[EM] minx(fpA-fpC) DMTBR counterexample?

[Kristofer Munsterhjelm]

On 17.04.2022 07:46, Kevin Venzke wrote:
> Hi Kristofer,
> 
> This method is a little tricky to double-check so I have a little less
> confidence than usual in offering scenarios...
> 
> Hmm, but try this one:
> 
> 0.345: C>D>B>A
> 0.320: A>C>D>B
> 0.243: B>A>C>D
> 0.090: D>B>C>A  -->  D>C>B>A
> 
> It seems to me this moves the win from C to A.
> 
> Defeats are A>C>B>A, A>D, C>D, D>B. They're unchanged by the vote.
> Changing B>C to C>B seems to help A by reducing B's effective first pref count
> for the three-set {A,B,C}.

Yeah, it checks out. The example can be reduced to:

7: C>D>B>A
6: A>C>D>B
5: B>A>C>D
2: D>B>C>A

so it shows a problem with my monotonicity checker.

Since my method is pretty similar to IFPP, it may be useful to look at
the latter to get an indication of whether the "wrong" winner is D
before the raising or A after.

IFPP elects C in both cases. OTOH, the set narrowing method I mentioned
earlier (where I was having problems with cycles) gives the outcome as a
tie between A, C, and D before; and as A uniquely after. So who knows?

>> Can anyone find a DMTBR counterexample for minx(fpA-fpC)? I haven't been
>> able to for either four or five candidates. Since DMTBR plus Condorcet
>> and either monotonicity or summability is a high bar in itself, I want
>> to be *very* sure before proclaiming that I've accomplished something
>> spectacular.
> 
> How about this:
> 
> 0.389: D>A>C>B
> 0.334: A>D>B>C  -->  A>B>C>D
> 0.162: B>A>D>C
> 0.114: C>B>D>A
> 
> Initially D is the CW and has over a third of first prefs. I think this vote
> change elects A.
> 
> Lowering D creates a B>D win. The other wins are A>B, A>C, C>B, D>A, D>C.
> D's worst score becomes -.107 (from {B,C,D}) while A's worst is -.055 (from
> {A,B,D}).

That seems right too. I managed to round it off to:

9: D>A>C>B
8: A>D>B>C
3: B>A>D>C
3: C>B>D>A

Again, my Monte-Carlo program doesn't seem to recognize the "before
burial" election to be vulnerable to burial.

(The exact manipulability results for 5 voters should still hold,
though; that's a different calculation and it doesn't check specifically
for burial.)

Thanks!

-km