# [EM] MaxMinPA

Forest Simmons fsimmons at pcc.edu
Thu Oct 13 15:09:51 PDT 2016

```Here’s an example (with variations):

41 C

30 A>B

29 B

Candidate A’s pairwise approval is 30 relative to every candidate.

Candidate B’s smallest pairwise approval is 29 against candidate A.

Candidate  C’s pairwise approval is 41 relative to every candidate.

So the MaxMinPA winner is C.

Now suppose that we replace 41 C with 41 C>B:

This raises B’s pairwise approval relative to A from 29 to 70 so that B’s
new min PA is 59 (relative to C)

Now B is the MaxMinPA winner.

It appears that if 41 C>B is the sincere preference, then truncation to 41
C is a successful maneuver to change the MaxMinPA winner from B to C.

Q. What defense is there to preserve the win for the sincere CW?

Ans. The 30 A>B voters can raise B to unconditional approval A=B.  This
works because now there are only two occupied levels (top and bottom), in
which case MaxMinPA reduces to ordinary Approval.

What if (in the original preference profile) we replace 29 B with 29 B>A ?

Then A’s PA relative to C increases to 59 and A’s PA relative to itself
increases to 44.5, but its PA relative to B stays at 30, so its MinPA does
not increase, so C is elected rather than the CW.

This example merely shows that the method does not satisfy the Condorcet
Criterion.  If the Condorcet Criterion is more important than the FBC, then
we could go with Smith//MaxMinPA.

If we’re not willing to give up the FBC, then at least eleven members of
the 29 B>A faction should raise A to the top tier: 11 B=A, 18 B>A.  Then A
and C are tied for the MaxMinPA=41, but A’s other PA’s dominate C’s.

So again the CW is rescued by raising it to top on some of the ballots.

It may turn out that approval strategy is optimal for MaxMinPA.

Another idea: instead of MaxMinPA apply game theoretic principles to the PA
matrix defined as the matrix whose entry in row i and column j is the
pairwise approval of candidate i relative to candidate j.

In this example (assuming that candidates one through three are
respectively candidates A through C) the three rows are respectively  [44.5,
30, 59], [29, 44, 59], and  [41, 41, 41].

Note that the third entry of the last row is dominated by the third entries
of the other two rows.

From a game theoretic point of view this allows us to eliminate the third
candidate.

The submatrix that remains has respective rows [44.5, 30] and [29, 44].
Their respective min’s are 30 and 29, so the first candidate (A) wins.

So I’m not sure that MaxMinPA is the best way to make use of the PA matrix,
but it is one way that guarantees FBC compliance.

In any case I think we should not throw out the PA matrix, because I
believe that its entries are useful estimates of how much approval
candidate i would receive if the contest were largely between candidates i
and j.  This information should be useful in deciding who the over-all
sincere approval winner should be.

Forest

On Thu, Oct 13, 2016 at 9:30 AM, Michael Ossipoff <email9648742 at gmail.com>
wrote:

> I meant W is the offensively strategizing faction.
>
> Michael Ossipoff
> On Oct 12, 2016 2:17 PM, "Forest Simmons" <fsimmons at pcc.edu> wrote:
>
>> The following method is based on score or range style ballots.  I
>> believe it satisfies the FBC, Plurality, the CD, Monotonicity,
>> Participation,  Clone Independence, and the IPDA.  It reduces to
>> ordinary Approval when only the extreme ratings are used for all candidates.
>>
>>
>>
>> I call it MinMaxPairwiseApproval or MinMaxPA for short.
>>
>>
>>
>> It is based on a concept of “pairwise approval.”
>>
>>
>>
>> A zero to 100% cardinal ratings ballot contributes the following amount
>> to the “pairwise approval of candidate X relative to candidate Y”:
>>
>>
>>
>> The amount is either …
>>
>> 100% if X is rated strictly above Y, or
>>
>> Zero if X is rated strictly below Y, or
>>
>> Their common rating if they are rated equally.
>>
>>
>>
>> According to this definition, the ballot’s contribution to the pairwise
>> approval of X relative to itself is simply the ballot’s rating of X, since
>> it is rated equally with itself.
>>
>>
>>
>> The method elects the candidate whose minimum pairwise approval (relative
>> to all candidates including self) is maximal.
>>
>>
>>
>> The motivation for this idea is the question, “If candidates X and Y were
>> the only two candidates with any significant chance of winning the
>> election, what is the probability that the ratings ballot voter would want
>> X approved (in a Designated Strategy Voting system, say)?”
>>
>>
>>
>> If the voter rated X over Y, this probability would be 100 percent.
>>
>> If the voter rated Y over X, this probability would be zero.
>>
>> If the voter rated both X and Y at 100 percent, this probability would be
>> 100 percent.
>>
>> If the voter rated them both at zero, she would want neither of the
>> approved.
>>
>> If she rated them both at 50%, then our best guess is that there is a
>> fifty-fifty chance that she would approve X.
>>
>> Etc.
>>
>>
>>
>> Whatever nice properties the method has depends solely on its definition,
>> not the motivation for the definition, so please explore it with an open
>> mind.
>>
>>
>>
>> Tomorrow, when I have more time, I’ll give some examples.
>>
>>
>> Enjoy,
>>
>>
>> Forest
>>
>>
>> P.S.
>>
>>
>> The rules can be modified for ranked preference ballots:
>>
>>
>> The amount (per ballot) of approval of X relative to Y  is either ...
>>
>>
>> 100 percent if X is ranked ahead of Y or equal top with Y
>>
>> zero if Y is ranked ahead of X or equal bottom with X
>>
>> 50 percent if both are ranked equally and strictly between top and bottom.
>>
>> Smith//MaxMinPA may be a nice method that trades the FBC and possibly
>> other nice properties for the Condorcet Criterion.
>>
>>
>> ----
>> Election-Methods mailing list - see http://electorama.com/em for list
>> info
>>
>>
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