[EM] "Possible Approval Winner" set/criterion (was "Juho--Margins fails Plurality. WV passes.")

Tue Mar 13 09:35:22 PDT 2007

Juho wrote:

> The "Possible Approval Winner" criterion looks actually quite natural 
> in the sense that it compares the results to what Approval voting 
> could have achieved.
>
I'm glad you think so.

> The definition of the criterion contains a function that can be used 
> to evaluate the candidates (also for other uses) - the possibility and 
> strength of an approval win. This function can be modified to support 
> also cardinal ratings.
>
> In the first example there is only one entry (11: A>B) that can vary 
> when checking the Approval levels. B can be either approved or not. In 
> the case of cardinal ratings values could be 1.0 for A, 0.0 for C and 
> anything between 0.0001 and 0.9999 for B. Or without normalization the 
> values could be any values between 0.0. and 1.0 as long as value(A) > 
> value(B) > value (C). With the cardinal ratings version it is possible 
> to check what the "original utility values" leading to this group of 
> voters voting A>B could have been (and if the outcome is achievable in 
> some cardinal ratings based method, e.g. max average rating).

This concept looks vulnerable to some weak irrelevant candidate being 
added to the top of some ballots, displacing a candidate down to
second preference and maybe thereby causing it to fall out of the set 
of  "possible winners". It probably has other problems regarding 
Independence
properties, and I can't see any use for it.

Chris Benham

> The "Possible Approval Winner" criterion looks actually quite natural 
> in the sense that it compares the results to what Approval voting 
> could have achieved.
>
> The definition of the criterion contains a function that can be used 
> to evaluate the candidates (also for other uses) - the possibility and 
> strength of an approval win. This function can be modified to support 
> also cardinal ratings.
>
> In the first example there is only one entry (11: A>B) that can vary 
> when checking the Approval levels. B can be either approved or not. In 
> the case of cardinal ratings values could be 1.0 for A, 0.0 for C and 
> anything between 0.0001 and 0.9999 for B. Or without normalization the 
> values could be any values between 0.0. and 1.0 as long as value(A) > 
> value(B) > value (C). With the cardinal ratings version it is possible 
> to check what the "original utility values" leading to this group of 
> voters voting A>B could have been (and if the outcome is achievable in 
> some cardinal ratings based method, e.g. max average rating).
>
> The max average rating test is actually almost as easy to make as the 
> PAW test. Note that my description of the cardinal ratings for 
> candidate B had a slightly different philosophy. It maintained the 
> ranking order of the candidates, which makes direct mapping from the 
> cardinal values to ordinal values possible. The results are very 
> similar to those of the approval variant but the cardinal utility 
> values help making a more direct comparison with the "original 
> utilities" of the voters.
>
> Now, what is the value of these comparisons when evaluating the 
> different Condorcet methods. These measures could be used quite 
> straight forward in evaluating the performance of the Condorcet 
> methods if one thinks that the target of the voting method is to 
> maximise the approval of the winner or to seek the best average 
> utility. This need not be the case in all Condorcet elections (but is 
> one option). There are several utility functions that the Condorcet 
> completion methods could approximate. The Condorcet criterion itself 
> is majority oriented. Minmax method minimises the strength of interest 
> to change the selected winner to one of the other candidates. Approval 
> and cardinal ratings have somewhat different targets than the majority 
> oriented Condorcet criterion and some of the common completion 
> methods, but why not if those targets are what is needed (or if they 
> bring other needed benefits like strategy resistance).
>
> I find it often useful to link different methods and criteria to 
> something more tangible like concrete real life compatible examples or 
> to some target utility functions (as in the discussion above). One key 
> reason for this is that human intuition easily fails when dealing with 
> the cyclic structures (that are very typical cases when studying the 
> Condorcet methods). In this case it seems that PAW and corresponding 
> cardinal utility criterion lead to different targets/utility than e.g. 
> the minmax(margins) "required additional votes to become the Condorcet 
> winner" philosophy. Maybe the philosophy of PAW is to respect clear 
> majority decisions (Condorcet criterion) but go closer to the 
> Approval/cardinal ratings style evaluation when the majority opinion 
> is not clear. You may have different targets in your mind but for me 
> this was the easiest interpretation.
>
> Juho
>
>
> P.S. One example.
> 1: A>B
> 1: C
> Here B could be an Approval winner (tie) but not a max average rating 
> winner in the "ranking maintaining style" that was discussed above 
> (since the rating of B must be marginally smaller than the rating of A 
> in the first ballot).
>
>
> On Mar 7, 2007, at 16:28 , Chris Benham wrote:
>
>>
>> Juho wrote (March7, 2007):
>>
>>>The definition of plurality criterion is a bit confusing. (I don't  
>>>claim that the name and content and intention are very natural  
>>>either :-).)
>>>- http://wiki.electorama.com/wiki/Plurality_criterion talks about  
>>>candidates "given any preference"
>>>- Chris refers to "above-bottom preference votes" below
>>>  
>>>
>>> /If the number of ballots ranking /A/ as the first preference is 
>>> greater than the number
>>> of ballots on which another candidate /B/ is given any preference, 
>>> then /B/ must not be elected./
>>>
>>>Electowiki definition could read: "If the number of voters ranking A  
>>>as the first preference is greater than the number of voters ranking  
>>>another candidate B higher than last preference, then B must not be  
>>>elected".
>>>
>> Yes it could and to me it in effect does (provided "last" means "last 
>> or equal-last") The criterion come
>> from Douglas Woodall who economises on axioms so doesn't use one that 
>> says that with three candidates
>> A,B,C a ballot marked A>B>C must always be regarded as exactly the 
>> same thing as  A>B truncates. He
>> assumes that truncation is allowed but above bottom equal-ranking isn't.
>>
>> A similar criterion of mine is the "Possible Approval Winner" criterion:
>>
>> "Assuming that voters make some approval distinction among the 
>> candidates but none among those
>> they equal-rank (and that approval is consistent with ranking) the 
>> winner must come from the set of
>> possible approval winners".
>>
>> This assumes that a voter makes some preference distinction among the 
>> candidates, and that truncated
>> candidates are equal-ranked bottom and so never approved.
>>
>> Looking at a profile it is very easy to test for: considering each 
>> candidate X in turn, pretend that the
>> voters have (subject to how the criterion specifies) placed their 
>> approval cutoffs/thresholds in the way
>> most favourable for X, i.e. just below X on ballots that rank X above 
>> bottom and on the other ballots
>> just below the top ranked candidate/s, and if that makes X the 
>> (pretend) approval winner then X is
>> in the PAW set and so permitted to win by the PAW criterion.
>>
>>11: A>B
>>07: B
>>12: C
>>
>> So in this example A is out of the PAW set because in applying the 
>> test A cannot be more approved
>> than C.
>>
>> IMO, methods that use ranked ballots with no option to specify an 
>> approval cutoff and rank among
>> unapproved candidates should elect from the intersection of the PAW 
>> set and the Uncovered set
>>
>> One of  Woodall's  "impossibility theorems" states that is impossible 
>> to have all three of  Condorcet,
>> Plurality and Mono-add-Top. MinMax(Margins) meets Condorcet and 
>> Mono-add-Top.
>>
>> Winning Votes also fails the Possible Approval Winner (PAW) 
>> criterion, as shown by this interesting
>> example from  Kevin Venzke:
>>
>>35 A
>>10 A=B
>>30 B>C
>>25 C
>>
>>A>B 35-30,  B>C 40-25, C>A 55-45
>>
>>Both Winning Votes and Margins elect B, but B is outside the PAW set{A,C}.
>>Applying the test to B, we get possible approval scores of A45, B40, C25.
>>
>>ASM(Ranking) and DMC(Ranking) and Smith//Approval(Ranking) all meet the Definite 
>>Majority(Ranking) criterion which implies compliance with PAW. The DM(R) set is
>>{C}, because interpreting ranking (above bottom or equal-bottom) as approval, both
>>A and B are pairwise beaten by more approved candidates.
>>
>>
>>Chris Benham
>>
>>
>>
>>
>>
>>    
>>
>>
>
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