[EM] Fwd: Ranked Pairs
Colin Champion
colin.champion at routemaster.app
Mon Oct 2 06:45:12 PDT 2023
And here, as promised, are some results for strategic voting.
* Constant truncation: WV beats margins for sincere voting, and also for
compromising and false cycles, but margins beats WV by quite a long way
(2.7%) for burial.
* Approval truncation: margins beats WV for sincere voting. The two
methods almost tie under compromising; margins wins by a long way under
false cycles (5.5%) and under burial.
* Candidate-specific truncation: WV beats margins for sincere voting; it
also wins (slightly more convincingly) under compromising; it loses
under false cycles and burial.
* Ignorance truncation: this was essentially a tie under sincere voting
and remains one under compromising; margins wins slightly under false
cycles and burial.
Approval truncation takes place before a voter's strategic reordering of
candidates; other forms of truncation take place after it. In each case
I measure the accuracy of a voting method in the presence of strategic
voting, not the vulnerability of the method to manipulation.
CJC
On 28/09/2023 13:00, Colin Champion wrote:
> 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
> 75.2630
> 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
> 46.7470
> 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.
> CJC
>
> 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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>>
>
>
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