# [EM] Favorite Betrayal and Condorcet

Forest Simmons forest.simmons21 at gmail.com
Sun Apr 17 17:21:52 PDT 2022

```It is well known that Range Voting, no matter its level of resolution, is
strategically equivalent to Approval. In particular, this means that under
perfect information conditions there always exists an optimal strategy that
makes no use of any intermediate ratings. [However, as in Linear
Programming, existence of an optimal "corner" solution in no way denies the
possible existence of other equally optimal non-corner solutions.]

Not so well known, but equally true, is that every Condorcet compliant,
Universal Domain (i.e. RCV) method reduces to  Approval when voters vote
only at the extremes.

Question 1. Does every perfect information UD Condorcet election have an
optimal strategy that makes no use of the intermediate rankings? This
certainly seems to be the tacit assumption of many Designated Strategy
Voting methods.

Question 2. Since Approval satisfies the Favorite Betrayal Criterion, does
it follow that any method that has an optimal strategy that makes no use of
the intermediate ranks in some sense satisfies the FBC? Could we call that
Strategic FBC?

And it seems possible that one DSV strategy for transforming a UD election
into an approval election might satisfy the FBC while another might not.

Suppose a DSV method M converts  UD elections into approval elections in a
CW preserving way, i.e. if X is the CW of some UD ballot set beta, then X
will also be the CW of the approval ballot set M(beta) and therefore the
approval winner. It seems like a voter voting through that DSV method M
would not be highly tempted to rank Favorite under Compromise, especially
if under M, candidates ranked top or equal top always get approved on
M(beta).

Here is an example of just such a DSV method M closely related to (but
better than) the flash in the pan method 2PFBCC:

First, for the ballot set beta, find and summarize the pairwise defeats and
ties in some convenient form.

Then convert each ballot B of beta into approval form by use of an
inclusive approval cutoff K defined by the lowest ranked candidate of B
that is not pairwise defeated by any candidate ranked ahead of it on ballot
B.

Since the inclusive cutoff is defined by a "ranked candidate" it cannot be
a truncated candidate. So a ballot that truncates all of the candidates
that are not ranked equal top makes approval the same as equal top. This
feature gives voters that don't trust M the ability to specify their own
approvals.

Note that if there is a CW it will define the cutoff on every ballot that
ranks it, a d since the cutoff is inclusive, the CW will be approved by
every ballot that ranks it.

It follows that the CW will be the approval winner, since it is ranked
(hence approved) above any rival on more ballots than not.

So here's my proposal ...

Given an RCV ballot set beta, elect the CW of beta if there is one, else
elect the CW of M(beta).

-Forest

El sáb., 16 de abr. de 2022 4:33 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> Kevin,
>
> Thanks for your clarifications, insights and insightful examples.
>
> -Forest
>
> El sáb., 16 de abr. de 2022 3:54 p. m., Kevin Venzke <stepjak at yahoo.fr>
> escribió:
>
>> Hi Forest,
>>
>> > It seems that my "proof" failed because I assumed that C's score could
>> not
>> > change by raising B equal to C ... but that's only true if we're talking
>> > majority defeat: raising B to equal with C  cannot change a defeat of B
>> by C (or
>> > a non-majority defeat of C by B) to a majority defeat of C by B.
>> >
>> > So let's try this fix:
>> >
>> > Elect the candidate that on the fewest ballots is outranked by any
>> candidate
>> > that majority defeats it.
>>
>> Stated like this, this does satisfy FBC, however it is not a Condorcet
>> method.
>>
>> If I dub this method MajBTP, MajBTP is very close to MDDA as you note. It
>> has
>> similar properties, including a Plurality failure risk with 4+
>> candidates. SFC
>> and SDSC/MD both seem to be preserved.
>>
>> It's pretty interesting that this works, and to consider how this resolves
>> differently from MDDA. One example:
>>
>> 40: A>B>C
>> 35: B>C>A
>> 25: C>A>B
>>
>> A>B>C>A majority cycle. MDDA elects B as the approval winner. MajBTP
>> effectively
>> picks the first pref winner A, as every second-ranked candidate has a maj
>> loss
>> to the first preference.
>>
>> > Or for lay person proposal completeness ...
>> >
>> > Lacking a Condorcet winner, elect the candidate that on the fewest
>> ballots is
>> > outranked by any candidate that outranks it on a majority of ballots.
>>
>> I might call this C//MajBTP. This can fail FBC in the same cases C//A
>> does:
>>
>> 0.394: C=A>B
>> 0.299: B=C>A  -->  B>A=C
>> 0.179: B>A>C
>> 0.126: A=B>C
>>
>> A>C>B>A cycle, no majorities. A wins on approval.
>>
>> When the .299 lower C, B becomes the CW.
>>
>> Kevin
>>
>
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