[EM] Simple Acceptable Ranked Choice Voting
Forest Simmons
forest.simmons21 at gmail.com
Sat Jan 14 13:54:43 PST 2023
Perhaps we need to use the entire finish order of the seed method to get an
appropriate uncovered winner:
Unc(Finish Order)
Initialize the variable X as the candidate highest in Finish Order.
Then ...
While X is covered, replace it with the highest Finish Order candidate that
covers it. EndWhile.
Elect the updated X.
I would like to suggest as the seed method the following version of
MaxMinPairwise Support:
Let M be the matrix whose entry in column k of row j is the number of
ballots on which candidate j is ranked ahead of or equal to candidate k
(but not truncated).
In particular, the i_th diagonal entry M(i,i) is the number of ballots on
which candidate i is ranked.
Sort the entries in each row in non-decreasing order from left to right.
Note that this sort moves the diagonal entries of M to the extreme right
of each row.
While keeping track of the correspondence between rows and candidates, sort
the rows lexicographically so that row R is above row R' iff R(i)>R'(i),
where i is the first place where the two rows disagree.
The order of the rows induces the modified finish order of the respective
candidates.
If I'm not mistaken, the following variant of the FBC is satisfied by this
version of MaxMinPairwise Support (before the uncovering modification):
If the winner W of this method is ranked top on ballot B, and the winner
changes when F is moved to equal top with W on ballot B, then the new
winner must be F.
Example:
48 C
28 A>B
24 B
The MaxMinPS finish order is C>A>B, and C is already uncovered.
Comments?
-Forest
On Sat, Jan 14, 2023, 12:53 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:
> Very Good!
>
> I have another more elaborate application of max gradient in mind, but I
> was hoping that some choice of seed would be good enough for a simple
> stand-alone max gradient.
>
> Too bad Brenham's Gross Loser Elimination is not monotone.
>
>
>
>
>
>
> On Fri, Jan 13, 2023, 5:39 AM Kevin Venzke <stepjak at yahoo.fr> wrote:
>
>> Hi Forest,
>>
>> Let's tackle this question first:
>>
>> > Also how does DSC do with regard to Chicken Defense?
>>
>> I'm glad you asked. DSC is a ""great"" CD method, maybe the only one.
>>
>> I have complained before that the CD criterion allows the larger faction
>> in a fragmented
>> majority to truncate without issue. That seems like a problem both
>> philosophically (i.e.,
>> what is the significance of faction size in a chicken game?) and also
>> practically (i.e. a
>> faction might truncate out of an erroneous belief that their faction is
>> the large one).
>>
>> Consider this election:
>>
>> 40 A
>> 35 B
>> 25 C>B
>>
>> IRV and Condorcet methods go soft and let B win. Maybe B was driving a
>> truck and C was
>> driving a car. But with DSC justice is blind. It stands its ground and
>> elects A, handing
>> the fragmented majority a well-earned punishment!
>>
>> And it goes much further:
>>
>> 100 A>B
>> 99 B
>> 98 C>B
>> 97 D>B
>> ... etc
>> 50 Z>B
>>
>> DSC will still elect A.
>>
>> > I wonder how this DSC-with-max-gradient-finisher would do:
>> >
>> > Initialize X as the DSC winner. Then ...
>>
>> I tried this with a few methods (DSC, DAC, FPP, approval) as I felt
>> unsure you really meant
>> to pick DSC for this. However, I seem to find that generally, no matter
>> the seed method,
>> this approach is violating monotonicity. Maybe this is creating ways to
>> rig the initial
>> chain head.
>>
>> > While X is covered, replace X with the candidate X' with the strongest
>> defeat against X
>> > among those candidates that cover X.
>> >
>> > Elect the last value of X... i.e. the first X that turns out to be
>> uncovered.
>> >
>> > Does any of the DSC burial resistance flavor make it through this
>> afterburner?
>>
>> DSC isn't particularly burial-resistant. Its significance is in lacking
>> truncation
>> incentive. But since it likes to agree with FPP, that might moderate the
>> burial issue.
>>
>> As far as simulations:
>>
>> Seeding with FPP or DSC made it a lot worse with minimal defense, so I
>> can't say I like
>> those ones.
>>
>> Some best-to-worst rankings for each seed type, and also C//A, with four
>> candidates:
>>
>> Compromise:
>> DAC > Approval > C//A > Gross score > DSC > FPP
>>
>> Truncation:
>> FPP > DSC > DAC > Gross score > Approval > C//A
>>
>> Burial:
>> C//A > FPP > Approval > DAC > DSC > Gross score
>>
>> For burial, FPP is the best seed probably because you can't manipulate
>> the initial chain
>> head through adjusting your lower preferences.
>>
>> In general it seems like the covering rule will introduce burial, because
>> it gives voters
>> some levers to indirectly attack a potential chain head.
>>
>> Kevin
>> votingmethods.net
>>
>
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