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<font face="Helvetica, Arial, sans-serif">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.
</font><br>
<font face="monospace"> random fptp dblv
seq conting nauru borda sbc2 bucklin sinkhorn
mj av coombs <br>
12.6630 35.6490 50.7000 44.9140 51.6650 54.5890
73.6530 - 66.3850 - - 53.3880 68.9630 <br>
clower knockout spe benham btr-irv baldwin
nanson minimax minimaxwv minisum rp river schulze
asm cupper <br>
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 <br>
condorcet+ random fptp dblv conting borda av <br>
70.6780 70.6580 70.9080 71.0760 72.2750 70.9920 <br>
llull+ randomr fptpf fptpr dblvf contingr bordaf
bordar avf avr minimaxf minimaxr<br>
71.6220 71.2570 71.9820 71.2600 71.9970 72.2020
72.0080 71.3300 72.0120 72.0510 72.0070 <br>
smith+ randomr fptpf fptpr dblvf contingr bordaf
bordar avf avr minimaxf minimaxr tideman <br>
71.3330 70.8970 71.5080 70.9620 71.5820 72.2730
71.6550 71.0270 71.6240 72.0990 71.6490 71.1760 <br>
</font><br>
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. <br>
<font face="monospace"> random fptp dblv
seq conting nauru borda sbc2 bucklin sinkhorn
mj av coombs <br>
12.5510 37.4290 43.1720 36.6340 41.2690 40.7330
34.6170 - 41.5260 - - 40.9330 42.4740 <br>
clower knockout spe benham btr-irv baldwin
nanson minimax minimaxwv minisum rp river schulze
asm cupper <br>
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 <br>
condorcet+ random fptp dblv conting borda av <br>
43.6260 44.0730 44.1880 43.9420 43.2570 43.5720 <br>
llull+ randomr fptpf fptpr dblvf contingr bordaf
bordar avf avr minimaxf minimaxr<br>
43.7980 43.9980 43.4990 44.0330 43.4980 43.3220
43.4960 43.6550 43.4950 43.9890 43.4980 <br>
smith+ randomr fptpf fptpr dblvf contingr bordaf
bordar avf avr minimaxf minimaxr tideman <br>
43.7660 44.1030 43.4060 44.1810 43.4080 43.2570
43.4000 43.5750 43.4000 44.0000 43.4100 43.5840 </font><br>
At risk of repetition... correctness of software is not guaranteed.<br>
CJC<br>
<br>
<div class="moz-cite-prefix">On 27/09/2023 12:45, Colin Champion
wrote:<br>
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<font face="Helvetica, Arial, sans-serif">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. <br>
<font face="monospace"><font face="Helvetica, Arial, sans-serif"><font
face="monospace"> random fptp dblv
seq conting nauru borda sbc2 bucklin
sinkhorn mj av coombs <br>
12.5820 35.9910 - 45.8790 -
53.6880 80.5090 - 67.5170 - -
55.7040 69.1810 <br>
clower knockout spe benham btr-irv
baldwin nanson minimax minimaxwv minisum rp
river schulze asm cupper <br>
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 <br>
condorcet+ random fptp dblv conting borda
av <br>
75.4610 75.5690 75.6860 75.8110 76.4530
75.8300 <br>
llull+ randomr fptpf fptpr dblvf contingr
bordaf bordar avf avr minimaxf minimaxr<br>
75.8750 75.8660 76.2610 75.8330 76.2600
76.3780 76.2620 75.9250 76.2590 75.9530 76.2620 <br>
smith+ randomr fptpf fptpr dblvf contingr
bordaf bordar avf avr minimaxf minimaxr
tideman <br>
75.7640 75.7470 76.2310 75.7630 76.2400
76.4530 76.2530 75.8650 76.2420 75.9680 76.2470
76.0700 </font><br>
</font></font>I will try a couple of other truncation models
and then look at strategic voting.<br>
CJC<br>
</font><br>
<div class="moz-cite-prefix">On 24/09/2023 13:41, Colin Champion
wrote:<br>
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<font face="Helvetica, Arial, sans-serif">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. <br>
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. <br>
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.<br>
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. <br>
I will try to extend my own evaluation software to allow a
less restrictive model of truncation.<br>
Colin<br>
</font><br>
<div class="moz-cite-prefix">On 23/09/2023 02:47, Kevin Venzke
wrote:<br>
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cite="mid:2129168188.8275917.1695433672418@mail.yahoo.com">
<pre class="moz-quote-pre" wrap="">Hi Colin,
Le vendredi 22 septembre 2023 à 02:57:42 UTC−5, Colin Champion <a class="moz-txt-link-rfc2396E" href="mailto:colin.champion@routemaster.app" moz-do-not-send="true"><colin.champion@routemaster.app></a> a écrit :
</pre>
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<pre class="moz-quote-pre" wrap="">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.
</pre>
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<pre class="moz-quote-pre" wrap="">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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