<div dir="auto"><div><br><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Thu, Jan 5, 2023, 9:24 AM Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de">km_elmet@t-online.de</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Happy new year!<br>
<br>
Now that it's a new year, I've done a bit more exploring around the <br>
low-manipulability generalizations of the contingent vote. My summary <br>
would be this:<br>
<br>
- I found one bug, but not all, that caused the Python vote simulator to <br>
give different results to quadelect. As I haven't found all, the stuff <br>
below still hasn't been fully verified.<br>
<br>
- There's a strange pattern where every method (if not augmented by <br>
Condorcet) seems to have almost all its strategy vulnerability as <br>
compromising. More later.<br>
<br>
- An interesting pattern seems to be: nonmonotone CV generalizations <br>
work by "If A and B are at the top by first preferences, then credit A <br>
with A>B and B with B>A". Monotone ones are about absolute comparisons <br>
instead: "if A and B combined are above some quota, then...".<br>
<br>
- For monotone "CW if there is one, otherwise method X", the best <br>
impartial culture generalization for four candidates goes like this:<br>
Let A and B be two candidates. Then if for every other candidate C, fpA <br>
+ fpB - 2 fpC > 0, let A's score in the (A,B) comparison be A>B and B's <br>
score be B>A. If they're all < 0, it's zero for both.<br>
Let every candidate A's score be the maximum of any comparison involving A.<br>
Highest score wins unless there's a CW. If there is, the CW wins.<br>
<br>
- According to quadelect, this achieves roughly 66% strategy <br>
vulnerability on four candidates and 97 voters. The variant with sum <br>
instead of max gets 70%.<br>
<br>
- The best nonmonotone method of this form is Condorcet//contingent <br>
vote. Its susceptibility is 60% with max and 57% with sum.<br>
<br>
- Including terms involving first preferences after elimination can <br>
improve the results quite a bit, but the resulting methods are <br>
completely incomprehensible. Here's an example for four candidates, that <br>
*should* be monotone although I haven't checked:<br>
- Let A and B be two candidates. Then if, for every pair {C, D} of <br>
other candidates, if fpA\C + fpB\C - fpD\C -fpD > 0, A's score is (A>B) <br>
and B's is (B>A). Otherwise if all are <0, it's zero for both. fpX\Y <br>
here is my terminology for "X's first preference count once Y has been <br>
eliminated".<br>
- The rest is as above (including the use of the max operator).<br>
<br>
- That method has 53% strategic susceptibility under the given conditions.<br>
<br>
----<br>
<br>
So lots of strange things. I'm kinda understanding the limitations of <br>
not having a theory; the simulator can come up with some generalization <br>
that seems to work, but I have no idea why it works. Leaving elections <br>
to a black box isn't really a good idea, either :-) What's necessary <br>
here, I think, is some kind of explanation for just why these are <br>
resistant to strategy and others are not, and I don't know what that <br>
would look like.<br>
<br>
The pattern seems to be that if A's score is a transformation of some <br>
linear combination of first preferences, and the method is monotone, <br>
then clearly there won't be any burial incentive because lowering A's <br>
rank can't harm A as nobody who prefers to B as A has A as their first <br>
preference anyway.<br>
<br>
And obviously sums of these and maxima of these retain the property.<br>
<br>
The surprising part is that sometimes, you can include pairwise counts <br>
and it's still all compromise all the time. E.g. the contingent vote:<br>
<br>
f(A) = sum over {A, B}:<br>
H(fpA - fpB) *<br>
product over C not in {A, B}:<br>
H(fpB - fpC)<br>
* A>B.<br>
<br>
where f(A) is A's score and H is the step function.<br>
<br>
Even the weird method<br>
f(A) = max over B: A>B<br>
<br>
seems to have a very low (though nonzero) rate of burial.</blockquote></div></div><div dir="auto"><br></div><div dir="auto"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"> </blockquote></div></div><div dir="auto">This weird fact suggests to me the following Banks efficient method:</div><div dir="auto"><br></div><div dir="auto">Start a pairwise defeat chain A>B with the greatest strength defeat pair A>B.</div><div dir="auto"><br></div><div dir="auto">Then of all the X that defeat both A and B, let X1 be the one with the greatest defeat against A, and then augment the chain with X1 to get X1>A>B.</div><div dir="auto"><br></div><div dir="auto">Then ...</div><div dir="auto"><br></div><div dir="auto">While some X defeats every member of the chain, augment the chain with the X that has the strongest defeat against the newest member of the chain. EndWhile </div><div dir="auto"><br></div><div dir="auto">Elect the head of the resulting maximal chain.</div><div dir="auto"><br></div><div dir="auto">Note that if there are only three candidates and those three are in a cycle ABCA, then this method will elect</div><div dir="auto">A=argmax f(X).</div><div dir="auto"><br></div><div dir="auto">-Forest</div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Why? You'd <br>
imagine that if A is tying C, and A>B is A's max pairwise victory over <br>
someone else, then the buriers can benefit by burying A under B; and <br>
that that would happen pretty frequently.<br>
<br>
On the other hand,<br>
<br>
f(A) = sum over B: H(A>B - B>A)<br>
<br>
is just Copeland, which has *plenty* of burial incentive. Yet it looks <br>
superficially like the other methods: monotone transformations of linear <br>
combinations of pairwise victories. How about if we only allow a single <br>
victory inside the monotone transformation, something like<br>
<br>
f(A) = sum over B: (A>B)^x<br>
<br>
with x>1? Is that also only compromise?<br>
<br>
-km<br>
----<br>
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</blockquote></div></div></div>