[EM] Modified Overall Preferences / margins, Smith set
Juho Laatu
juho.laatu at gmail.com
Mon Jul 8 04:39:34 PDT 2019
Few more words on using margins and Minmax in MOP-F2. These words are intended for readers who normally do not like margins, and who prefer methods that always elect the winner from the Smith set.
The previous mail already demonstrated that margins can be used, and still provide protection against some strategic attempts. Margins provide good results with sincere votes, so why not use margins, and leave the strategic concerns to the cutoff part of the method.
The question of electing from the Smith set is another interesting question. I'm not a proponent of Smith as a strict requirement since I think it makes sometimes sense to elect outside of Smith set (sincere votes assumed). There are however also cases where the top looped candidates are indeed clones in the sense that all voters consider them to be closely related. In such case their defeats to each others could be considered meaningless, and the winner should be one of them, even if there are candidates outside the Smith set with smaller defeats than the looped candidates have between themselves.
MOP-F2 can separate these different cases from each other by using the cutoff. This makes Smith set criterion compatibility a matter of the voters to decide. If voters vote 17: A>B>C>>D, 17: B>C>A>>D, 17: C>A>B>>D, 16: D>A>B>C, 16: D>B>C>A, 16: D>C>A>B, one of A, B and C will win. If A, B and C supporters will not use cutoffs, D wins in MOP-F2.
The point here is that use of Minmax as part of MOP-F2 makes it possible to voters to express their opinions more extensively (rankings + favourites) than in methods that use only rankings, and that this approach allows voters to indicate which candidates should be treated as clones and which ones not. Explicitly indicated clones will be treated as one fixed group of candidates (clones), without paying too much attention to their defeats to each others, in the final results.
One typical case where looped candidates would not be treated as if they were one candidate (group of clones) is 17: A>B>D>C, 17: B>C>D>A, 17: C>A>D>B, 16: A>D>B>C, 16: B>D>C>A, 16: C>D>A>B. In these votes A, B and C are clearly not clones since they are not next to each others in any of the ballots. D wins because its worst defeat is the smallest. With these votes, voters can not identify A, B and C as one uniform group on favourites (since they are not next to each others). Some voters might however use the cutoffs in some way, and change the result in some direction.
Margins and Minmax can thus be used, and the additional cutoffs may provide additional properties that plain margins and plain Minmax do not provide.
> On 08 Jul 2019, at 13:36, Juho Laatu <juho.laatu at gmail.com> wrote:
>
> The MOP approach is quite good in thwarting possible strategic burying attempts. I'll address briefly the basic case of having three candidates that will be looped strategically.
>
> I'll use one particular MOP method that we might call MOP-F2 (identify Favourites using a cutoff, stretch factor = 2). It will be symmetric (both A>B and B>A will be weakened if A and B are marked as favourites), and it will be simply based on use of margins and Minmax. The modification function will be f(m) = m + sign(m) * (1-P/N) * 2 * N, where N is the number of voters, and P is the number of voters that identified the two candidates as their (Protected) favourites (= candidates above the cutoff). Parameter m is the original margin, and the output of the function will be the modified value to be fed to Minmax. Note that with 100 voters the original margin values are in range [-100, 100], and the modified values are in range [-300, 300].
>
> I chose stretch factor to be 2, since that gives us some quite natural defensive properties. it seems that when certain number of of voters try to bury the sincere CW to make their favourite the winner (or vote sincerely that way), it is enough if a "corresponding" number of the other voters indicate the other two candidates as their favourites, to thwart the strategic attempt. In the example below 100% of the A supporters are strategic, and 100% of those voters that can easily defend against the strategy will do so. Stretch factor 1 would not give us such "full protection", and stretch factor 3 might be considered already an overkill.
>
> 49: A>B>>C --> A>C>B large number of strategists that try to bury B
> 15: B>A>>C --> B>A>C sincere CW, and a member of the AB group
> 36: C>B>A --> C>B>>A the other group that decides to protect the CW
>
> The opinions on the left are sincere. The actual votes are on the right. Note that all defensive votes (B and C supporters) can maintain the original ranking order. There is thus no need to falsify their rankings. The only required change is that they change the position of their favourite cutoff. B supporters stop protecting A, once they hear about the strategic plans of the A supporters. C supporters extend their protection to cover B (the CW), once they hear about the strategic plans of the A supporters.
>
> The worst defeats with the original (non-modified) margins are A:-2, B:2 (CW), and C:-28. If A voters apply their strategy, the worst defeat margins become A:-2, B:-70, and C:-28. If the votes are modified, and B supporters still protect A, and C supporters do not protect B, the modified worst defeats become A:-172, B:-270, and C:-228. If B supporters stop protecting A, the modified worst defeats become A:-202, B:-270, and C:-228. And if C supporters protect B, the modified worst defeats become A:-202, B:-198, and C:-228. CW (B) wins again. It seems that one can effectively defend against any such (single) strategic burying attempt. And if you can, maybe nobody ever even tries such strategic attempts. This would make use of the cutoffs even more sincere, and more unnecessary, but maybe still good sincere information for the election analysts.
>
> I like the idea that rankings of the non-starategic voters may stay the same, and also that the favourite cutoffs may stay quite natural, even when used defensively. The non-strategic voters may use the protective cutoff in a quite natural way, while it is of no use to the strategists. (If the A supporters would vote A>C>>B, they would just support the election of C that is the worst alternative to them.) There is also no need to resort to use of winning votes or something similar to provide means to defend against strategies (margins are a more natural way to measure sincere preferences).
>
>
>
>
> P.S. I note that in an earlier mail I forgot to include the sign function in one of the the equations. That equation should have been sign(initial)*(1-mod)*k*max + initial. I'm sorry if someone tried to calculate the results and got confused, and didn't dare to ask why that function seemed strange. I hope at least the function I gave above is correct. I think that one (MOP-F2) is a good one to start with, if someone wants to test the MOP methods in practice.
>
>> On 08 Jul 2019, at 10:06, Juho Laatu <juho.laatu at gmail.com> wrote:
>>
>> James Green-Armytage's Cardinal-Weighted Pairwise method (CWP) has also a variant, Approval-Weighted Pairwise (AWP), that uses only approvals (instead of full ratings) to determine the strength of the pairwise preferences (https://electowiki.org/wiki/Cardinal_pairwise). AWP comes actually quite close to the Modified Overall Preferences approach (MOP) when one uses only one cutoff to tell which candidates are to be protected.
>>
>> Those two methods are similar in that they both use ballots that have one approval like cutoff (in addition to the basic rankings). They both determine the (potentially cyclic) ranking order of the candidates based on the rankings of the ballots only (not using the cutoffs). They both use the cutoff only in determining the strength (not direction) of the pairwise preferences. They both then proceed to use some Condorcet method (pick your favourite) to find the winner, based on the newly calculated defeat strengths.
>>
>> Now to the differences. I'll use abbreviation MOP-F to refer to a MOP method that is uses only one cutoff to indicate a set of "preferred favourites" whose defeats to each other should be treated as "friendly" defeats.
>>
>> - AWP uses the cutoff to determine the defeat strengths, while MOP (and MOP-F too) uses the cutoff to modify the defeat strengths that were derived from the rankings. The basic philosophy in MOP is to allow voters to tell which defeats should be treated as "friendly" / "weak" defeats. This means that although most voters may agree on the direction of some pairwise preference, that defeat may still be considered weak, if voters want it that way.
>>
>> - In AWP every vote that approves A and does not approve B, increases the strength of that defeat (A>B). In MOP-F every vote that has A and B above the cutoff, indicates that defeats between A and B should be considered "friendly" (or only A>B if the method is asymmetric). These approaches are quite similar in the sense that having the two candidates above the cutoff leads to weak defeats, and having the candidates at different sides of the cutoff leads to strong defeats. But defeats between two candidates below the cutoff are weak in AWP but strong in MOP-F.
>>
>> In most elections all well known Condorcet methods may well be good enough as they are, without any additions. If additions are however used, one additional cutoff may not be too complex to use. (Also CWP's ratings may be easy enough in some elections.) In AWP one should place the cutoff so that one's strongest or most important preferences between candidates cross the cutoff point. In MOP-F one should put those candidates that one sees as "all good choices" above the cutoff. The MOP-F approach is good in the sense that the placement of the cutoff is very sincere. In AWP one may have to consider the strategic placement of the cutoff more. In order to make the vote as efficient as possible in MOP-F, one should however consider positioning the cutoff so that at least two potential winners (= candidates that may end up in the top loop) will be above the cutoff. Otherwise one might defend only candidates that have no chances to win. If one doesn't have two such candidates that one want
> s
>> to defend (sincerely, or potentially in some cases to defend against strategic voting), then setting the cutoff higher (and probably having no influence on the outcome), or not using the cutoff at all, makes sense. This adds a little bit of strategic thinking also to MOP-F. A good simple advice to the MOP-F voters could be to "mark those candidates that you want to defend".
>>
>> In most typical (large public) elections cutoffs are quite irrelevant in both methods, since they will influence the outcome only when there is a top loop. Voters may thus ignore them if they so wish. If voters do not use the cutoff feature, MOP-F uses the original strengths that were derived from the rankings. In AWP placing the cutoff somewhere in the vote is somewhat more important since otherwise all defeats would have the same strength. Getting additional information on the voter preferences may be useful and interesting for statistical purposes, even if they don't have any influence on the final outcome in most elections.
>>
>> I note that CWP may be easier to the voters to adopt than AWP, since rating the candidates may be more natural than trying to identify the correct approval cutoff point. On the MOP side MOP-F may be the easiest option since naming one's favourites, or those candidates that one wants to defend, seems much easier than trying to determine preference strengths separately to all given preference relations. I mean that although I make a technical comparison between AWP and MOP-F (to point out the differences in their philosophy), in real life the choice might be between using CWP and MOP-F. When comparing CWP to AWP, the main difference is that in CWP the voter will share the strength of the used range (e.g. from 0 to 100) between the given preferences (in fractions of the range). In MOP methods the philosophy is rather to give maximal protection to some pairs (= weaken their defeats to each other) and none to others. (Also intermediate values would be possible but probably not practical
> ,
>> since they would be just "weakened opinions".)
>>
>>
>>> On 26 Jun 2019, at 10:55, Juho Laatu <juho.laatu at gmail.com> wrote:
>>>
>>> I studied the Modified Overall Preferences approach a bit more. Here's one new simple pairwise preference modification function. I'll explain the MOP approach again below, and describe the new simple modification function too.
>>>
>>> The modification process starts from (1) some initial set of preferences. I'll use margins again, but also e.g. winning votes would be possible, although maybe not as natural as margins since the modification process already does quite similar things to the results that also the winning votes approach is supposed to do.
>>>
>>> The next step (2) is to modify the strengths of the initial pairwise preferences. The simple modification function is (1-mod)*k*max + initial, where "initial" is the initial pairwise comparison result, "max" is the largest possible defeat size (= number of voters in margins), "k" refers to the easiness of influencing the preferences (e.g. k=4 means that if 1/4 of the voters (25%) think that A and B are "clones", then defeats between them are always weaker than any defeat between two candidates that nobody claims to be "clones"), "mod" refers to the proportion of voters that think that the compared candidates should be considered "clones" (protected from strong defeats to each others), and that the pairwise defeats should be modified / weakened (0 = nobody says so, 1 = all voters say so).
>>>
>>> The third and final phase (3) is to find the winner based on the modified pairwise preferences. In this mail I'll use Minmax. Other approaches work too.
>>>
>>> I'll describe the philosophy of the modification function briefly. Vote A>B>>C will be read as a normal ranked preference, but in addition to that the voter wants to say that A and B should be seen as "clones" (and defeats between them should be weakened). The presence of ">>" is interpreted so that the ">" preferences will be read as indicating "clones" / protection. In this modification function these opinions are symmetric (i.e. also (possible) B>A defeat should be weakened). If k=4 and we have 100 voters, that means that the modified preference strength values of defeats (that were initially margins in range 1..100) may range from 1 to 400. If mod=0, the strength range of defeats is from 301 to 400. If mod=1/4=1/k, the range is from 201 to 300. If mod=1, the range is from 1 to 100. This modification function can be said to use a "spread spectrum" technique since it expands the range of pairwise preference strengths (k times wider than the range of the initial strengths) to cre
> a
>> te
>>> space also for the weakened preferences.
>>>
>>> Note that only the strengths of the defeats are modified. The (cyclic and transitive) preferences will stay just as they are in the initial preferences.
>>>
>>> Rankings and protection ("clone indications") are in principle two separate opinions that could be given separately, but in practice protection is probably derived from enhanced rankings that may contain e.g. cutoffs, numeric gaps, positional gaps or strong and weak preference relations.
>>>
>>> Next, few examples (with numeric results, in case you want to check the operation of the method). The first one describes a burying strategy example.
>>>
>>> 45: A > B >> C
>>> 15: B > A >> C
>>> 40: C > B > A
>>>
>>> A and B clearly form a uniform group that has 60% support. B would win because it is a Condorcet winner, but then all the 45 A supporters decide to bury B and vote A>C>B. That would make A the winner. If k=4 and 31 C supporters (out of the 40) decide to protect B by voting C>B>>A, then B wins despite of the deceptive A supporters (A:-350, B:-346, C:-420). The method can thus defend against this kind of burying strategy in a quite natural way. Also the 15 B supporters could stop supporting A (as a "clone" of B) and vote B>A>C instead, in which case already 16 C supporters (voting C>B>>A) would make A supporters' strategy void (A:-410, B:-406, C:-420).
>>>
>>> 40: A (no "clones")
>>> 35: B>>C>A (A and C indicated as "clones")
>>> 25: C (no "clones")
>>>
>>> This is the first challenge of Forest Simmons. A wins if k=4, as it should (A:-280, B:-405, C:-410). A wins also with k=1, k=2 and k=3.
>>>
>>> 40: A
>>> 35: B>C>>A
>>> 25: C
>>>
>>> This is the second challenge. C wins if k=4, as it should (A:-420, B:-405, C:-270). C wins also with k=1, k=2 and k=3.
>>>
>>> 40: A
>>> 35: B>>C>A
>>> 25: C>>B>A
>>>
>>> This is the third challenge. B wins if k=4, as it should (A:-320, B:320, C:-410). C wins also with k=0 (no modification), k=1, k=2 and k=3. Any number of C supporters could change their vote to C>B>A, and/or B supporters to B>C>A, and B will still win. That is because B is a Condorcet winner, and the modification function will keep initial defeats (and ties) as defeats (and ties), just with different strengths.
>>>
>>> If you have examples where this kind of pairwise overall defeat modifying / weakening approach would not work well, please tell. I'm eager to evaluate MOP methods against such challenges.
>>>
>>>
>>>
>>>> On 18 Jun 2019, at 09:01, Juho Laatu <juho.laatu at gmail.com> wrote:
>>>>
>>>> Chris Benham and I discussed (in private mail) on how well the example method below complies with the challenge that Forest Simmons proposed.
>>>>
>>>>> Forest Simmons fsimmons at pcc.edu
>>>>> Thu May 30
>>>>>
>>>>>> In the example profiles below 100 = P+Q+R, and?? 50>P>Q>R>0.??
>>>>>>
>>>>>> I am interested in simple methods that always ...
>>>>>>
>>>>>> (1) elect candidate A given the following profile:
>>>>>> P: A
>>>>>> Q: B>>C
>>>>>> R: C,
>>>>>>
>>>>>> and
>>>>>> (2) elect candidate C given
>>>>>> P: A
>>>>>> Q: B>C>>
>>>>>> R: C,
>>>>>>
>>>>>> and
>>>>>> (3) elect candidate B given
>>>>>> P: A
>>>>>> Q: B>>C?? (or B>C)
>>>>>> R: C>>B. (or C>B)
>>>>
>>>>
>>>> I copy my answers to those questions below.
>>>>
>>>> Note that I'm struggling a bit with presentation of votes since Forest Simmons' challenge seems to talk about strengthening some preferences, while my mail talked about weakening some preferences (= making the defeats friendlier). When interpreting Forest's presentation of votes I assumed weakened preferences to be present only in places where the stronger (">>") preferences make the presence of weaker preferences obvious. I thus assumed that e.g. vote B>C (= B>C>A) in Case (3) to not contain any weakened preferences. Same e.g. with vote A in all of the cases. Only votes like B>>C (= B>>C>A) were assumed to contain weakened preferences (between C and A).
>>>>
>>>>> I tried to see how it relates to Forest's requests.
>>>>>
>>>>> Case (2)
>>>>> Since all candidates lose to one of the others, and the only defeat to be weakened is the B>C defeat, and there are always more than 25% of the voters demanding that (Q>25%), then the "4*" version always lowers the strength of that defeat to 0. Therefore C always wins with that method.
>>>>>
>>>>> Case (1)
>>>>> The example method didn't have any special "dislike" cutoff (only a "near clones this far" cutoff). I.e. only votes like A>B>C>>D>>E were allowed. The easiest way to introduce richer use of preference strengths would be to allow any preference (in the ranked vote) to be either ">" or ">>". In this case that would lead to votes Q: B>>C>A. That would weaken A's defeat to C. Again, the number of Q voters is higher than 25%, and with the "4*" factor the strength of A's defeat to C would become 0, and A would win.
>>>>>
>>>>> Case (3)
>>>>> Also here I must assume that votes Q: B>>C>A and R: C>>B>A are allowed. Since (Q+R)>25%, strength of A's defeat would be 0. Candidate B has however no defeats nor ties, so B will win in any case. Having votes of form B>C>A or C>B>A would make A's defeat more severe (at least once we get below the 25% "clone treatment recommendation" limit). Candidate B (Condorcet winner) wins thus also in the original example method.
>>>>>
>>>>> In summary, the method seems to meet Forest's requirements if we allow also weakening of the lower preferences.
>>>>
>>>> I note that the method doesn't even have a name yet. Maybe MMM-MOP will do. It refers to the base method (Minmax(margins)) and the modification to it (Modified Overall Preferences). That leaves some space to using different parameters and approaches in modifying the pairwise preferences in the matrix.
>>>>
>>>> P.S. A bit more on weakened preferences / indicated clones (vs strengthened preferences). I'll use square brackets to indicate clones/weakening here. One can vote [A>B]>C (corresponds to A>B>>C), which means that B's possible defeat to A should be just a friendly one. One could vote also [A=B]>C if ties are allowed. This means that also ties can be weak. Or maybe one should rather say that also tied candidates can be considered clones/friendly (maybe even as default). One option would be to allow also truncated candidates (equal last) be indicated as being friendly. As noted in the beginning of my original mail, the friendliness indicators / weak preferences can be seen as a separate opinion that doesn't have much to do with the original ranking of the candidates (those rankings are as strong as ever in determining the (possibly looped) order, even though the final strength of preferences will be weakened, once the (possibly looped) order has been determined). In typical Condorce
> t
>>
>>> methods weakening of overall pairwise preferences has no meaning except when there are loops, and strengths of different pairwise defeats need to be compared. In the example method of the original mail I stuck with one approval like cutoff only, but in order to respond to the challenge of Forest Simmons we need a bit richer preference structure. If we would go for three preference strengths (>, >>, >>>), maybe that would lead to a "clones among clones" approach in the weakened votes terminology.
>>>>
>>>> P.P.S. In the original mail, ignore words "which means".
>>>>
>>>> Juho
>>>>
>>>>
>>>>> On 16 Jun 2019, at 16:51, Juho Laatu <juho.laatu at gmail.com> wrote:
>>>>>
>>>>> The point of this mail is to promote the idea of separating the strength of different pairwise preferences from the ordinary rankings (of ranked or Condorcet methods). Instead of giving voters the ability to adjust the strength of their own vote or pairwise preferences (the more common approach), they can influence the strength of the final preferences of the whole electorate. In this approach the pairwise preferences of the electorate will not be changed from A>B to B>A. Only the strength of that preference (A>B or B>A) will be modified.
>>>>>
>>>>> You could make many complex tricks with this kind of tools, but I'll concentrate here on one example method that is very simple to the voters. It will consist of adding just one approval like threshold to a basic Condorcet method. I'll use Minmax(margins) as the base method to be modified. Instead of voting A >> B >> C >> D >> E (a normal ranked vote) voters can vote also A > B > C >> D >> E (i.e. one or more of the first preferences may be weakened/moderated). The philosophy of the latter vote is that A, B and C are promoted as "favoured candidates" or "near clones" or "protected candidates" (with the full strength of one vote).
>>>>>
>>>>> In the counting process, first count the normal pairwise preferences (i.e. the matrix). After that, some of the pairwise preferences are made weaker. And then the final results are counted, (almost) as in Minmax(margins).
>>>>>
>>>>> Our first approach is to count the number of votes that had preference A>B (moderated preference) (= Mab). Then the A over B preference count (of the whole electorate) will be weakened by multiplying it with 1-Mab/N (where N = number of votes). Moderated preferences are transitive in the sense that A > B > C increases also the Mac count. The idea is that if voters consider A and B to be "near clones", their defeats to each other should be seen and treated as "friendly defeats". Their strength can thus be weakened, although a large number of voters may have preferred one over another (in a friendly way).
>>>>>
>>>>> (I note that some alternatives to the presented example method could be to make it symmetrical by using factor 1-(Mab+Mba)/N, or one could use the number of votes that preferred either A to B or B to A instead of N.)
>>>>>
>>>>> I'll add one more trick to the example method. The "friendliness factor" can be stronger than presented above. Let's say that 50% of the voters (maybe one of two parties) think that A and B should be treated as "near clones". Half of those voters have voted A > B, and half B > A. This means that 25% of the voters have given (moderated) preference A > B. Maybe already this 25% is enough to convince us that A and B indeed are to be treated as full clones. If so, we can use factor max(1-4*Mab/N,0) instead of 1-Mab/N. Here reaching strength 0 means that those candidates will be declared as "clones" and treated as such.
>>>>>
>>>>> The counting process may have some problems with ties if multiple preferences will be equal to 0. I will not discuss this question much more in this mail. Let's just say that also those preferences (although they all seem to be equal to 0) can be seen to have different strength, e.g. based on the initial margins. And that if there are two parties with 50 votes, and there are some "clones" in one of the parties, they could be seen as one alternative when flipping the coin. (Winner among the clones to be decided separately if the clone party wins.)
>>>>>
>>>>> Here's one example set of votes.
>>>>>
>>>>> 45 A>B>>C --> A>>C>>B
>>>>> 15 B>A>>C
>>>>> 40 C>B>>A
>>>>>
>>>>> B is the sincere Condorcet winner. Supporters of A will however strategically bury B under C. In Minmax(margins) the worst defeats are A:-10, B:-70, C:-20. The strategic voters will get what they wanted. There were however 40 voters that said Mcb. This means that in the example method the strength of B's defeat to C will be 0 (using the "4*" in the factor). There are also 15 voters that said Mba. This helps A a bit (-10 --> -4), but not too much. If all C supporters (or 17 or more of them) had voted C>>B>>A, A would win. C supporters may thus vote C>B>>A sincerely or as a (still quite sincere) defensive tactic (after hearing about A supporters' plans).
>>>>>
>>>>> Another question where this ability to moderate the defeats of favourite near clones is interesting is whether to elect from the Smith set or outside of it.
>>>>>
>>>>> 17 A>>B>>C>>D
>>>>> 17 B>>C>>A>>D
>>>>> 17 C>>A>>B>>D
>>>>> 16 D>>A>>B>>C
>>>>> 16 D>>B>>C>>A
>>>>> 16 D>>C>>A>>B
>>>>>
>>>>> A, B and C are not "clones" in the described sense (which means no weakening of their mutual defeats since not a single voter proposed that). D wins.
>>>>>
>>>>> 17 A>B>C>>D
>>>>> 17 B>C>A>>D
>>>>> 17 C>A>B>>D
>>>>> 16 D>>A>>B>>C
>>>>> 16 D>>B>>C>>A
>>>>> 16 D>>C>>A>>B
>>>>>
>>>>> A, B and C are "clones" (weight of mutual defeats = 0, with the "4*" moderation). One of them wins.
>>>>>
>>>>>
>>>>> ----
>>>>> Election-Methods mailing list - see https://electorama.com/em for list info
>>>>
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