I'm referring to MMT2, the MMT version that I discussed above here.
I'm curious about MMMPO's compliance with FBC, LNHa and Mono-Add-Plump,
and its
compliance in Kevin's MMPO bad-example--a previously unattainable
combination of
properties. If MMMPO can be presented to the public in a simple,
naturally and obviously
motivated manner, then it would have the advantage that it wouldn't even
be necessary
to answer any Mono-Add-Plump criticism.
Mike Ossipoff
Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Complete MMT definition
Mike, I think this fails the FBC. Say sincere is: 45: C 06: DA 39: AB 20: BA There is no mutual majority set (by your latest definition) so C wins. That is also true if the 6 DA voters change to D=A or D=A=B or D=AB or anything else except AB or A=B or BA in which case the winner changes to A. It also fails Mono-add-Plump. 49: C 27: AB 24: BA Your latest version of MMT elects A, but if we add between 2 and 21 ballots that plump for A then there is no longer a majority candidate set and so the MMT winner changes from A to C. 49: C 21: A (new voters, whose ballots switch the MMT winner from A to C) 27: AB 24: BA (121 ballots, majority threshold = 61) I think all reasonable methods will elect A in both cases. Electing C in the second case will have voters wondering why they bothered switching from FPP, and is a very bad case of failing Condorcet and Mutual Dominant Third (DMT). A is voted above all other candidates on nearly 40% of the votes, and AC 72-49 and AB 48-24. Chris Benham Mike Ossipoff wrote (6 Dec 2011): Complete new definition of Mutual-Majority-Top (MMT): A mutual-majority candidate set is a set of candidates who are each rated above-bottom by each member of the same majority of voters--where that set of candidates contains every candidate rated above bottom by any member of that majority of the voters. If there are one or more mutual-majority candidate sets, then the winner is the most top-rated candidate who is in a mutual-majority candidate set. If there are no mutual-majority candidate sets, then the winner is the most top-rated candidate. [end of latest definition of MMT] Election-Methods mailing list - see http://electorama.com/em for list info
[EM] How to vote in IRV
Mike, Similar to the good Approval strategy approve the candidate A you would vote for in FPP, plus all the candidates you like as much or better than A as an IRV strategy guide is vote in first place the candidate A you would vote for in FPP and in second place the candidate B that you would vote for in FPP if A wasn't on the ballot and in third place the candidate C you would vote for in FPP if neither A or B was on the ballot, and so on. So barring rare and risky Push-over strategy opportunities, I don't see how IRV voting strategy is qualitatively more difficult than FPP strategy. When there are completely unacceptable candidates who might win (I call that condition u/a, for “unacceptable/acceptable”) IRV, like many methods, has a relatively simple strategy: When the voter's over-riding priority is to prevent the election of an unacceptable candidate, the voter should rank the acceptable candidates in order of estimated pairwise strength versus the likely unacceptable finalist (i.e. the unacceptable candidate that isn't eliminated before the final virtual run-off). Ideally, then rank the unacceptables in order of some complicated combination of their disutility and (some guessed or complicatedly-calculated measure of) their popularity. Actually, ignore that last paragraph. There is no reason at all to not rank the unacceptables sincerely. If your ranking among the unacceptables is ever counted it means that your strategy (aimed at preventing the election of an unacceptable candidate) has failed, and if any are even slightly less bad than the very worst you might as well help the lesser evil (unless you are concerned about your vote's symbolic gesture and want to deny any unacceptable winner legitimacy). What’s that you say? You might get lucky, even if you don’t top-rank a compromise? I think the voter very probably will, and most of the time will have sufficient information to know that s/he can safely vote hir sincere ranking. “Step right up, folks, and pick a card!” IRV, a game of chance, should only be allowed in states that allow gambling. I won't bother, but I think it is at least as easy to argue that Approval is a game of chance. Chris Benham Mike Ossipoff wrote (6 Dec 2011): How to vote in IRV: When there are completely unacceptable candidates who might win (I call that condition u/a, for “unacceptable/acceptable”) IRV, like many methods, has a relatively simple strategy: Rank the acceptable candidates in order of (some guessed or complicatedly-calculated measure of) their popularity. Ideally, then rank the unacceptables in order of some complicated combination of their disutility and (some guessed or complicatedly-calculated measure of) their popularity. Actually, ignore that last paragraph. In u/a, all the unacceptables are just unacceptable. What matters is the election of an acceptable instead of an unacceptable. In u/a, IRV is just ranked Plurality. In Plurality you vote for the acceptable candidate who is most popular (most likely to get the most votes). The difference is that the needed measure of popularity is simpler in Plurality (Which of the acceptables will be the best votegetter?). In that (decisive) regard, Plurality is better than IRV. Oh, and, by the way, our public political elections are u/a. IRV’s LNHa and LNHe: Some boast that IRV meets those two criteria. Well, if your 2nd choice gets your vote, your favorite, by that time, is beyond help or harm, isn’t s/he. Let’s protect hir from harm from your 2nd choice vote, by expelling hir from the election. :-) …A sort of electoral euthanasia. In IRV, you don’t have to be afraid to vote your 2nd choice necessary compromise at least in 2nd place. In fact you have to be afraid to not rank hir alone in 1st place. So, you see, IRV takes LNHa one step farther :-) What’s that you say? You might get lucky, even if you don’t top-rank a compromise? “Step right up, folks, and pick a card!” IRV, a game of chance, should only be allowed in states that allow gambling. Mike Ossipoff Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Sorry--One more revision of MMT
Mike, I was a bit confused about this for a while, because your definition of MMT doesn't make clear that a majority candidate set may contain only one candidate. Given that this uses 3-slot ballots, isn't it just (interpreting any above-Bottom rating as approval) Majority Approval//Top Ratings? *If no candidate is majority-approved elect the most top-rated candidate. Otherwise elect the most top-rated majority-approved candidate.* But of course that fails Later-no-Harm, because it could be that if some voters vote A truncate then no candidate will have majority approval and A wins but if they vote AB then B will have majority approval and the win will change to B. Chris Benham Mike Ossipoff wrote (5 Dec 2011): Mutual-Majority-Top (MMT): A set of candidates who are each rated above bottom by each member of the same majority of the voters is a majority candidate set. If there are one or more majority candidate sets, then the winner is the most top-rated candidate who is in a majority candidate set. If there are no majority candidate sets, then the winner is the most top-rated candidate. [end of MMT definition] The previous definition didn't allow for the fact that there can be overlapping majorities of the voters. MMT has the properties that I want (FBC, LNHa, ABE-non-failure, 3P), and avoids the not-really-valid criticisms of Mono-Add-Plump failure and electing C in Kevin's MMPO bad-example. That also seems to be true of Forest's FBC/ABE-passing method, which seems to act quite similarly to MMT. MTAOC too, with the added advantage of optional unconditional middle-rating support for a lesser-evil. We're always seeking better methods, and I'd like to find out if there's a simple wording that would allow voters in MMT to have the option of giving unconditional middle-rating support. But if I find that, I won't make it a replacement for the current MMT. I'll give it a different name. Likewise, it would be interesting if MTAOC, or something like it, could be written with a complete description in a short paragraph (though there's nothing wrong with its full definition being a computer program, while having a brief verbal description). Those two goals probably amount to about the same thing. Mike Ossipoff Election-Methods mailing list - see http://electorama.com/em for list info
[EM] This might be the method we've been looking for:
Forest, I don't understand the algorithm's definition. It seems to be saying that it's MinMax(Margins), only computing X's gross pairwise score against Y by giving X 2 points for every ballot on which X is both top-rated and voted strictly above Y, and otherwise giving X 1 point for every ballot on which X is top-rated *or* voted strictly above Y. But from trying that on the first example it's obvious that isn't it. Can someone please explain it to me? Chris Benham Forest Simmons wrote (2 Dec 2011): Here’s a method that seems to have the important properties that we have been worrying about lately: (1) For each ballot beta, construct two matrices M1 and M2: In row X and column Y of matrix M1, enter a one if ballot beta rates X above Y or if beta gives a top rating to X. Otherwise enter a zero. IN row X and column y of matrix M2, enter a 1 if y is rated strictly above x on beta. Otherwise enter a zero. (2) Sum the matrices M1 and M2 over all ballots beta. (3) Let M be the difference of these respective sums . (4) Elect the candidate who has the (algebraically) greatest minimum row value in matrix M. Consider the scenario 49 C 27 AB 24 BA Since there are no equal top ratings, the method elects the same candidate A as minmax margins would. In the case 49 C 27 AB 24 B There are no equal top ratings, so the method gives the same result as minmax margins, namely C wins (by the tie breaking rule based on second lowest row value between B and C). Now for 49 C 27 A=B 24 B In this case B wins, so the A supporters have a way of stopping C from being elected when they know that the B voters really are indifferent between A and C. The equal top rule for matrix M1 essentially transforms minmax into a method satisfying the FBC. Thoughts? Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Approval vs IRV
Mike, Someone said that IRV lets you vote more preferences than Approval does. But what good does that do, if it doesn't count them? The term count here can be a bit vague and propagandistic. Also you imply that it is always better to count preferences (no matter how) than to not. Also you seem to imply that all the voters care nothing about anything except affecting (positively from their perspective) the result and perhaps how their vote will do it. I reject that. A lot of voters want to know details of the result besides just who won, and want to see how some or all the candidates went, perhaps with the perspective of thinking about their voting strategy in the next election. And some people get some satisfaction from giving their full ranking of the candidates, even though most of that information will be ignored by the voting method algorithm. As a thought experiment, consider this method: voters strictly rank from the top however many candidates they like and also give an approval cut-off, the winner is the most approved candidate, exact ties resolved by random ballot (doesn't matter if drawn ballot doesn't show approval for any of the tied candidates). After the election each candidates' top rankings scores (and preferably other voted preference information) is made known along with their approval scores. I as a voter would happier with this than plain Approval. But I think after a while, say if the published results showed a failure of Majority Favourite, some voters might wonder why they have to gamble or use guess work in deciding where to put their approval cut-off and why the voting method can't use some algorithm that usefully uses more of the information on the ballots To say that IRV fails FBC is an understatement. IRV fails FBC with a vengeance. IRV thereby makes a joke any election in which it is used. That is an exaggeration. Regarding the proper version of IRV I earlier defined (that allows voters to strictly rank from the top however many candidates they want), most of the time none of the voters wouldn't even notice any FBC failure (and so incentive to use the Compromise strategy). As I've already said, all it takes is for favoriteness-support to taper moderately gradually away from the middle, something that is hardly unusual. Eliminations from the extremes will send transfers inward to feed the candidates flanking a middle CW, resulting in hir elimination. Yes, but if the wing voters' pairwise preference for the middle CW over their opposite wing's candidate is weak, then arguably that doesn't matter much. Also, even though Approval has a strong centrist bias, it is possible that Approval will fail to elect a CW that IRV would have. After all, IRV meets Mutual Dominant Third and Condorcet Loser. (So for your example to work, the middle CW has to be solidly supported by fewer than a third of the voters). That said, though Approval or MTA is incomparably better than Plurality, and would be completely adequate, I'd prefer, if electorally-attainable, a method that meets LNHa. I like MTA and IBIFA (preferably with 4-slot ballots), and some of the Condorcet methods. I wouldn't say that Approval would be completely adequate (but of course a big improvement on FPP). Chris Benham Mike Ossipoff wrote (1 Dec 2011): Someone said that IRV lets you vote more preferences than Approval does. But what good does that do, if it doesn't count them? Approval counts every preference that you vote. Since Approval doesn't let you vote all of your preferences, it doesn't count all of your preferences. But, unlike IRV, you can choose which of your preferences will be counted. You can divide the candidate-set into two parts in any way you choose. You, and only you, choose among which two sets of candidates your preferences will be counted. As I've said, our elections have completely unacceptable candidates. Under those conditions, most methods reduce to Approval anyway. When, in Approval, you approve all of the acceptable candidates and none of the unacceptable candidates, you're doing all that you'd want to do anyway. - Yes, Approval has the ABE problem, the co-operation/defection problem. We've discussed two solutions for that problem that could be used in Approval: 1. Your faction makes it known that they will, from principle, refuse to support some inadequate alleged lesser-evil compromise. The other greater-evil-opposers including the supporters of that lesser-evil will understand that, if they need the votes of a more principled faction, and aren't going to get their votes, then they had better approve that faction's candidates if they don't want a greater evil to win. Of course, no one who prefers your faction's policies to those of that lesser-evil would have any pragmatic reason to approve the lesser-evil but not
[EM] Approval vs. IRV
Ted Stern wrote (29 Nov 2011): 47: A 05: AB (sincere is AB) 41: B 07: BC Approvals: B53, A52, C7 I find this example contrived. * If mass polling is available, many people will be aware of the 52/48 split between A and B ahead of time. * Corruption is a separate issue. With proper election funding control, support for C would be restricted. Ted, I reject your criticisms of my example. Of course it's contrived. So what? How could it not be? In my example many people as you say are aware of the 52/48 split between A and B ahead of time. 95% of them vote as if they are aware of it. Approval-Bucklin (AKA ER-Bucklin) has the advantage in your contrived example of allowing the A B voters to add B at a lower rank, which would not count unless neither A nor B achieves a majority. In many cases, it would not be necessary to rate candidates at the second (or lower) choice option, but having that option increases the available nuance of the vote. Yes. My favourite similar methods are IBIFA and MTA. However IRV does impose a false choice -- that you must rank your preferences separately, no equal ranks allowed. In the case of methods that would fail FBC if they did allow equal-top ranking, I don't consider this to be a big deal. In the case of IRV, allowing it would make Push-over strategising easier and the method more complex to count/implement. In my opinion IRV is one of the reasonable algorithms to use with ranked ballots, and the best for those who prefer things like Later-no-Harm and Invulnerability to Burial to either the Condorcet or FBC criteria. But are these the criteria we really want to achieve in a single-winner election? Invulnerability to Burial is a very attractive property to me, but perhaps not. To say that LNH is the most important criterion is, at its most basic level, an emotional argument. I don't say that, but some people definitely like LNHarm. I prefer its LHHelp compliance, and regard its LNHarm compliance as only excusable because of it. I think what we really want to look for is a method that does a good job of finding the candidate closest to the center of the electorate, while resisting strategic manipulation. I am mostly in sympathy with that aim. Probably the best methods meet one of Condorcet and the FBC. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Approval vs. IRV
Juho Laatu wrote (29 Nov 2011): We may compare IRV also to the other commonly used single-winner method TTR. To be brief, one could say that IRV is better than TTR since it has more elimination rounds. IRV's problem in this comparison is that it collects so much information that one can, after the election, see what strategies would have paid off. In TTR one may have very similar problems but people stay happier since they can not see the problems. They can't see for example what would have happened if some other pair of candidates would have made it to the second round. Spoilers may exist but they remain undetected, or at least unverified. Yes, IRV is much better than TTR partly for that reason. IRV simulates everyone gets one vote each, eliminate one candidate, repeat until one remains (a process I think is called the Exhaustive Ballot) except that voters can't sit out a round or two and then come back in, and they have to keep voting consistent with their ranking that they give at the beginning (so if in the first round they vote for X they have to keep voting for X until X is eliminated or wins). This last feature is a big positive because it makes using the devious Push-over strategy much more difficult and risky. In TTR if you are confidant that your favourite F will make the second round without your vote (but not make the majority threshold even with your vote) you might be able to improve F's chance of winning by voting in the first round for a turkey T that you are sure that F can pairwise beat with your vote. In IRV if you try that and you succeed in causing the final (virtual) runoff to be between F and T, F has to win with you still voting for T. I'd like to add that IRV is an algorithm for those that want to favour the large parties. The main thing that favours large parties is legislators elected in single-member districts versus some form of PR in multi-member districts. But yes, IRV is a bit biased towards slightly off-centre candidates whereas Approval has a strong bias toward centrist candidates. In Approval it is just possible to have a surprise centrist winner, by getting all the approvals of voters in the centre (with maybe some being exclusive approvals) and approvals from some of the wing voters who fear the opposing wing candidate more than they like (or are hopeful about) their own. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Approval vs. IRV
Matt Welland wrote (26 Nov 2011): Also, do folks generally see approval as better than or worse than IRV? To me Approval seems to solve the spoiler problem without introducing any unstable weirdness and it is much simpler and cheaper to do than IRV. If we are talking about the classic version of IRV known as the Alternative Vote in the UK and Optional Preferential Voting in Australia, then I see IRV on balance as being better than Approval. The version of IRV I'm referring to: *Voters strictly rank from the top however many or few candidates they wish. Until one candidate remains, one-at-a time eliminate eliminate the candidate that (among remaining candidates) is highest-ranked on the fewest ballots.* The unstable weirdness of Approval is in the strategy games among the rival factions of voters, rather than anything visible in the method's algorithm. Approval is more vulnerable to disinformation campaigns. Suppose that those with plenty of money and control of the mass media know from their polling that the likely outcome of an upcoming election is A 52%, B 48% and they much prefer B. In Approval they can sponsor and promote a third candidate C, one that the A supporters find much worse than B, and then publish false polls that give C some real chance of winning. If they can frighten/bluff some of A's supporters into approving B (as well as A) their strategy can succeed. 47: A 05: AB (sincere is AB) 41: B 07: BC Approvals: B53, A52, C7 Approval is certainly the bang for buck champion, and voters never have any incentive to vote their sincere favourites below equal-top. But to me the ballots are insufficiently expressive by comparison with the strict ranking ballots used by IRV. IRV has some Compromise incentive, but it is vastly less than in FPP. Supposing we assume that there are 3 candidates and that you the voter want (maybe for some emotional or long-term reason) to vote your sincere favourite F top even if you think (or know) that F can't win provided you don't thereby pay too high a strategic penalty, i.e. that the chance is small that by doing that you will lose some (from your perspective positive) effect you might otherwise have had on the result. In FPP, to be persuaded to Compromise (i.e.vote for your compromise might win candidate C instead of your sincere favourite F) you only have to be convinced that F won't be one of the top two first-preference place getters. In IRV if you are convinced of that you have no compelling reason to compromise because you can expect F to be eliminated and your vote transferred to C. No, to have a good reason to compromise you must be convinced that F *will* be one of the top 2 (thanks to your vote) displacing C, but will nonetheless lose when C would have won if you'd top-voted C. In my opinion IRV is one of the reasonable algorithms to use with ranked ballots, and the best for those who prefer things like Later-no-Harm and Invulnerability to Burial to either the Condorcet or FBC criteria. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] An ABE solution
Forest Simmons, responding to questions from Mike Ossipff, wrote (19 Nov
2011):
4. How does it do by FBC? And by the criteria that bother some
people here about MMPO (Kevin's MMPO bad-example) and MDDTR
(Mono-Add-Plump)?
I think it satisfies the FBC.
Forest's definition of the method being asked about:
Here’s my current favorite deterministic proposal: Ballots are Range
Style, say three slot for simplicity.
When the ballots are collected, the pairwise win/loss/tie relations are
determined among the candidates.
The covering relations are also determined. Candidate X covers
candidate Y if X
beats Y as well as every candidate that Y beats. In other words row X
of the
win/loss/tie matrix dominates row Y.
Then starting with the candidates with the lowest Range scores, they are
disqualified one by one until one of the remaining candidates X covers
any other
candidates that might remain. Elect X.
Forest,
Doesn't this method meet the Condorcet criterion? Compliance with
Condorcet is incompatible with FBC, so
why do you think it satisfies FBC?
http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-June/016410.html
Hello,
This is an attempt to demonstrate that Condorcet and FBC are incompatible.
I modified Woodall's proof that Condorcet and LNHarm are incompatible.
(Douglas R. Woodall, Monotonicity of single-seat preferential
election rules,
Discrete Applied Mathematics 77 (1997), pages 86 and 87.)
I've suggested before that in order to satisfy FBC, it must be the case
that increasing the votes for A over B in the pairwise matrix can never
increase the probability that the winner comes from {a,b}; that is, it
must
not move the win from some other candidate C to A. This is necessary
because
if sometimes it were possible to move the win from C to A by increasing
v[a,b], the voter with the preference order BAC would have incentive to
reverse B and A in his ranking (and equal ranking would be inadequate).
I won't presently try to argue that this requirement can't be avoided
somehow.
I'm sure it can't be avoided when the method's result is determined solely
from the pairwise matrix.
Suppose a method satisfies this property, and also Condorcet. Consider
this
scenario:
a=b 3
a=c 3
b=c 3
ac 2
ba 2
cb 2
There is an ACBA cycle, and the scenario is symmetrical, as based on
the submitted rankings, the candidates can't be differentiated. This means
that an anonymous and neutral method has to elect each candidate with
33.33%
probability.
Now suppose the a=b voters change their vote to ab (thereby
increasing v[a,b]).
This would turn A into the Condorcet winner, who would have to win
with 100%
probability due to Condorcet.
But the probability that the winner comes from {a,b} has increased
from 66.67%
to 100%, so the first property is violated.
Thus the first property and Condorcet are incompatible, and I contend
that FBC
requires the first property.
Thoughts?
Kevin Venzke
Chris Benham
Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Votes-only criteria vs preference criteria. IRV squeeze-effect. Divulge IRV election specifics?
Mike refers to this scenario: The Approval bad-example is an example of that. I'll give it again here: Sincere preferences: 49: C 27: AB 24: BA A majority _equally strongly_ prefer A and B to C. Actual votes: The A voters defect, in order to take advantage of the co-operativeness and responsibility of the A voters: 49: C 27: AB 24: B I agree that *if* the sincere preferences are as Mike specifies then a just interventionist mind-reading God should award the election to A. But a voting method's decisions and philosophical justification should be based on information that is actually on the ballots, not on some guess or arbitrary assumption about some maybe-existing information that isn't. I think a very reasonable tenet is that if, based on the information on the ballots, candidate X utterly dominates candidate Y then we should not elect Y. For several reasons (for those who can pooh-pooh this as merely aesthetic): electing Y gives the supporters of X a very strong post-election complaint with no common-sense or philosophically cogent answer, X is highly likely to be higher Social Utility (SU), Y's victory will have compromised legitimacy. The Plurality criterion is one very reasonable criterion that says that C is so much stronger than A that the election of A can't be justified. . There are other criteria I find reasonable that say the same thing: Strong Minimal Defense: If the number of ballots on which both X is voted above bottom and Y isn't is greater than the total number of ballots on which Y is voted above bottom, then don't elect Y. And 2 that only use information from the normal gross pairwise matrix: Pairwise Plurality: If X's smallest pairwise score is larger than Y's largest pairwise score, then don't elect Y. Pairwise Strong Minimal Defense : If X's pairwise score versus Y is larger than Y's largest pairwise score, then don't elect Y. The election of A is unacceptable because C's domination of A is vastly more impressive than A's pairwise win over B. The Plurality criterion plus the three other criteria I define above all loudly say not A. Minimal Defense and Strong Minimal Defense and Pairwise Strong Minimal Defense all say not C (due to B), and I find that message very reasonable but nothing like as compelling as the not A message. The A voters defect, in order to take advantage of the co-operativeness and responsibility of the A voters: The plausibility of arbitrary claims about the voters' sincere preferences and motivations can weighed in the light of the used election method's incentives. How is it so co-operative and responsible of the A voters to rank B when doing so (versus truncating) can only help their favourite? And why would the B voters be insincerely truncating (defecting) when doing so can only harm their favourite? Given the incentives of the MDD,TR method that Mike is advocating, it is only reasonable to assume that the truncators are all sincere and that the AB voters' sincere preferences could be AB or AC or A. It's a bit like Mike is assuming that the voters were all deceived into thinking that their votes would be counted using a method like Bucklin or MCA (which have truncation and defection incentives, failing Later-no-Harm and meeting Later-no-Help). (I might comment on IRV in another post). Chris Benham Mike Ossipoff wrote (16 Nov 2011): Votes-only criteria vs preference criteria: Kevin, you objected to my preference-mentioning criteria on the grounds that no one knows what the voters' true preferences really are. But so what? As I said before, my criteria indirectly stipulate votes. They do that when they stipulate that people have a certain preference and vote sincerely; or have a certain preference and don't vote anyone equal to or over their favorite. Etc. Are you saying that methods meeting my preference-mentioning criteria can act wrongly when the preferences aren't as stipulated? If so, then say so explicitly, and show how that can happen. As a matter of fact, that _can_ happen with some votes-only criteria, such as the Plurality Criterion: The Approval bad-example is an example of that. I'll give it again here: Sincere preferences: 49: C 27: AB 24: BA A majority _equally strongly_ prefer A and B to C. Actual votes: The A voters defect, in order to take advantage of the co-operativeness and responsibility of the A voters: 49: C 27: AB 24: B Now, in MDDTR and MMPO, A wins. According to the Plurality Criterion, that's wrong. But it's only wrong if the B voters aren't voting for A because they don't prefer A to C as much as the A voters prefer B to C. Given the preferences, and the explanation for the actual votes, the Plurality Criterion is wrong when it calls the election of A a wrong result. So yes: A criterion can rule wrongly, based on an incorrect built-in assumption about true preferences.
[EM] Descending Acquiescing Coalitions versus Nested Acquiescing Coalitions
Forest,
This NAC method suggestion of yours fails my Descending Solid Coalitions
bad example:
49: C
48: A
03: BA
NAC, like DSC and FPP, elects C while DAC elects the MDT (Mutual
Dominant Third) winner A.
DAC goes AC96 (disqualify B), AB51 (disqualify C), A wins.
NAC skips AB because that includes an already disqualified candidate and
next goes to C49 and
disqualifies A.
Which of the good properties of DAC are retained by NAC?
I think Majority for Solid Coalitions and probably Clone Independence
and maybe some others.
I'd be surprised if it meets Participation.
Chris Benham
Forest Simmons wrote (9 Nov 2011):
DAC (descending acquiescing coalitions) disappointed Woodall because of
the following example:
03: D
14: A
34: AB
36: CB
13: C
The MDT winner is C, but DAC elects B.
DAC elects B even though the set {B} has a DAC score of zero, because
the descending order of
scores includes both the set {C,B} (with a score of 49) and the set
{A,B} (with a score of 48), and the
only candidate common to both sets is B, so B is elected by DAC.
But suppose that we change DAC to NAC (Nested Acquiescing Coalitions) so
that sets in the sequence
of descending scores are not only skipped over when the intersection is
empty, but also skipped over
when the set with the lower score is not a subset of the previously
included sets. Then, in the above
example, C is elected.
I want to point out that this NAC method also solves the bad approval
problem by electing C, B, and A
respectively, given the respective ballot sets
49 C
27 AB
24 B,
and
49 C
27 A=B
24 B,
and
49 C
27 AB
24 BA .
Which of the good properties of DAC are retained by NAC?
Thanks,
Forest
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[EM] Reply to Chris regarding the Approval bad-example
49: C 27: AB 24: B I agree that *if* the sincere preferences are as Mike specifies then a just interventionist mind-reading God should award the election to A. [endquote] Fine. But can Chris say what's wrong with that outcome in other instances? Yes. If the method used meets Later-no-Harm but fails Later-no-Help, i.e has a strong random-fill incentive like the MDD,TR method that Mike is advocating, there isn't any good reason to assume that the Middle ratings are sincere. So it could be that all the voters really have no interest in any candidate except their favourites and sincere is 49: C 27: A 24: B in which case C is the strong sincere Condorcet winner, or as Jameson pointed out it could be worse still and the A voters were Burying against C so sincere is 49: C 27: AC 24: B Chris continues: Given the incentives of the MDD,TR method that Mike is advocating, it is only reasonable to assume that the truncators are all sincere [endquote] Wait a minute: I'm not saying that B truncation is a problem in MDDTR or MMPO. In fact, my point is that it is _not_. Truncation isn't a problem (for the full-rankers) as an offensive strategy. The problem is that it isn't fair to the sincere truncators. Wait a minute. These candidates in this example are A, B, and C. How does A lack legitimacy? Among the candidates not majority-defeated, A has more favoriteness-supporters than any other candidate. Translation: I love this arbitrary algorithm, so any winner it produces is by definition legitimate. A's win lacks legitimacy simply because there is another candidate that was vastly better supported on the ballots. If we add between 2 and 21 ballots that plump for A, then C's majority-defeatedness goes away and the winner changes from A to C, another failure of Mono-add-Plump. If we nonetheless accept that C but not A should be immediately disqualified, electing the undisqualified candidate with the most top-ratings is just another arbitrary feature of the algorithm. Why that candidate and not the one that is most approved? Based on the information actually on the ballots, no faction of voters has a very strong post-election complaint against B. Chris Benham 49: C 27: AB 21: A (new voters, whose ballots change the MDD,TR winner from A to C) 24: B Mike Ossipoff wrote (17 Nov 2011): Chris said: Mike refers to this scenario: The Approval bad-example is an example of that. I'll give it again here: Sincere preferences: 49: C 27: AB 24: BA A majority _equally strongly_ prefer A and B to C. Actual votes: The A voters defect, in order to take advantage of the co-operativeness and responsibility of the A voters: 49: C 27: AB 24: B I agree that *if* the sincere preferences are as Mike specifies then a just interventionist mind-reading God should award the election to A. [endquote] Fine. But can Chris say what's wrong with that outcome in other instances? Chris continued: But a voting method's decisions and philosophical justification should be based on information that is actually on the ballots, not on some guess or arbitrary assumption about some maybe-existing information that isn't. [endquote] Why? Why shouldn't a voting system avoid a worst-case, if, by so doing, it hasn't been shown to act seriously wrongly in other cases? And MMPO MDDTR don't just bring improvement in the Approval bad-example. They, in general, get rid of any strategy dilemma regarding whether you should middle-rate a lesser-evil instead of bottom-rating hir. For instance, consider the A 100, B 15, C 0 utility example. In MCA, there's a question about whether you should middle-rate or bottom-rate B. In MDDTR and MMPO, that dilemma is completely eliminated. In those methods, middle rating someone can never help hir against your favorite(s). Chris continues: I think a very reasonable tenet is that if, based on the information on the ballots, candidate X utterly dominates candidate Y then we should not elect Y. [endquote] Yes, there are many reasonable tenets among the aesthetic criteria. Chris continues: For several reasons (for those who can pooh-pooh this as merely aesthetic): electing Y gives the supporters of X a very strong post-election complaint with no common-sense or philosophically cogent answer, X is highly likely to be higher Social Utility (SU), Y's victory will have compromised legitimacy. [endquote] Wait a minute. These candidates in this example are A, B, and C. How does A lack legitimacy? Among the candidates not majority-defeated, A has more favoriteness-supporters than any other candidate. Chris continues: The Plurality criterion is one very reasonable criterion that says that C is so much stronger than A that the election of A can't be justified. . [endquote] There are lots of aesthetic criteria that say things like that, and they all sound aesthetically reasonable. How great is their practical strategic
[EM] ABucklin doesn't meet Mono-Add-Top or Participation, but meets Mono-Add-Plump. MDDTR and Mono-Add-Plump.
Mike, You continued: (Also it looks like you have some other method in mind [endquote] How so? As I said, I'm referring to MDDTR. Because in the description of your example you referred to information that MDDTR ignores: Say the method is MDDTR, and your favorite candidate is F. F doesn't have a winning approval (top + middle) score, because x has significantly more approvals. MDDTR takes no account of approval scores. It is only interested in majority-strength pairwise defeats and TR scores. It looks more like you were describing MDDA. But x is disqualified by having a (bare) majority voting y over hir. With x disqualified, F wins with the most approvals of any undisqualified candidate. F isn't close to having a top-rating majority. Nor is MDDTR (explicitly) interested in a top-rating majority. Now it looks more like you are describing MDD,ABucklin But I'll post an example of that particular kind of Mono-Add-Plump failure within the next few days. I look forward to seeing it. Chris Benham Chris Benham wrote: It isn't possible for a method to both meet Mono-add-Top and fail Mono-add-Plump. [endquote] I hope that I didn't say that ABucklin fails Mono-Add-Plump. If I did, it was an error and I retract the statement. In the subject-line, I said that ABucklin passes Mono-Add-Plump. So yes, that was a typo. I meant what I said in the subject-line: ABucklin doesn't meet Mono-Add-Top, but it meets Mono-Add-Plump. You (Chris) said: (Just before posting this I've noticed that your quoted text isn't consistent with your Subject line) [endquote] Yes, there was a typo in my message, regarding that. I'd said: MDDTR and Mono-Add-Plump: Say the method is MDDTR, and your favorite candidate is F. F doesn't have a winning approval (top + middle) score, because x has significantly more approvals. But x is disqualified by having a (bare) majority voting y over hir. With x disqualified, F wins with the most approvals of any undisqualified candidate. F isn't close to having a top-rating majority. Then you and a few other people show up, and plump for F. (You top rate F, and don't rate anyone else). Now your presence in the election increases the requirement for a majority, with the result that x no longer has a majority ranking y over hir. Now, x wins instead of F, because x has significantly more approvals (F was behind x in approvals by more than the number of newly-arrived voters. By plumping for F, you and the other newly-arrived voters have made F lose. You wrote: Mike, I'd like to see an example election of what you are talking about. If this way of MDD,TR failing Mono-add-Plump is possible it isn't the one I know about. [endquote] I admit, not assert, that MDDTR fails Mono-Add-Plump. We agree that it does. But I'll post and example of that particular kind of Mono-Add-Plump failure within the next few days. You continued: (Also it looks like you have some other method in mind [endquote] How so? As I said, I'm referring to MDDTR. Here's my definition of MDDTR: 3-slot method: top, middle, and bottom (unrated) Disqualify every candidate who has another candidate voted over hir by a majority. The winner is the undisqualified candidate with the most top ratings. [end of MDDTR definition] That is the method that I was referring to when I said MDDTR. You wrote: 25: AB 26: BC 23: CA 04: C (78 ballots) BC 51-27, CA 53-25, AB 48-26 TR scores: C27, B26, A25. Approval scores: C53, B51, A48. All candidates have a majority-strength pairwise defeat, so no candidate is disqualified. MDD,TR and MDD,A and MDD,ABucklin (as you call it) all elect C. Now say we add 22 ballots which plump for C. 25: AB 26: BC 23: CA 26: C (100 ballots) BC 51-49, CA 75-25, AB 48-26 TR scores: C49,B26,A25. Approval scores: C75, B51, A48. Now there is one candidate (B) without a majority-strength pairwise defeat, so all except B are disqualified and B wins. [endquote] Thank you for that example showing the MDDTR Mono-Add-Plump scenario that I described. No, your example is not different from my scenario. It's a numerical example of my scenario. The plump-ballots took away B's majority defeat, allowing B to win. The only difference was that, my scenario, B beat C by higher Approval score, whereas, in your example, B wins by being the only undisqualified candidate. Unimportant difference. In both stories, the plump-ballots take away B's majority defeat by raising the requirement for a majority. You wrote: BTW, unrelated to the Mono-add-Plump issue, C in both elections is uncovered and positionally dominant so I think a method needs a much better excuse for not electing C in both cases than any that the MDD methods can offer. [endquote] Cetainly, if uncoveredness and positional dominance can be shown to have great practical importance, as opposed to aesthetic appeal. Any method will fail many
[EM] ABucklin doesn't meet Mono-Add-Top or Participation, but meets Mono-Add-Plump. MDDTR and Mono-Add-Plump.
Mike Ossipoff wrote (12 Nov 2011): ABucklin and Mono-Add-Top: In the criterion-compliance table that I posted, I said that ABucklin meets Mono-Add-Plump, Mono-Add-Top and Participation. Actually, it only meets Mono-Add-Top. It isn't possible for a method to both meet Mono-add-Top and fail Mono-add-Plump. Ballots that plump for X are also ballots that top-vote X. (Just before posting this I've noticed that your quoted text isn't consistent with your Subject line) ABucklin meets Mono-add-Plump and fails (as shown in my last post) Participation. MDDTR and Mono-Add-Plump: Say the method is MDDTR, and your favorite candidate is F. F doesn't have a winning approval (top + middle) score, because x has significantly more approvals. But x is disqualified by having a (bare) majority voting y over hir. With x disqualified, F wins with the most approvals of any undisqualified candidate. F isn't close to having a top-rating majority. Then you and a few other people show up, and plump for F. (You top rate F, and don't rate anyone else). Now your presence in the election increases the requirement for a majority, with the result that x no longer has a majority ranking y over hir. Now, x wins instead of F, because x has significantly more approvals (F was behind x in approvals by more than the number of newly-arrived voters. By plumping for F, you and the other newly-arrived voters have made F lose. Mike, I'd like to see an example election of what you are talking about. If this way of MDD,TR failing Mono-add-Plump is possible it isn't the one I know about. (Also it looks like you have some other method in mind, but my comments still apply). 25: AB 26: BC 23: CA 04: F (78 ballots) BC 51-27, CA 53-25, AB 48-26 TR scores: C27, B26, A25. Approval scores: C53, B51, A48. All candidates have a majority-strength pairwise defeat, so no candidate is disqualified. MDD,TR and MDD,A and MDD,ABucklin (as you call it) all elect C. Now say we add 22 ballots which plump for C. 25: AB 26: BC 23: CA 26: C (100 ballots) BC 51-49, CA 75-25, AB 48-26 TR scores: C49,B26,A25. Approval scores: C75, B51, A48. Now there is one candidate (B) without a majority-strength pairwise defeat, so all except B are disqualified and B wins. BTW, unrelated to the Mono-add-Plump issue, C in both elections is uncovered and positionally dominant so I think a method needs a much better excuse for not electing C in both cases than any that the MDD methods can offer. So you storm into the Department of Elections office, to complain about that. The person at the counter says, Excuse me, but do you think that the election was a Plurality election? You see, in Plurality, 1st choice votes are what decide the election. Rank methods look at more than that. They look at your other preferences too. Maybe it's tempting to want 1st choice ratings to decide the election in rank methods too. But they're rank methods, and rank methods needn't act like Plurality. This explanation might be acceptable if we were just talking about a failure of Mono-add-Top where the complainers provided some extra information that the voting-method algorithm might have reasonably construed as strengthening not just their favourite but also the winner, or even just extra information that might have caused the algorithm to be (perhaps) forgivably confused. Yes, it's aesthetically nice if the win is monotonically related to addition of 1st choice ballots, but there is no reason why it should or must be. Rank methods aren't Plurality. Here again it sounds more like you are talking about Mono-add-Top instead of Mono-add-Plump. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] ABucklin doesn't meet Mono-Add-Top or Participation, but meets Mono-Add-Plump. MDDTR and Mono-Add-Plump.
Sorry, a small mistake in my first election corrected. Mike Ossipoff wrote (12 Nov 2011): ABucklin and Mono-Add-Top: In the criterion-compliance table that I posted, I said that ABucklin meets Mono-Add-Plump, Mono-Add-Top and Participation. Actually, it only meets Mono-Add-Top. It isn't possible for a method to both meet Mono-add-Top and fail Mono-add-Plump. Ballots that plump for X are also ballots that top-vote X. (Just before posting this I've noticed that your text that I quote above isn't consistent with what you wrote in the Subject line.) ABucklin meets Mono-add-Plump and fails (as shown in my last post) Participation. MDDTR and Mono-Add-Plump: Say the method is MDDTR, and your favorite candidate is F. F doesn't have a winning approval (top + middle) score, because x has significantly more approvals. But x is disqualified by having a (bare) majority voting y over hir. With x disqualified, F wins with the most approvals of any undisqualified candidate. F isn't close to having a top-rating majority. Then you and a few other people show up, and plump for F. (You top rate F, and don't rate anyone else). Now your presence in the election increases the requirement for a majority, with the result that x no longer has a majority ranking y over hir. Now, x wins instead of F, because x has significantly more approvals (F was behind x in approvals by more than the number of newly-arrived voters. By plumping for F, you and the other newly-arrived voters have made F lose. Mike, I'd like to see an example election of what you are talking about. If this way of MDD,TR failing Mono-add-Plump is possible it isn't the one I know about. (Also it looks like you have some other method in mind, but my comments still apply). 25: AB 26: BC 23: CA 04: C (78 ballots) BC 51-27, CA 53-25, AB 48-26 TR scores: C27, B26, A25. Approval scores: C53, B51, A48. All candidates have a majority-strength pairwise defeat, so no candidate is disqualified. MDD,TR and MDD,A and MDD,ABucklin (as you call it) all elect C. Now say we add 22 ballots which plump for C. 25: AB 26: BC 23: CA 26: C (100 ballots) BC 51-49, CA 75-25, AB 48-26 TR scores: C49,B26,A25. Approval scores: C75, B51, A48. Now there is one candidate (B) without a majority-strength pairwise defeat, so all except B are disqualified and B wins. BTW, unrelated to the Mono-add-Plump issue, C in both elections is uncovered and positionally dominant so I think a method needs a much better excuse for not electing C in both cases than any that the MDD methods can offer. So you storm into the Department of Elections office, to complain about that. The person at the counter says, Excuse me, but do you think that the election was a Plurality election? You see, in Plurality, 1st choice votes are what decide the election. Rank methods look at more than that. They look at your other preferences too. Maybe it's tempting to want 1st choice ratings to decide the election in rank methods too. But they're rank methods, and rank methods needn't act like Plurality. This explanation might be acceptable if we were just talking about a failure of Mono-add-Top where the complainers provided some extra information that the voting-method algorithm might have reasonably construed as strengthening not just their favourite but also the winner, or even just extra information that might have caused the algorithm to be (perhaps) forgivably confused. Yes, it's aesthetically nice if the win is monotonically related to addition of 1st choice ballots, but there is no reason why it should or must be. Rank methods aren't Plurality. Here again it sounds more like you are talking about Mono-add-Top instead of Mono-add-Plump. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Criterion-compliance table. Method merit order. Polling and proposing methods.
Mike Ossipoff wrote (11 Nov 2011): Let me know if there are errors in the following table: MAP is Mono-Add-Plump. MAT is Mono-Add-Top. ABE means that the method passes in the Approval Bad Example. = FBC...3P...1CM...SDSC...UP...MAP...MAT...Participation...SFC...ABE -- ApprovalYes...No...NoNo.No...Yes...Yes...Yes.NoNo MTA.Yes...Yes..Yes...No.No...Yes...Yes...No..NoNo MCA.Yes...Yes..Yes...No.No...Yes...Yes...No..NoNo SMDTR...Yes...Yes..Yes...No.No...Yes?.?..NoNo IBIFA...Yes...Yes..Yes...No.No...Yes...NoNo..NoNo MDDAYes...Yes..Yes...No.No...NoNoNo..Yes...No ABucklinYes...Yes..Yes...YesYes..Yes...Yes...Yes.NoNo MDD,ABucklinYes...Yes..Yes...YesYes..NoNoNo..Yes...No MDDTR...Yes...No...NoNo.No...NoNoNo..Yes...Yes Mike, A quick partial reply. SMD,TR fails Mono-add-top and so therefore also Participation. 8: C 3: F 2: XF 2: YF 2: ZF F wins after all other candidates are disqualified, but if 2 FC ballots are added C wins ER-Bucklin(whole), ABucklin on your chart, fails Participation as shown by this demonstration from Kevin Venzke (which also applies to MCA, MTA, and MDD,ABucklin): 5: AB 4: BC A is a majority favorite and wins. But add these in: 2: CA There is no majority favorite and B wins in the second round. IBIFA meets UP provided ballots with enough slots to enable voters to strictly rank all the candidates are used. I strongly disagree with your suggested method merit order, and I'll explain how and why in a later post. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Approval Bad Example
Jameson, In response to Forest asking if there was a method that satisfies something plus FBC you responded: Yes. 321 voting http://wiki.electorama.com/wiki/321_voting 321 voting From Electowiki Jump to: navigation #column-one, search #searchInput 3-level rated ballots. Of the 3 candidates with the most ratings, take the 2 candidates with the most top-ratings, and then take the 1 pairwise winner among those. This fails FBC in the same way that ER-IRV(whole) does. From my 2 Nov. EM post: snip Here is Kevin Venzke's example from a June 2004 EM post: 6: A 3: CB 2: C=B (sincere is CB) 2: B The method is ER-IRV(whole). If the 2 C=B voters sincerely vote CB then the first-round scores are A6, C5, B2. B is eliminated and A wins. As it is the first-round scores are A6, C5, B4. B is still eliminated and A wins. To meet FBC no voters should have any incentive to vote their sincere favourite below equal-top. 6: A 3: CB 2: BC (sincere is CB) 2: B But if those 2 voters (sincere CB, was C=B) do that and strictly top-rank their compromise candidate B, then the first-round scores are A6, B4, C3. C is eliminated and B wins: B7, A6. By down-ranking their sincere favourite those 2 voters have gained a result they prefer that they couldn't have got any other way, a clear failure of the Favorite Betrayal Criterion (FBC). snip Even if 321 voting met FBC with 3 candidates it it wouldn't with more, because sincerely rating your sincere favourite Top instead of Bottom could mean that your favourite displaces your compromise candidate from the top 3 most rated candidates and goes on to lose when your compromise would have won. Chris Benham . Forest Simmons wrote (9 Nov 2011): I'm assuming approval bad example is typified by the implicit approval order in the scenario 49 C 27 AB 24 B It seems to me that IF we (1) want to respect the Plurality Criterion, (2) discourage chicken strategy, (3) stick with determinism, and (4) not take advantage of proxy ideas, then our method must allow equal-rank-top and elect C in the above scenario, but elect B when B is advanced to top equal with A in the middle faction: 49 C 27 A=B 24 B Then if sincere preferences are 49 C 27 AB 24 BA, the B faction will be deterred from truncating A. While if the B supporters are sincerely indifferent between A and C, the A supporters can vote approval style (A=B) to get B elected. Do we agree on this? Note that IRV (=whole) satisfies this, but now the question remains ... is there a method that satisfies this which also satisfies the FBC? Forest Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Approval Bad Example
Mike Ossipoff wrote (9 Nov 2011): Here's the definition of MDD,TR: 3-slot method: Top, Middle, Bottom (unmarked) Disqualify any candidate(s) having a majority pairwise defeat. The winner is the un-disqualified candidate with the most top ratings. [end of MDD,TR definition] This definition isn't complete. As it is, it isn't decisive because it's possible that *all* the candidates can be disqualified. You need to specify that if all the candidates have a majority-strength defeat then none of them are disqualified. I'm not a fan of this method for reasons I may elaborate on in a later post. It has a strong random-fill incentive, and fails the Plurality and Mono-add- Plump criteria. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] ER-IRV(whole) fails FBC (was no subject)
Mike Ossipoff wrote (2 Nov 2011): Kevin-- You wrote: ER-IRV(whole) doesn't satisfy FBC. You may need to demote your favorite in order to get a preferable elimination order. [endquote] How? Say that there's a particular candidate whom you need to have win. You can give him a vote by downrating your favorite in order to get hir eliminated soon, so that your ballot will give a vote to the compromise. But you could also just give the compromise an immediate vote, by ranking hir in 1st place. Why would you need to do otherwise in order to help hir win? Here is Kevin Venzke's example from a June 2004 EM post: 6: A 3: CB 2: C=B (sincere is CB) 2: B The method is ER-IRV(whole). If the 2 C=B voters sincerely vote CB then the first-round scores are A6, C5, B2. B is eliminated and A wins. As it is the first-round scores are A6, C5, B4. B is still eliminated and A wins. To meet FBC no voters should have any incentive to vote their sincere favourite below equal-top. 6: A 3: CB 2: BC (sincere is CB) 2: B But if those 2 voters (sincere CB, was C=B) do that and strictly top-rank their compromise candidate B, then the first-round scores are A6, B4, C3. C is eliminated and B wins: B7, A6. By down-ranking their sincere favourite those 2 voters have gained a result they prefer that they couldn't have got any other way, a clear failure of the Favorite Betrayal Criterion (FBC). http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-June/013434.html Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Enhanced DMC
Forest Simmons wrote (15 Aug 2011):
Here's a possible scenario:
Suppose that approval order is alphabetical from most approval to least A, B,
C, D.
Suppose further that pairwise defeats are as follows:
CADBA together with BCD .
Then the set P = {A, B} is the set of candidates neither of which is pairwise
beaten by anybody with greater approval.
Since the approval winner A is not covered by B, it is not covered by any
member of P, so the enhanced version of DMC elects A.
But A is covered by C so it cannot be elected by any of the chain building
methods that elect only from the uncovered set.
Forest,
Is the Approval Chain-Building method the same as simply electing the
most approved uncovered candidate?
I surmise that the set of candidates not pairwise beaten by a more
approved candidate (your set P, what I've
been referring to as the Definite Majority set) and the Uncovered set
don't necessarily overlap.
If forced to choose between electing from the Uncovered set and electing
from the DM set, I tend towards
the latter.
Since Smith//Approval always elects from the DM set, and your suggested
enhanced DMC (elect the most
approved member of the DM set that isn't covered by another member)
doesn't necessarily elect from the Uncovered set;
there doesn't seem to be any obvious philosophical case that enhanced
DMC is better than Smith//Approval.
(Also I would say that an election where those two methods produce
different winners would be fantastically unlikely.)
A lot of Condorcet methods are promoted as being able to give the
winner just from the information contained in the
gross pairwise matrix. I think that the same is true of these methods
if we take a candidate X's highest gross pairwise
score as X's approval score. Can you see any problem with that?
Chris Benham
- Original Message -
From:
Date: Friday, August 12, 2011 3:12 pm
Subject: Enhanced DMC
To: election-methods at lists.electorama.com,
From: C.Benham
To: election-methods-electorama.com at electorama.com
Subject: [EM] Enhanced DMC
Forest,
The D in DMC used to stand for *Definite*.
Yeah, that's what we finally settled on.
I like (and I think I'm happy to endorse) this Condorcet method
idea, and consider it to be clearly better than regular DMC
Could this method give a different winner from the (Approval
Chain Building ?) method you mentioned in the C//A thread (on 11
June 2011)?
Yes, I'll give an example when I get more time. But for all practical
purposes they both pick the highest approval Smith candidate.
Here's a possible scenario:
Suppose that approval order is alphabetical from most approval to least
A, B, C, D.
Suppose further that pairwise defeats are as follows:
CADBA together with BCD .
Then the set P = {A, B} is the set of candidates neither of which is
pairwise
beaten by anybody with greater approval.
Since the approval winner A is not covered by B, it is not covered by any
member of P, so the enhanced version of DMC elects A.
But A is covered by C so it cannot be elected by any of the chain building
methods that elect only from the uncovered set.
Forest Simmons wrote (12 June 2011):
I think the following complete description is simpler than anything
possible for ranked pairs:
1. Next to each candidate name are the bubbles (4) (2) (1). The
voter rates a candidate on a scale from
zero to seven by darkening the bubbles of the digits that add up to
the desired rating.
2. We say that candidate Y beats candidate Z pairwise iff Y is rated
above Z on more ballots than not.
3. We say that candidate Y covers candidate X iff Y pairwise beats
every candidate that X pairwise
beats or ties.
[Note that this definition implies that if Y covers X, then Y beats X
pairwise, since X ties X pairwise.]
Motivational comment: If a method winner X is covered, then the
supporters of the candidate Y that
covers X have a strong argument that Y should have won instead.
Now that we have the basic concepts that we need, and assuming that
the ballots have been marked
and collected, here's the method of picking the winner:
4. Initialize the variable X with (the name of) the candidate that
has a positive rating on the greatest
number of ballots. Consider X to be the current champion.
5. While X is covered, of all the candidates that cover X, choose the
one that has the greatest number of
positive ratings to become the new champion X.
6. Elect the final champion X.
7. If in step 4 or 5 two candidates are tied for the number of
positive ratings, give preference (among the
tied) to the one that has the greatest number of ratings above level
one. If still tied, give preference
(among the tied) to the one with the greatest number of ratings above
the level two. Etc.
Can anybody do a simpler description of any other Clone Independent
Condorcet method?
Election-Methods mailing list - see http://electorama.com/em
[EM] A variant of DSC
Forest,
Your suggested variant of DSC doesn't address DSC's bad failures of
Mutual Dominant Third and Minimal Defense.
49: A
48: B
03: CB
The biggest solid coalition is {A}49, so both DSC and your suggestion
elect A. But MD says not A and MDT says B.
As near as I can tell, my version still has all of the advantages of DSC, including later-no-harm, clone
independence, monotonicity, etc.
Your version fails Later-no-Harm:
49: A
27: BA
24: CB
It (like DSC) elects A, but if the 49 A voters change to AB your
version eliminates C and then elects B.
DSC (like DAC, DHSC and SC-DC) meets Participation.
31: ACB
33: BAC
36: C
Your version (like DSC) elects C, but if we add 6 CAB ballots the
winner changes to A (a failure of both Participation and Mono-add-Top).
31: ACB
33: BAC
36: C
06: CAB
And then if 2 of those CAB ballots changes to ABC the winner changes
back to C, failing Mono-raise.
31: ACB
33: BAC
36: C
04: CAB
02: ABC
the only advantage of DSC over DAC is that DAC does not satisfy later-no-harm.
DSC meets Independence from Irrelevant Ballots, but DAC badly fails it,
as shown from this old example from
Michael Harman (aka Auros):
03: D
14: A
34: AB
36: CB
13: C
B wins, but if the 3D ballots are removed then C wins.
(Also B is an absurd-looking unjustified winner.)
I regard DSC as FPP elegantly fixed up to meet Clone-Winner and
Majority for Solid Coalitions, but it's shortcomings
help to show that its set of of criterion compliances isn't sufficient
(and that Participation is 'expensive').
I still think IRV (Alternative Vote, no above-bottom equal-ranking,
voters can strictly rank from the top as many or few
candidates as they like) is the best of the single-winner methods that
meets Later-no-Harm.
Chris Benham
Forest Simmons wrote (Sun 7 Aug 2011):
That Q in the previous subject heading was a typo.
Here's an example that illustrates the difference in Woodall's DSC and my
modified version:
25 A1A2
35 A2A1
20 BA1
20 CA1
Woodall's DSC assigns 60 points to {A1, A2} and then the only other positive point coalitions that have
non-empty intersections with this set are {A2}, {A1}, {A1, B}, and {A1, C}, with respective points of 35,
25, 20 and 20. The 35 point set {A2} decides the result: A2 wins.
In my version, the 60 point coalition is the highest point proper coalition {A1, A2}, so candidates B and C
are struck from the ballots and we are left with
25 A1A2
35 A2A1
40 A1
This time A1 wins.
As near as I can tell, my version still has all of the advantages of DSC, including later-no-harm, clone
independence, monotonicity, etc.
Note that Woodall and I get the same result for
25 A1A2
35 A2A1
40 DA1
namely, that A1 wins. But if you split the D faction in half, you get the original scenario above. It seems
to me that A1 should continue to win, but classical DSC switches to A2 without any good reason. In
other words, it lacks a certain kind of consistency that our modified version has.
Jameson,
the only advantage of DSC over DAC is that DAC does not satisfy later-no-harm. In the context of
chicken this would keep the bluffer from truncating, but to no avail; the plurality winner (with 48 points)
would win, since (singleton) it would form the highest point solid coalition all by itself.
Under DAC the bluffer would truncate but would still form an assenting coalition with the guy who did not
truncate her, but not a solid coalition. An even bigger assenting coalition would be the plurality winner
together with the bluffer. Of these two, only the bluffer would be in the second largest coalition, so the
bluffer would win under DAC.
- Original Message -
From:
Date: Saturday, August 6, 2011 3:13 pm
Subject: AQ variant of DSC
To: election-methods at lists.electorama.com
http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com,
/ One way of looking at Woodall's DSC method is that it is
// designed to elect from the clone set that
// extends up to the top rank on the greatest number of ballots,
// i.e. kind of the plurality winner among
// clone sets.
//
// There are two ways in which this description is not precise, but
// maybe we would get a better method if
// we follwed this description more closely.
//
// (1) The solid coalitions look like clone sets on the ballots
// that reach up to the top, but they don't have to
// look like clone sets on the other ballots.
//
// (2) This description doesn't tell how DSC narrows down after
// finding the plurality winner solid coalition.
// In fact the entire set of candidates is automatically the solid
// coalition that extends to the top rank on
// 100% of the ballots, so for starter we need to narrow down to a
// proper sub-coalition.
//
// With regard to (1), imagine a one dimensional issue space with
// the candidates distributed as follows:
//
//
[EM] Enhanced DMC
Forest, The D in DMC used to stand for *Definite*. I like (and I think I'm happy to endorse) this Condorcet method idea, and consider it to be clearly better than regular DMC Could this method give a different winner from the (Approval Chain Building ?) method you mentioned in the C//A thread (on 11 June 2011)? Initialize a variable X to be the candidate with the most approval. While X is covered, let the new value of X be the highest approval candidate that covers the old X. Elect the final value of X. For all practical purposes this is just a seamless version of C//A, i.e. it avoids the apparent abandonment of Condorcet in favor of Approval after testing for a CW. Assuming cardinal ballots, candidate A covers candidate B, iff whenever B is rated above C on more ballots than not, the same is true for A, and (additionally) A beats (in this same pairwise sense) some candidate that B does not. Your newer suggestion (enhanced DMC) seems to have an easier-to-explain and justify motivation. Chris Benham Forest Simmons wrote (12 July 2011): One of the main approaches to Democratic Majority Choice was through the idea that if X beats Y and also has greater approval than Y, then Y should not win. If we disqualify all that are beaten pairwise by someone with greater approval, then the remaining set P is totally ordered by approval in one direction, and by pairwise defeats in the other direction. DMC solves this quandry by giving pairwise defeat precedence over approval score; the member of P that beats all of the others pairwise is the DMC winner. The trouble with this solution is that the DMC winner is always the member off P with the least approval score. Is there some reasonable way of choosing from P that could potentially elect any of its members? My idea is based on the following observation: There is always at least one member of P, namely the DMC winner, i.e. the lowest approval member of P, that is not covered by any member of P. So why not elect the highest approval member of P that is not covered by any member of P? By this rule, if the approval winner is uncovered it will win. If there are five members of P and the upper two are covered by members of the lower three, but the third one is covered only by candidates outside of P (if any), then this middle member of P is elected. What if this middle member X is covered by some candidate Y outside of P? How would X respond to the complaint of Y, when Y says, I beat you pairwise, as well as everybody that you beat pairwise, so how come you win instead of me? Candidate X can answer, That's all well and good, but I had greater approval than you, and one of my buddies Z from P beat you both pairwise and in approval. If Z beat me in approval, then I beat Z pairwise, and somebody in P covers Z. If you were elected, both Z and the member of P that covers Z would have a much greater case against you than you have against me. Election-Methods mailing list - see http://electorama.com/em for list info
[EM] A Comparison of the Two Known Monotone, Clone Free Methods for Electing Uncovered Alternatives
Forest Simmons wrote (31 Dec 2010): Chris, You are right that since Chain Climbing does not satisfy IPDA, neither does the method that takes the parwise victor of it and the Covering Chain winner. I was more thinking out loud than pushing that idea. Do you think that Approval Sorted Pairwise and the Covering Chain process are simple enough for use in a public proposal? Happy New Year! Forest Forest, Regarding your first paragraph above, the method you suggested before was to elect whichever of the Chain Climbing and Covering Chain winners was higher on the list L (made by some method that meets mono-raise), not whichever of the two pairwise beats the other; but I assume the same applies. In answer to your question, I'm afraid probably not. For a sceptical electorate accustomed to essentially *no* voting algorithm, I doubt that Approval-Sorted Margins by itself is simple enough. And yet it is nice to be able to do without the concept of the Smith set, necessary for Smith//Approval. Regarding Chain Covering, does the extra complexity of using ASM instead of Approval to make the list L really gain much? Happy New Year to you too. :) Chris Benham The second method, the covering chain method, starts at the top of the list and works downward. A variable X is initialized as the alternative highest on the list. While some alternative covers X, the highest such alternative on the list becomes the new value of X. The final value of X is the covering chain winner. Election-Methods mailing list - see http://electorama.com/em for list info
[EM] A Comparison of the Two Known Monotone, Clone Free Methods for Electing Uncovered Alternatives
Forest Simmons wrote (16 Dec 2010): Chris, Thanks for reminding me of Approval-Sorted Margins. The covering chain method applied to the list obtained by approval sorted margins certainly has a maximal set of nice properties, in that any additional nice property would entail the loss of some highly desireable property. Do you think it is better, in this context, to base approval on ranked-above-last, or by use of an explicit approval cutoff marker? Forest, I like both versions. I think the version that uses an approval cut-off (aka threshold) marker is a bit more philosophically justified. (It seems arbitrary to assume that ranked-above-bottom signifies approval, or putting it the other way, unpleasantly restrictive to not allow voters to rank among candidates they don't approve.) On the other hand the other version is simpler, and probably normally elects higher SU winners and resists burial strategy a bit better. From your December 2 post: I do suggest the following: In any context where being as faithful as possible to the original list order is considered important, perhaps because the only reason for not automatically electing the top of the list is a desire to satisfy Condorcet efficiency, then in this case I suggest computing both the chain climbing winner and the covering chain winner for the list L, and then going with which ever of the two comes out higher on L. Have you since retreated from this idea? Would using this on the list obtained from Approval-Sorted Margins lose (compared to just using the covering chain method) compliance with Independence from Pareto-Dominated Alternatives? Could the two ever give different winners? Sorry to be a bit tardy in replying, Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] A Comparison of the Two Known Monotone, Clone Free Methods for Electing Uncovered Alternatives
After pasting below, I fixed up the MCA instead of DMC mistake, in accordance with what Forest has since wrote that he meant. The covering chain method is not guaranteed to pick from Banks, but it has a nice property that chain climbing lacks, namely it satisfies independence from pareto dominated alternatives (IPDA). So we cannot say that one of these methods is uniformly better than the other. I vastly prefer the covering chain method. When the Smith set has 3 members (more would be very uncommon), it is the same as Smith//Approval (which I like and endorse). 25: AB 06: AC 32: BC 27: CA 10: C CA 69-31, AB 58-32, BC 57-43 Approval (ranking) scores: C 75, A 58, B 57. As I pointed out in an earlier thread, the climbing chain method elects A. The complaint of the voters who prefer C would in my mind be unanswerable. Now suppose that we bubble sort the list L to get the list L' = [D, C, A, B] If the original list L were the approval order, then this list would be the Definite Majority Choice order after the bubble sort, and we would discover alternative D to be the DMC winner. The list L' is fair game for our two methods, since bubble sorting preserves monotonicity and clone independence,... That would also be true of Approval-Sorted Margins, which I prefer: http://wiki.electorama.com/wiki/Approval_Sorted_Margins First seed the list in approval order. Then while any alternative X pairwise defeats the alternative Y immediately above it in the list, find the X and Y of this type that have the least difference D in approval, and modify the list by swapping X and Y. I do suggest the following: In any context where being as faithful as possible to the original list order is considered important, perhaps because the only reason for not automatically electing the top of the list is a desire to satisfy Condorcet efficiency, then in this case I suggest computing both the chain climbing winner and the covering chain winner for the list L, and then going with which ever of the two comes out higher on L. I am happy with this, if the Approval-Sorted Margins order is used for L' . More simply just using the plain approval order for L would also probably be fine. On the other hand, when the list L is just considered a convenient starting place, with no other special importance, then I suggest bubble sorting L to get L', and then flip a coin to decide which of the two methods to use for processing this sorted list. But not with this. That would still give the dominated (by C) candidate A in my example a 50% chance of winning. Chris Benham Forest Simmons wrote (2 Dec 2010): To my knowledge, so far only two monotone, clone free, uncovered methods have been discovered. Both of them are ways of processing given monotone, clone free lists, such as a complete ordinal ballot or a list of alternatives in order of approval. The first method, chain climbing, starts at the bottom of the list and works upward. It initializes a chain with the lowest alternative of the list and while there is any alternative that pairwise beats all the current members of the chain, the lowest such member of the list is added to the chain. The last alternative added to the chain is the chain climbing winner. The second method, the covering chain method, starts at the top of the list and works downward. A variable X is initialized as the alternative highest on the list. While some alternative covers X, the highest such alternative on the list becomes the new value of X. The final value of X is the covering chain winner. Both methods pick from the uncovered set, so they are both Condorcet efficient. Suppose, for example, that the alternatives A, B, C, D, and E are arranged along a line in alphabetical order with C at the voter median position. The sincere, rational ordinal ballot of a voter near alternative A would likely list the alternatives in alphabetical order. If this ballot were taken as the list L, and the two methods were applied to L, the first method (chain climbing) would build up the chain in the order E,D, C, so C would be the chain climbing winner. If the second method were used, the successive values of X would be A, B, and C, so C would also be the covering chain winner. The two methods approach the voter median alternative from opposite sides. If C were replaced with the cycle C1 beats C2 beats C3 beats C1, and the list order became A, B, C1, C2, C3, D, E, then the chain climbing chain would build up in the order E, D, C3, C2, with C2 winning, while the successive covering chain values of X would be A, B, C1, respectively, with C1 becoming the covering chain winner. This illustrates the general principle that when the Smith set is a cycle of three alternatives, chain climbing is more penetrating than the covering chain method. In this context the covering chain method will always stop at the first member of the top cycle that it encounters, thus
[EM] election strategy paper, alternative Smith, web site relaunch
From James Green-Armytage's paper on election strategy: I focus on the nine single-winner voting rules that I consider to be the most widely known, the most widely advocated, and the most broadly representative of single-winner rules in general: these are plurality, runoff, alternative vote, minimax, Borda, Bucklin, Coombs, range voting, and approval voting8. I would think that Schulze(Winning Votes) is more widely advocated than minimax, aka MinMax(Margins). 2. Preliminary definitions 2.1. Voting rule definitions In this paper, I analyze nine single-winner voting methods. I follow Chamberlin (1985) in including plurality, Hare (or the alternative vote), Coombs, and Borda, and to these I add two round runoff, minimax (a Condorcet method), Bucklin, approval voting, and range voting. My assumption about incomplete ranked ballots is that candidates not explicitly ranked are treated as being tied for last place, below all ranked candidates. My assumption about votes that give equal rankings to two or more candidates is that they are cast as the average of all possible orders allowed by the rankings that they do specify. http://www.econ.ucsb.edu/~armytage/svn2010.pdf I find these assumptions about ballots that are truncated or have equal-ranking to be very unsatisfactory. It means that the version of Bucklin you are considering is a strange one (advocated by no-one) that fails the Favorite Betrayal criterion. It would also fail Later-no-Help, which is met by normal Bucklin. It means that the only version of minimax you can consider is Margins, and you can't consider Schulze(Winning Votes). Unlike minimax(margins), Schulze(WV) meets the Plurality, Smith and Minimal Defense criteria. Alternative vote, or Hare: Each voter ranks the candidates in order of preference. The candidate with the fewest first choice votes (ballots ranking them above all other candidates in the race) is eliminated. The process repeats until one candidate remains. Coombs12: This method is the same as Hare, except that instead of eliminating the candidate with the fewest first-choice votes in each round, it eliminates the candidate with the most last-choice votes in each round. Surely this is a museum curiosity that no-one currently advocates? This fails Majority Favourite, but I think there is another version with a 'majority stopping rule'. http://wiki.electorama.com/wiki/Coombs%27_method http://en.wikipedia.org/wiki/Coombs'_method http://www.fact-index.com/c/co/coombs__method.html 6.2.2. Compromising strategy results Tables 9-11 and figures 10-12 show the voting rules‘ vulnerability to the compromising strategy, given various specifications. As shown in proposition 4, Coombs is immune to the compromising strategy Of course the version with the majority stopping rule isn't immune to that strategy (Compromise). Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] election strategy paper, alternative Smith, web site relaunch
James Green-Armytage wrote (20 Nov 2010):
snip
In addition to the nine methods listed above, I tried some of my
analyses with six other Condorcet methods: beatpath, ranked pairs,
Smith/Hare, alternative Smith, and two versions of cardinal pairwise.
Beatpath and ranked pairs generally seem to perform like minimax, and
cardinal pairwise usually but not always performs somewhat better than
these, but the really striking news in my opinion is how well the
Hare-Condorcet hybrids perform.
That is, given a preliminary analysis, they seem to be as resistant to
strategic voting as Hare (and possibly slightly more resistant), and
they are distinctly less vulnerable to strategic nomination (because
they are Smith efficient, and therefore only vulnerable to strategic
nomination when there is a majority rule cycle). So, for single-winner
public elections, alternative Smith and Smith/Hare seem to have a lot
to recommend them, i.e. the combination of Smith efficiency with
strong resistance to both types of election strategy.
I should define these methods here, for clarity. Smith/Hare eliminates
all candidates not in the Smith set (minimal dominant set, i.e. the
smallest set of candidates such that all members in the set pairwise
beat all members outside the set), and then holds an IRV tally among
remaining candidates. This method has been floating around this list
for a while, yes? Does anyone know of an academic publication that
mentions it? I seem to remember reading something that said that it
had been named after a person at some point, but I no longer know
where I read that.
Alternative Smith is a closely related method, which Nic Tideman made
up when he was writing Collective Decisions and Voting. It (1)
eliminates all candidates not in the Smith set, then (2) eliminates
the candidate with the fewest top-choice votes. Steps 1 and 2
alternate until only one candidate remains. (See page 232 of the
book.) I focus on this rule rather than Smith/Hare in the paper,
because I find it marginally more elegant, but the difference between
the two is very minor.
James,
We discussed these Hare-Condorcet hybrids on EM in the months of
October and
November 2005. Then I quoted Douglas Woodall's demonstration that both
the versions
you discuss fail Mono-add-Plump and Mono-append.
abcd 10
bcda 6
c 2
dcab 5
All the candidates are in the top tier, and the AV winner is a. But
if you add two extra ballots that plump for a, or append a to the two
ballots, then the CNTT becomes {a,b,c}, and if you delete d from all
the ballots before applying AV then c wins.
Translating to a more familiar EM format:
10: ABCD
06: BCDA
02: C
05: DCAB
All candidates are in the Smith set (Woodall's Condorcet-Net Top
Tier), and the Hare
(aka Alternative Vote, aka IRV) winner is A.
But if you add 2 ballots that bullet-vote (plump) for A, or change the
two C ballots to CA,
the Smith set becomes {A B C}, and if you delete D from all the ballots
from all the ballots before
applying Hare (i.e. properly eliminate D and not just disqualify D
from winning) then C wins.
Smith,Hare (which Woodall called CNTT,AV) meets those criteria and has
a simpler algorithm:
Begiinining with their most preferred candidate, voters strictly rank
however many candidates they wish.
Before each (and any) elimination, check for a candidate X that
pairwise beats all (so far uneliminated)
candidates. Until such an X appears, one-at-a-time eliminate the
candidate that is voted favourite
(among uneliminated candidates) on the fewest ballots. As soon as an X
appears, elect X.
So why put up with failures of mono-add-plump and mono-append? What
advantage (if any) do you think
the two versions you discuss have over Smith,Hare to compensate for that?
Chris Benham
Election-Methods mailing list - see http://electorama.com/em for list info
[EM] My Favorite Deterministic Condorcet Efficient Method: TACC
Forest wrote (13 Nov. 2010): snip I'm not a die hard Condorcet supporter. In fact my truly favorite methods are neither Condorcet efficient nor deterministic; hence the title of this thread is intended to connote a deliberate restriction of attention to lesser evil methods that might be acceptable to Condorcet enthusiasts. So far most Condorcet supporters seem to think that we have to have cycles, and therefore.the important thing is how to deal with them rather than how to prevent them. Nor am I a die-hard Condoret supporter, but I'm intolerant of methods that aren't deterministic. I have sympathy for the philosophical view that the winner must come from the Smith or Schwartz set., but not for the view that there aren't other desirable representative criteria regarding which member of that set we elect. 25: AB 06: AC 32: BC 27: CA 10: C TACC's election of A here is unacceptably silly because C is so dominant over A. I consider not electing C here somewhat embarrassing, but I have defended a couple of methods that elect B: IRV and Smith,IRV. But IRV is completely invulnerable to Burial strategy, and Smith,IRV is a Condorcet method that keeps some of that IRV quality: Mutual Dominant Third candidates are invulnerable to Burial. In the example above we can see that C could be a sincere DMT candidate that has been successfully buried by the 25 AB voters (sincere may be A or AC) in TACC. I think that if for the sake of defensive strategy and/or higher Social Utility we encourage voters to truncate, then it is better to dump the Condorcet criterion in favour of the Favourite Betrayal criterion (while making do with other representative criteria compliances.) So I certainly prefer IBIFA (my favourite FBC method) to TACC and Winning Votes and Margins. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] My Favorite Deterministic Condorcet Efficient Method: TACC
Regarding my example 31: AB 32: BC 37: CA Forest: I've come around to the belief that most Condorcet cycles in ordinary elections are artificial, so chances are that this cycle was created from the burial of B by the C faction. Giving C the win only rewards this manipulation. Chris: I can't see any remotely rational justification for assuming that this is the case rather than, say, the cycle was created by the A voters burying C. Forest: Well, usually the largest faction is the one with the best chance of getting away with it, and would get away with it under Beatpath, Ranked pairs, etc. unless the Condorcet supporters took defensive action. With TACC no defensive action was necessary. Now let's consider the possibility that you suggest, namely that the true preferences were 31 AC 32 BC 37 CA If there is enough information for the A faction to think it is safe to bury C, then there is enough information for the C faction to take the precaution of truncating A defensively. The we have 31 AB 32 BC 37 C That's why I think this scenario is less likely than the one I suggested. Chris: Yes, but only somewhat. This all assumes that the A faction's pairwise preferences are all about equally strong, that accurate pre-poll data is available to all the factions, and that the C faction voters are strategically minded. It is nothing like a sufficient counter to my original central point, that on the ballots as voted C is the strongest candidate and solidly dominates the TACC winner A. Interpreting the ballots as 3-slot ratings, C has the highest Approval score and the the highest Top-Ratings score. And C pairwise beats A Let me modify my example to further strengthen C and weaken A: 25: AB (sincere is A) 06: AC 32: BC 27: CA 10: C Approvals: B57, A58, C75. (Top-Ratings as before). AB 58-32, BC 57-43, CA 69-31. TACC still elects A. C is the sincere CW. C's voted pairwise defeat is the weakest (as measured by both Winning Votes and Margins) while A's is (by those same measures) is the strongest. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] TACC (KM, CB)
Chris wrote: BTW, I also like the version of Smith//Approval that allows voters to indicate an approval threshold so they can rank among unapproved candidates. Kevin responded (10 Nov 2010): I still don't. I don't understand why you should be allowed to vote nonsense rankings and not have to stand by them when you succeed in creating an artificial cycle. It means burial strategy only backfires when the pawn candidate becomes the CW, which basically means burial is safe as long as only one faction is doing it. I think it's arguable that encouraging truncation goes against the spirit of the Condorcet criterion, and I hate random-fill incentives. I just think that the winner of Smith//Approval (threshold) can never be too bad (SU-wise) or silly. Arguing against results arising from nonsense rankings to me is almost an implicit criticism of the Condorcet criterion itself. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Re : TACC (KM, CB)
Kevin wrote: I think it's arguable that encouraging truncation goes against the spirit of the Condorcet criterion, and I hate random-fill incentives. So, you don't like the implicit version. That's fine. I like (and endorse) them both. I prefer them both to Winning Votes. I have no inherent problem with Condorcet methods on threshold ballots. There has just got to be a better-designed option than Smith//Approval. I gave a lot of thought to this a few years ago. I reject anything that fails the Definite Majority criterion. I just think the alternatives to Smith//Approval (on threshold ballots) are in general too complicated and too hard to justify. My alternative favourite in this group (which used to be my favourite) is Approval-Sorted Margins: http://wiki.electorama.com/wiki/Approval_Sorted_Margins First seed the list in approval order. Then while any alternative X pairwise defeats the alternative Y immediately above it in the list, find the X and Y of this type that have the least difference D in approval, and modify the list by swapping X and Y. Kevin wrote: I do wonder how fishing through disapproving rankings will aid SU. It may not, but it won't stifle voters from expressing all their sincere rankings so as not to conceal any sincere CW (regarded by many as in principle the best winner no-matter-what). Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] My Favorite Deterministic Condorcet Efficient Method: TACC
Chris wrote ... / 31: AB // 32: BC // 37: CA // // Approvals: B63, A68, C69. ABCA. // // TACC elects A, but C is positionally the dominant candidate and // pairwise beats A. // // For a Condorcet method with pretension to mathematical elegance, // I don't/ /see how that/ /can be justified. // // Chris Benham // // PS: Could someone please refresh our memories: What is the // Banks Set? / Forest Replies: As you know C is the DMC winner, and would be a slightly better winner, given that the ballots are sincere. But DMC is not as burial resistant and truncation resistant as TACC. It is interesting that DMC and TACC have opposite rules for which of the top two approval members of the top cycle (of three) wins. DMC awards the win to the one (of these two) that beats the other. TACC awards the win to the one that is beaten by the other. Chris: I have long since abandoned the Definite Majority Choice (DMC) method in favour of Smith//Approval (as my preferred Condorcet method), which also elects C here. I still like the Definite Majority criterion, which says that no candidate that is pairwise beaten by a more approved candidate is allowed to win. I think that (in isolation) meeting the Condorcet criterion is desirable, but not so holy that on discovering there is no voted CW the method should proceed on the assumption that there is really a sincere CW that has been victimised by strategists the method should try to frustrate or punish. Condorcet methods are vulnerable to Burial, period. Futile attempts to address this should not be at the expense of producing winners that can have no philosophical justification on the assumption that all the votes are sincere (or are all equally likely to be sincere). The TACC winner A simply has no shred of justification versus the Smith//Approval winner C. Forest: I've come around to the belief that most Condorcet cycles in ordinary elections are artificial, so chances are that this cycle was created from the burial of B by the C faction. Giving C the win only rewards this manipulation. Chris: I can't see any remotely rational justification for assuming that this is the case rather than, say, the cycle was created by the A voters burying C. BTW, I also like the version of Smith//Approval that allows voters to indicate an approval threshold so they can rank among approved candidates. I think on balance I prefer IBIFA to any of the Condorcet methods. Thanks for explaining the Banks Set. I'll look more into it. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] My Favorite Deterministic Condorcet Efficient Method: TACC
31: AB 32: BC 37: CA Approvals: B63, A68, C69. ABCA. TACC elects A, but C is positionally the dominant candidate and pairwise beats A. For a Condorcet method with pretension to mathematical elegance, I don't see how that can be justified. Chris Benham PS: Could someone please refresh our memories: What is the Banks Set? From Jobst Heitzig (March 2005): ROACC (Random Order Acrobatic Chain Climbing): -- 1. Sort the candidates into a random order. 2. Starting with an empty chain of candidates, consider each candidate in the above order. When the candidate defeats all candidates already in the chain, add her at the top of the chain. The last added candidate wins. The good thing about ROACC is that it is both - monotonic and - the winner is in the Banks Set, in particular, the winner is uncovered and thus the method is Smith-, Pareto-, and Condorcet-efficient. Until yesterday ROACC was the only way I knew of to choose an uncovered candidate in a monotonic way. But Forest's idea of needles tells us that it can be done also in another way. The only difference is that in step 1 we use approval scores instead of a random process: TACC (Total Approval Chain Climbing): 1. Sort the candidates by increasing total approval. 2. Exactly as above. The main differences in properties are: TACC is deterministic where ROACC was randomized, and TACC respects approval information where ROACC only uses the defeat information. And, most important: TACC is clone-proof where ROACC was not! That was something Forest and I tried to fix without violating monotonicity but failed. More precisely, ROACC was only weakly clone-proof in the sense that cloning cannot change the set of possible winners but can change the actual probabilites of winning. With TACC, this makes no difference since it is deterministic and so the set of possible winners consists of only one candidate anyway. Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Guaranteed Majority criterion on Electowiki
http://wiki.electorama.com/wiki/Majority_Choice_Approval#Criteria_compliance MCA-AR satisfies the Guaranteed majority criterion /wiki/Guaranteed_majority_criterion, a criterion which can only be satisfied by a multi-round (runoff-based) method. http://wiki.electorama.com/wiki/Guaranteed_majority_criterion The *guaranteed majority criterion* requires that the winning candidate always get an absolute majority /wiki/Absolute_majority of valid votes in the last round of voting or counting. It is satisfied by runoff voting /wiki/Runoff_voting, MCA-AR /wiki/MCA, and, if full rankings are required, IRV /wiki/IRV. However, if there is not a pairwise champion (aka CW), there could always be some candidate who would have gotten a majority over the winner in a one-on-one race. Since, unlike most criteria, this criterion can depend on both counting process and result, there could be two systems with identical results, with only one of them passing the guaranteed majority criterion. This is an example of what Mike Ossipoff used to rightfully excoriate as a rules criterion. To me if two voting systems/methods always give the same results with the same impute, then they are really just one method (which perhaps has alternative algorithms) and so they both meet and fail all the same (non-silly) criteria. A voting method criterion should relate to some desirable standard. Is IRV that doesn't allow truncation somehow better that IRV that does? Why can't normal IRV (that allows truncation) just have a rule that says that exhausted ballots in the last round of counting are no longer valid? Or better yet, since IRV meets Woodall's Symmetric Completion criterion, why can't it include a rule that all ballots are symmetrically completed so then the winner in the final round of counting will certainly have an absolute majority of valid votes? Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] MCA on electowiki (re Later-no-help and Favorite Betrayal criteria)
http://wiki.electorama.com/wiki/Majority_Choice_Approval#Criteria_compliance The Later-no-help criterion /wiki/Later-no-help_criterion and the Favorite Betrayal criterion /wiki/Favorite_Betrayal_criterion are satisfied by MCA-P They are also met by MCA-A, MCA-M and MCA-S. I consider it desirable that methods should have Later-no-Harm and Later-no-Help in at least approximate probabilistic balance. These methods all (badly) fail Later-no-Harm, so meeting LNHelp contributes to the strong truncation incentive. They're also satisfied by MCA-AR if MCA-P is used to pick the two finalists That method does not meet the Favourite Betrayal criterion. 25: A 24: AC 02: BA 22: B 25: CB 02: C=B (sincere is CB) No candidates' TR (or P) score reaches the majority threshold of 51 and all their Approval scores exceed it, so a resolution method is needed. Of the candidates that reached a majority score, I gather the method selects the two with the highest TR scores for a runoff. TR scores: A49,B26,C27. The method selects A and C for the runoff, which A wins 51-27. If the 2 C=B voters vote sincerely CB the result is the same. But if they change to BC the TR scores change to A49, B26, C25 and the method then selects A and B for the runoff which B wins 51-49, a result those two voters prefer. 25: A 24: AC 02: BA 22: B 25: CB 02: BC (was C=B, sincere is CB) Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] MCA on electowiki
Jameson Quinn wrote (18 Oct 2010): I edited Electowiki to essentially replace the Bucklin-ER article with a new, expanded MCA article. In this article, I define 6 MCA variants. I find that as a class, they do surprisingly well on criteria compliance. Please check my work: http://wiki.electorama.com/wiki/Majority_Choice_Approval#Criteria_compliance Now quoting from the referred-to Electowiki page: Majority Choice Approval (MCA) is a class of rated voting systems which attempt to find majority support for some candidate. It is closely related to Bucklin Voting, which refers to ranked systems using similar rules. In fact, some people consider MCA a subclass of Bucklin, calling it ER-Bucklin http://wiki.electorama.com/wiki/ER-Bucklin (for Equal-Ratings-[allowed] Bucklin). Who are these people? As I understand it, ER-Bucklin is a method that uses ranked ballots that allow equal-ranking whereas MCA is a method that uses 3-slot ratings ballots (but could be extended to more than 3 rating slots). Voters rate candidates into a fixed number of rating classes. There are commonly 3, 4, 5, or even 100 possible rating levels. The following discussion assumes 3 ratings, called preferred, approved, and unapproved. If one and only one candidate is preferred by an absolute majority http://wiki.electorama.com/wiki/Absolute_majority of voters, that candidate wins. If not, approvals are added to preferences, and again if there is only one candidate with a majority they win. If the election is still unresolved, one of two things must be true. Either multiple candidates attain a majority at the same rating level, or there are no candidates with an absolute majority at any level. In either case, there are different ways to resolve between the possible winners - that is, in the former case, between those candidates with a majority, or in the latter case, between all candidates. The possible resolution methods include: * MCA-A: Most approved candidate (most votes above lowest possible rating) Until I read this, the only versions of MCA that I was aware of were this one and another that differs only by using a hybrid FPP-Approval ballot that restricts voters to indicating one candidate as most preferred plus they can approve as many candidates as they like. (The latter version was an early suggestion that seem to quickly fall out of favour). MCA-P: Most preferred candidate (most votes at highest possible rating) I've heard of this, as a 3-slot method with a different name. The strategic incentive for voters to not use any rating-slot other than the top one is even higher than it is with MCA-A. A note on terminology Majority Choice Approval was first used to refer to a specific form, which would be 3-level MCA-AR in the nomenclature above (specifically, 3-MCA-AR-M). Later, a voting system naming poll http://betterpolls.com/v/1189 chose this term as a more-accessible replacement for ER-Bucklin in general. As I previously implied, this is news to me. How exactly does this mysterious 3-MCA-AR-M method work? Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] MCA fails Irrelevant Ballots (and therefore Jameson's mono-add-antiplump)
Jameson Quinn wrote (19 Oct 2010): Indeed, all forms of MCA satisfy mono-add-plump (unless a non-compliant method is used to choose the finalists for the runoff in MCA-IR or MCA-VR). Yes. In fact, they satisfy an slightly stronger criterion, let's call it mono-add-antiplump. You cannot cause Y to win by adding a ballot which doesn't approve Y (that is, votes them at the lowest rating possible). Since they all fail Independence from Irrelevant Ballots, this claim can't be correct. 51: AB 40: B 09: C They all elect A, but if we add 3 ballots that plump for X the winner changes from A to B. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] MCA on electowiki
Jameson Quinn wrote (18 Oct 2010): I edited Electowiki to essentially replace the Bucklin-ER article with a new, expanded MCA article. In this article, I define 6 MCA variants. I find that as a class, they do surprisingly well on criteria compliance. Please check my work: http://wiki.electorama.com/wiki/Majority_Choice_Approval#Criteria_compliance Criteria compliances All MCA variants satisfy the Plurality criterion http://wiki.electorama.com/wiki/Plurality_criterion, the Majority criterion for solid coalitions http://wiki.electorama.com/wiki/Majority_criterion_for_solid_coalitions, Monotonicity http://wiki.electorama.com/wiki/Monotonicity_criterion (for MCA-AR, assuming first- and second- round votes are consistent), and Minimal Defense http://wiki.electorama.com/wiki/Minimal_Defense_criterion (which implies satisfaction of the Strong Defensive Strategy criterion http://wiki.electorama.com/wiki/Strong_Defensive_Strategy_criterion). It is well known that in general run-off methods fail mono-raise (aka Monotonicity), and these methods are no exception. 22: A 23: AC 24: B 27: CB 02: DC 06: E (104 ballots) TR scores: A45, B24, C27, D2, E6. Approval scores: A45, B51, C52, D2, E6. I am assuming that 3-slot ballots are used, and since no candidate has either a Top Ratings or Approval score that reaches the majority threshold the runoff will be between the TR winner A and the Approval winner C. A wins that runoff 45-29, but if the 2 DC ballots change to DA the Approval winner changes to B and now A loses that runoff 47-49. 22: A 23: AC 24: B 27: CB 02: DA (was DC) 06: E (104 ballots) TR scores: A45, B24, C27, D2, E6. Approval scores: A47, B51, C50, D2, E6. Also I would quibble that methods that use ballots that don't allow voters to express a full ranking of the candidates really properly meet Majority for Soild Coalitions, but instead just meet a restricted form of it (which is nonetheless very valuable). And I'm surprised that a MCA advocate doesn't mention the Favourite Betrayal criterion. Of course the suggested runoff variants of MCA also fail that. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Schulze (Approval-Domination prioritised Margins)
On 18 Jan 2009 I proposed a Condorcet method, Approval-Domination Prioritised Margins: I have an idea for a new defeat-strength measure for the Schulze algorithm (and similar such as Ranked Pairs and River), which I'll call: Approval-Domination prioritised Margins: *Voters rank from the top however many candidates they wish. Interpreting ranking (in any position, or alternatively above at least one other candidate) as approval, candidate A is considered as approval dominating candidate B if A's approval-opposition to B (i.e. A's approval score on ballots that don't approve B) is greater than B's total approval score. All pairwise defeats/victories where the victor approval dominates the loser are considered as stronger than all the others. With that sole modification, we use Margins as the measure of defeat strength.* This aims to meet SMD (and so Plurality and Minimal Defense, criteria failed by regular Margins) and my recently suggested Smith- Comprehensive 3-slot Ratings Winner criterion (failed by Winning Votes). http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023595.html Here is an example where the result differs from regular Margins, Winning Votes and Schwartz//Approval. 44: A 46: BC 07: CA 03: C AB 51-46 = 5 * BC 46-10 = 36 CA 56-44 = 12 Plain Margins would consider B's defeat to be the weakest and elect B, but that is the only one of the three pairwise results where the victor approval-dominates the loser. A's approval opposition to B is 51, higher than B's total approval score of 46. So instead my suggested alternative considers A's defeat (with the next smallest margin) to be the weakest and elects A. Looking at it from the point of view of the Ranked Pairs algorithm (MinMax, Schulze, Ranked Pairs, River are all equivalent with three candidates), the AB result is considered strongest and so locked, followed by the BC result (with the greatest margin) to give the final order ABC. Winning Votes considers C's defeat to be weakest and so elects C. Schwartz//Approval also elects C. Margins election of B is a failure of Minimal Defense. Maybe the B supporters are Burying against A and A is the sincere Condorcet winner. I've discovered that this actually fails my suggested Smith- Comprehensive 3-slot Ratings Winner criterion. 20: AB 20: A=B 15: BC 45: C CA 60-40 = 20 * AB 20-15 = 5 BC 55-45 = 10 In this example borrowed from Kevin Venzke, C is in the Smith set, has the highest Top-Ratings score, the highest Approval score and the lowest Maximum Approval Opposition score and yet B wins. So I withdraw my endorsement of this method. I no longer see any real justification for preferring it to the much simpler Smith//Approval, which I continue to endorse. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Holy grail: a condorcet compliant cardinal method (MCA/Bucklin variant)
Jameson Quinn wrote (5 Sep 2010): Here's my latest Bucklin variant, which, pending the results of the naming poll http://betterpolls.com/do/1189, I'm calling RMCA (because of the catchy music). (Of course, if it's OK to appropriate the name MCA, the editorial headline writes itself...) Start with two-rank Bucklin ballots: Preferred, Approved, or Unapproved. The highest majority preferred, if any, wins it. If not, find the highest number of approved-or-preferred (approval winner, AW), and the highest range score (range winner, RW), counting 2/1/0 for P/A/U. If those are the same candidate, that candidate wins; otherwise, those two go into a runoff. (If either of these measures gives an exact tie, then the two tied candidates go to runoff.) The first (to me, surprising) result is that any Condorcet winner which is determinable from the ballots must get into the runoff. Proof: Say that the AW is not the CW. Then the number of ballots n with CWAW is greater than m with AWCW. On a ballot where X beats Y, X has a range advantage of either 1 (XY) or 2 (XY). Sf n2 where CWAW is greater than (m2 where AWCW)+(n-m), then the CW is the RW. And if n2 m2, then there are more ballots which approve the CW and not the AW than the reverse, which contradicts the assumption that the AW is the AW. QED. Adapting an example from Douglas Woodall: 4: AB 6: AC 6: BA 2: BC 3: CB The Condorcet winner is B, but Jameson's suggested condorcet compliant method elects A. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] A completely idiotic Instant Runoff Voting (IRV) election
C.Benham wrote: / Score voting http://rangevoting.org/RangeVoting.html considers this // election an easy call. It would elect B if all voters gave score X to // their first choice, Y to their second, // and Z to their third, for /any/ X?Y?Z, not all equal. // // Really? // // 18: A9, B1, C0 // 24: B9, C1, A0 // 15: C9, A8, B0 // // A wins. Doesn't this example qualify? / Kristofer M. wrote: I don't think so. For the first two ballot groups, you have X = 9, Y = 1, Z = 0, but then you change them to X = 9, Y = 8, Z = 0 for the last. So what does the phrase not all equal refer to then? Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Thoughts on Burial
Jameson Quinn wrote (23 July 2010): In Australia (IRV), the clear strategy is to vote in a plurality-like way. That is, between two near clones A and B who share a majority, supporters of A, the one with least support from C voters, should betray and vote BAC. In the absence of such a strategy from A voters, C voters should dishonestly vote CAB, under the assumption that BCA and ABC are more common than BAC and ACB. Both of these strategies are simple enough to describe, especially if there's a pseudo-one-dimensional issue space. The favorite-betrayal one, if correctly applied, increases social utility and would probably dominate and suppress the burial strategy (since it's an effective defense). But as we can see with plurality, it also decreases incentives for conciliation from candidate B towards the A voters, allowing party B to become more corrupt over time. Jameson, What exactly do you mean by the phrase share a majority? I assume that in your scenario there are only three candidates. Is that right? IRV is invulnerable to Burial strategy, and meets Majority for Solid Coalitions. If the A and B supporters (a majority of the voters) all vote both A and B above C then C can't win. But if they don't then it is the supporters of the member of the pair of near-clones with the least support from the other one that has the incentive to betray their favourite by using the Compromise strategy. 49: CB 21: AB 03: A 27: BA Of the A-B pair of near clones it is A who has the least support form the C voters, but it is the supporters of B with the incentive to betray by Compromising. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] SMD,TR fails the Plurality criterion.
Kevin Venzke has come up with an example that shows that my Strong Minimal Defense, Top Ratings (SMD,TR) method fails the Plurality criterion,contrary to what I've claimed. 21: AC 08: BA 23: B 11: C Approval scores: A29, B31, C32 Maximum Approval Opposition scores: A11, B32, C31 Top-Ratings scores: A21, B31, C11. By the rules of SMD,TR B is disqualified because B's MAO score (of 32, C's approval score on ballots that don't approve B) is greater than B's approval score. Then A (as the undisqualified candidate with the highest TR score) wins. But since B has more first-place votes than A has total votes, or in the language of this method B's TR score is greater than A's total approval score, the Plurality criterion says that A can't win. This seems to show that compliance with my Unmanipulable Majority criterion is a bit more expensive than I thought. I still endorse SMD,TR as a good Favourite Betrayal complying method, but with less enthusiasm. (My UM criterion says that if A is a winner and on more than half the ballots is voted above B, it is impossible to make B the winner by altering any ballots on which B is voted above A without raising on them B's ranking or rating.) I was wrong to claim that compliance with Strong Minimal Defense implies compliance with the Plurality criterion. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] The general form of Quick Runoff
Juho wrote (23 May 2010): snip / 1. Rank the candidates. Truncation is allowed. Equal ranking is not // planned for (but we could come up with something). // 2. Label the candidates A, B, C, ... Z in descending order of first // preference count. // 3. Let the current leader be A. // 4. While the current leader has a majority pairwise loss to the very // next candidate, set the current leader to the latter candidate. (In // other words step 4 must be repeated until there is no loss or no other // candidates.) // 5. Elect the current leader. / How about this example and LNH. 6: AC 5: BA 2: CB 2: C Candidate names indicate the order in first preferences. B beats A. C beats B. C wins. 6: AC 5: BA 2: CB 2: CA Two C voters have changed their vote to CA. B does not beat A. A wins. The C voters were harmed when they included their later preferences. Juho Juho, The key word you missed in the definition is majority. In both your elections there are 15 ballots, so a majority pairwise loss requires a winning score of at least 8. In both cases the FPP winner A wins, in the first because B's pairwise score against A is 7, a pairwise win but not a majority pairwise win (and so of course not a majority pairwise loss for A). Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] Proposal: Majority Enhanced Approval (MEA)
Forest Simmons wrote (12 May 2010): Here's another proposal. Let M be the matrix whose (i,j) element is the number of ballots on which candidate i is ranked ahead of candidate j. I think that this is what you mean by the normal gross pairwise matrix that you mention below. For each candidate i, let d(i) be the difference of the maximum number in column i and the minimum number in row i. In other words d(i) is the difference is the maximum number of points scored against candidate i in a pairwise contest and the minimum number of points that candidate i scored in a pairwise contest. Generally speaking, the smaller d(i), the stronger candidate i. So list the candidates in increasing order of d(i) instead of the order of decreasing approval, and apply the enhancement as before: Let D1 be the candidate i with the smallest difference d(i). Elect D1 if uncovered, else let D2 be the smallest d(i) candidate among those that cover D1, etc. This method wastes the diagonal slots of matrix M just like all of the other standard Condorcet methods. But I would be interested if you would run it by your standard test cases. Forest, Your suggested method fails both the Minimal Defense and Plurality criteria. 49: A 24: B 27: CB Forest scores A: 51-49 = 2,C: 49-27 = 22, B: 49-24 = 25. A has the lowest score and is uncovered and so wins, violating Minimal Defense (which says that A can't win because on more than half the ballots A is ranked below B and not above equal bottom). 7: AB 5: B 8: C Forest scores A: 8-7 = 1, B: 8-5 = 3, C: 12-8 = 4. A has the lowest score and is uncovered and so wins, violating the Plurality criterion (which says that A can't win because C has more top-preference votes than A has above-bottom votes). Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
[EM] WMA
Forest Simmons wrote (24 April 2010): I want to thank Markus for keeping me from going too far off track. And the link he gave below to a great message of Chris Benham was valuable for more than showing us that Bucklin violates mono-add-top: Chris also pointed out that WMA (weighted median approval) does satisfy Participation. I never fully appreciated before what a good method WMA was. Forest, In the April 2004 message you refer to I can't see that I claimed that WMA satisfies Participation. The closest to that is: It seems to me that WMA and WMA-STV meet Mono-add-top. In 2004 I was still new to analysing single-winner voting methods, so the opinions I voiced then shouldn't be taken as authoritative pronouncements. :) As I explained how WMA works in 2004: Voters rank the candidates, equal preferences ok. Each candidate is given a weight of 1 for each ballot on which that candidate is ranked alone in first place, 1/2 for each ballot on which that candidate is equal ranked first with one other candidate, 1/3 for each ballot on which that candidate is ranked equal first with two other candidates, and so on so that the total of all the weights equals the number of ballots. Then approval scores for each candidate is derived thus: each ballot approves all candidates that are ranked in first or equal first place (and does not approve all candidates that are ranked last or equal last). Subject to that, if the total weight of the approved candidates is less than half the total of number of ballots, then the candidate/s on the second preference-level are also approved, and the third, and so on; stopping as soon as the total weight of the approved candidates equals or exceeds half the total mumber of ballots. Then the candidate with the highest approval score wins. 2: ka=x 1: kab 4: akb 1: dx=b 1: ex=b 1: fx=b 1: gx=b 1: hx=b 1: ix=b Weights: a4, k3, defghi 1 each, bx 0 each. All ballots approve their top two preference-levels, giving these final scores: x8, a7, k7, b6, defghi 1 each. The winner is x. Now say we add two ballots that bullet-vote (plump) for x. 2: ka=x 1: kab 4: akb 1: dx=b 1: ex=b 1: fx=b 1: gx=b 1: hx=b 1: ix=b 2: x Weights: a4, k3, x2, defghi 1 each. Now all ballots approve all their ranked candidates, giving these scores: b11, x10, a7, k7, defghi 1 each. The new winner is b. This is a failure of Mono-add-Plump and so also a failure of Mono-add-Top and Participation. I reject all methods that fail mono-add-plump as unacceptably silly. Also Douglas Woodall has shown that WMA fails Clone-Winner, so I don't consider WMA to be a good method. Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)
Jameson Quinn wrote (28 March 2010): / What does MCV stand for? // / Ooops. I garbled your term, didn't I? It's supposed to be Majority Choice Approval, not Majority Choice Voting. Majority Choice Approval was invented and introduced a few years ago by Forest Simmons, and I think he coined the term. For a short time I endorsed it as something simple that meets Favorite Betrayal, the voted 3-slot ballot version of Majority for Solid Coalitions and Mono-raise. / Does top-two runoffs mean a second trip to the polls? // / Yes. I regard this as an advantage. If the situation is divisive enough to prevent a majority choice in two rounds of approval, then a further period of campaigning is a healthy thing. It's the only way to guarantee a majority. (I don't think that mandating full ranking counts as a true majority). / // How are the candidates scored to determine the top two? Is it based on the //candidates' scores after the second Bucklin round? // /That's the simplest answer, and I'd support it. It's also the best answer with honest voters. Actually, the best answer for discouraging strategy is to use the two first-round winners. That tends to discourage strategic bullet voting, since expanding your second-round approval can not keep your favorite candidate from a runoff. Unfortunately these top-two runoff versions break MCA's compliance with Favorite Betrayal and Mono-raise. Top-rating your favourite F could cause F to displace your compromise C in the runoff with your greater-evil E, and then F loses to E when C would have beaten E. Also, like plain Approval followed by a runoff between the two most approved candidates, it is *very* vulnerable to turkey-raising Push-over strategy. Voters who are fairly confident that their favourite can get into the final runoff have an incentive to also approve (or top-rate, depending on the version) all the candidates they are confident their favourite can pairwise beat in the runoff. The Push-over incentive is stronger than it is in normal TTR, because the strategists don't have to abandon their favourite in the first round (and so taking a much greater risk, if there are too many trying the strategy, of their favourite not getting into the final without their votes when without their strategising their favourite would have got into the second round and won it). Also some people might object that parties that run a pair of clones have an advantage over parties that run a single candidate. From your (Jameson's) earlier (26 March 2010) message, I gather you consider likely to elect the CW a big positive. For something simple then, why not 3-slot Condorcet//Approval? *Voters give each candidate a Top, Middle or Bottom rating. Default rating is Bottom. If one candidate X (based on these maybe constrained ballots) pairwise beats all others, elect X. Otherwise, interpreting Top and Middle rating as approval, elect the most approved candidate.* Chris Benham Election-Methods mailing list - see http://electorama.com/em for list info
