[EM] Fwd: Ranked Pairs

Colin Champion colin.champion at routemaster.app
Thu Sep 28 05:00:21 PDT 2023

I tried two other forms of truncation. Under "candidate-specific 
truncation" the m candidates have associated truncation levels which are 
a random permutation of the numbers 1...m. A ballot is truncated to the 
level corresponding to its first candidate. I expected this to be a hard 
case for WV, but in fact it does appreciably better than margins.
             random    fptp     dblv seq    conting   nauru    borda     
sbc2   bucklin  sinkhorn mj       av     coombs
            12.6630  35.6490  50.7000  44.9140  51.6650  54.5890 
73.6530     -     66.3850     -        -     53.3880  68.9630
             clower  knockout   spe     benham  btr-irv  baldwin nanson  
minimax minimaxwv minisum     rp     river   schulze asm     cupper
            70.0190  71.5400  71.7760  71.2680  70.9510  71.4700 
71.8440  72.0970  72.9090  72.1000  71.5630  71.9420  71.3330 72.2980  
condorcet+  random    fptp     dblv   conting   borda      av
            70.6780  70.6580  70.9080  71.0760  72.2750  70.9920
     llull+ randomr   fptpf    fptpr    dblvf   contingr  bordaf 
bordar    avf      avr    minimaxf minimaxr
            71.6220  71.2570  71.9820  71.2600  71.9970  72.2020 
72.0080  71.3300  72.0120  72.0510  72.0070
     smith+ randomr   fptpf    fptpr    dblvf   contingr  bordaf 
bordar    avf      avr    minimaxf minimaxr tideman
            71.3330  70.8970  71.5080  70.9620  71.5820  72.2730 
71.6550  71.0270  71.6240  72.0990  71.6490  71.1760

The other form I tried was 'ignorance truncation'. Each candidate has a 
prominence - i.e. probability of being recognised by an arbitrary voter 
- drawn (separately for each election) from a Beta(r,s) distribution. 
Voters rank the candidates they recognise in order of proximity, 
truncating after the last candidate they recognise. I used r=2, s=1, 
giving a recognition probability of 2/3. This was essentially a tie 
between the two minimax variants. Borda, which looked good against other 
forms of truncation, did badly this time. Evidently ignorance truncation 
is more damaging than the other sorts.
            random    fptp     dblv seq    conting   nauru    borda     
sbc2   bucklin  sinkhorn mj       av     coombs
            12.5510  37.4290  43.1720  36.6340  41.2690  40.7330 
34.6170     -     41.5260     -        -     40.9330  42.4740
             clower  knockout   spe     benham  btr-irv  baldwin nanson  
minimax minimaxwv minisum     rp     river   schulze asm     cupper
            43.1770  43.8040  44.4050  43.5870  44.0050  44.0480 
43.9970  43.9990  43.9330  44.0170  43.8610  44.0040  43.7660 43.6000  
condorcet+  random    fptp     dblv   conting   borda      av
            43.6260  44.0730  44.1880  43.9420  43.2570  43.5720
     llull+ randomr   fptpf    fptpr    dblvf   contingr  bordaf 
bordar    avf      avr    minimaxf minimaxr
            43.7980  43.9980  43.4990  44.0330  43.4980  43.3220 
43.4960  43.6550  43.4950  43.9890  43.4980
     smith+ randomr   fptpf    fptpr    dblvf   contingr  bordaf 
bordar    avf      avr    minimaxf minimaxr tideman
            43.7660  44.1030  43.4060  44.1810  43.4080  43.2570 
43.4000  43.5750  43.4000  44.0000  43.4100  43.5840
At risk of repetition... correctness of software is not guaranteed.

On 27/09/2023 12:45, Colin Champion wrote:
> I have some preliminary results for "approval truncation" in which a 
> voter truncates at the largest gap between cardinal rankings. Minimax 
> (margins) does slightly better than minimax (WV). Voting is sincere; 
> there are 8 candidates and 10001 voters (a ballot is truncated on 
> average to 4.6 entries). Full figures follow (which won't be very 
> readable in a variable-width font). It's noticeable that the results 
> are worse than for fixed truncation, even though the average ballot 
> length is slightly greater.
>             random    fptp     dblv seq    conting   nauru    
> borda     sbc2   bucklin sinkhorn    mj       av     coombs
>            12.5820  35.9910     -     45.8790     - 53.6880  
> 80.5090     -     67.5170     -        - 55.7040  69.1810
>             clower  knockout   spe     benham  btr-irv baldwin   
> nanson  minimax minimaxwv minisum     rp river   schulze    asm     
> cupper
>            75.1840  75.8440  76.2830  76.0300  75.8900 75.8700  
> 75.9440  75.9660  75.9580  75.9680  75.8200 -     75.7640  75.9200  
> 77.3430
> condorcet+  random    fptp     dblv   conting   borda av
>            75.4610  75.5690  75.6860  75.8110  76.4530 75.8300
>     llull+ randomr   fptpf    fptpr    dblvf   contingr bordaf   
> bordar    avf      avr    minimaxf minimaxr
>            75.8750  75.8660  76.2610  75.8330  76.2600 76.3780  
> 76.2620  75.9250  76.2590  75.9530  76.2620
>     smith+ randomr   fptpf    fptpr    dblvf   contingr bordaf   
> bordar    avf      avr    minimaxf minimaxr tideman
>            75.7640  75.7470  76.2310  75.7630  76.2400 76.4530  
> 76.2530  75.8650  76.2420  75.9680  76.2470 76.0700
> I will try a couple of other truncation models and then look at 
> strategic voting.
>     CJC
> On 24/09/2023 13:41, Colin Champion wrote:
>> Kevin – thanks for this helpful reply. I'm inclined to favour viewing 
>> a tie as two half-voters with opposed preferences. I admit that this 
>> can only be a rule of thumb, but I find it quite persuasive. After 
>> all, the whole point of ranked voting is that voters start out, I 
>> assume, with nebulous cardinal judgements in their heads, and that 
>> turning these judgements into rankings puts them onto a common basis 
>> (albeit with loss of information) which allows them to be 
>> meaningfully combined. The WV rule could easily undermine the premise 
>> of this procedure.
>>    I believe that asymmetric treatment of ties in the Borda count 
>> leads quite directly to errors of the sort I described, but I don't 
>> know if this is widely accepted.
>>    It's true that Darlington models ties as genuine expressions of 
>> indifference. In practice ties can mean almost anything; 
>> indifference, laziness, ignorance... Quite possibly voting methods 
>> which work well for one sort of tie will work less well for another. 
>> The result I produced myself is probably genuine, and indicates that 
>> WV is more accurate than margins for mandatory truncation; but I was 
>> wrong to suppose that it could be interpreted more generally since it 
>> omits the effect which is most likely to work against WV.
>>    As for the positive arguments you put forward, well they might 
>> justify a rule of thumb but I wouldn't find them compelling. I don't 
>> find the Condorcet principle persuasive on its own merits (and do not 
>> believe it generally sound), but I accept it as a working principle 
>> because I don't know any other way of obtaining simple accurate 
>> voting methods under a spatial model.
>>    I will try to extend my own evaluation software to allow a less 
>> restrictive model of truncation.
>>       Colin
>> On 23/09/2023 02:47, Kevin Venzke wrote:
>>> Hi Colin,
>>> Le vendredi 22 septembre 2023 à 02:57:42 UTC−5, Colin Champion<colin.champion at routemaster.app>  a écrit :
>>>> A possible explanation for the discrepancy between my result and Darlington's is that
>>>> in my evaluation every ballot had the same number of ties and in Darlington's the
>>>> numbers differed.
>>>> On the face of it, WV doesn't treat voters equally. If we defined "winning votes" as
>>>> "the number of voters who prefer A to B plus half the number who rank them equally",
>>>> then every voter would contribute m(m-1)/2 winning votes and WV would be equivalent
>>>> (I think) to Margins. But instead we define winning votes asymmetrically so that WV
>>>> is *not* equivalent to margins but voters contribute different numbers of winning
>>>> votes depending on the number of ties in their ballots. I can imagine this leading to
>>>> artefacts which Darlington's evaluation would pick up and mine would miss. If this is
>>>> what happened, then even Darlington's evaluation must be too lenient to WV since he
>>>> doesn't include effects which would in fact arise, such as voters truncating
>>>> differentially according to their political viewpoint.
>>>> Maybe these things have been taken into account; I have no idea, having never seen the
>>>> thinking behind WV.
>>> I am not sure what to make of Darlington's defeat strength comparison. It sounds like
>>> it was basically a simulation of sincere voters who vote equality because they actually
>>> consider the candidates equal. That premise is fine but somewhat far removed from how
>>> this topic is usually discussed, i.e. with some consideration of comparative strategy.
>>> I notice incidentally that Darlington says incorrectly on page 22 that MinMax(PO) is a
>>> Condorcet method. I wonder whether he implemented it as one to get his numbers on that.
>>> In any case:
>>> To find the motivation for WV I would start with first principles. How should we design
>>> a Condorcet completion method to minimize strategic incentives? A motivation behind
>>> Condorcet itself is that voters should not vote sincerely only to find that they
>>> should've voted another way.
>>> What could this mean here? Well, a full majority can always get what they want by
>>> changing their votes. Therefore if a majority votes A>B yet B is elected, we have
>>> *probably* done something wrong, because the majority certainly did have the power to
>>> make A win instead. The election of B gives the A>B voters an incentive to vote
>>> differently to change the outcome. The voters obtain a "complaint," I will call it.
>>> Since majorities will most predictably obtain such complaints when we override their
>>> preference, we should prioritize locking majorities.
>>> With WV, there is no special heed paid to majorities, it just goes down the list of
>>> contests starting with the largest winning blocs. But this achieves the goal. It
>>> applies its principle to sub-majority contests as well, and maybe this is good bad or
>>> neutral, but maybe we can believe that if it was helpful (for our end goal) to favor
>>> majorities over sub-majorities then it could also be helpful to favor larger
>>> sub-majorities over smaller sub-majorities. It certainly stands to reason that the more
>>> voters you have sharing some stance, the more likely it is that a vote change on their
>>> part could change the outcome.
>>> (On my website I describe a different approach focused on compromise incentive, and
>>> measuring the potential for this more directly, and one can take that as me suggesting
>>> that WV actually leaves some room for improvement.)
>>> You notice that adding half-votes to equal rankings under WV will turn it into margins.
>>> This would give every contest a full majority on the winning side, and seemingly we can
>>> trivialize this requirement of mine to prioritize majorities.
>>> But I think it's clear, in the context of this analysis, that adding half-votes for
>>> equal rankings doesn't make sense. The voter who says A=B doesn't turn into a pair of
>>> opposing "half-complaints," where one of the complaints has the potential to be voiced
>>> when *either* of A or B is elected. The A=B voter has no possible complaint either way,
>>> as neither result can incentivize them to change their vote.
>>> Additionally, I think that voters expect and want it to be the case that abstaining
>>> from a pairwise contest does not mean the same thing as saying they rate both
>>> candidates equal. I touched on this in my previous post.
>>> Consider this election:
>>> 7 A>B
>>> 5 B
>>> 8 C
>>> Margins elects A, which is very unusual across election methods, and I think most
>>> people would find this result surprising due to a sense of what truncation ought to
>>> mean.
>>> (Consider copying it into votingmethods.net/calc to see margins and MMPO stand alone
>>> here.)
>>> Perhaps with enough education people can *understand* that the method takes seriously
>>> the apparent equality of the truncated preferences. But I don't think voters will find
>>> it comfortable to vote under those circumstances. I think voters want to be able to
>>> identify the set of candidates that they believe they are trying to defeat, leave them
>>> out of their ranking, and not have to think any further about it.
>>> Kevin
>>> votingmethods.net
>> ----
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