[EM] Approval ballots. Two to elect. Best method? (Andy Jennings)

[Forest Simmons]

Continuing as promised...

Now let's allow range ballots with greater resolution than mere approval
ballots.

For each w-tuple of candidates we ...

(1) first divide the ballots into two sets, the set S of ballots that have
positive support for at least one candidate of our w-tuple, and the
complementary set S' of ballots that rate every member of our w-tuple at
zero.

(2) Then we imagine an experiment of drawing a ballot B at random from the
entire set of ballots.  Let p0 be the probability that B is a member of S',
and let p1, p2, ... pw be the respective probabilities for the choices of
the respective members of our w_tuple given that B is a member of S.
[Multiplication by (1-p0) would give the respective unconditioned
probabilities.]

(3)  Let a1, a2, ... be the respective candidate score averages for ballots
in S.

(4)  elect the w-tuple with the greatest value of the expression

min(a1*p1, a2*p2, ...) - p0.

There are at least three good ways of defining the details of the random
ballot experiment.

(1)  The value p3, say, is the probability that candidate 3 would be the
highest rated candidate (from our w-tuple) on a randomly drawn ballot from
the set S.  If several candidates are tied in this respect, divide up the
probability (from Ballot B) equally among them..

(2)  The positive scores on every ballot B are normalized with respect to
the members of our w-tuple.  These normalized scores are averaged over S to
get the respective "random ballot" probabilities.

(3)  Use a Toby Pereira transformation to convert each range style ballot
into one hundred approval ballots.  Then use a random approval ballot
lottery on that set. I'll explain this in my next post ...

On Thu, Dec 3, 2015 at 1:44 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

> Continuing as promised ..
>
> Suppose that there are to be w winners, and the ballots are still approval
> style:
>
> For each w_tuple of candidates, let p1, p2, ... be the respective
> probabilities of selection of the respective candidate by random ballot
> (restricted to the w-tuple), and let p0 be the probability that a random
> ballot would not approve any of the candidates in the w-tuple.
>
> Elect the w-tuple with the largest value of  min(p1, p2, ...) - p0.
>
> To Be Continued ...
>
>
> On Thu, Dec 3, 2015 at 1:30 PM, Forest Simmons <fsimmons at pcc.edu> wrote:
>
>>
>>
>> This query has lead me to some interesting ideas:
>>
>> Approval Ballots.  Two to elect:
>>
>> For each pair of candidates {X1, X2}, let p1 and p2 be the respective
>> probabilities that X1 or X2 would be selected by a random approval ballot
>> drawing (restricted to our pair of candidates), and let p0 be the
>> probability that a random ballot would approve neither X1 nor X2.  Elect
>> the pair with the greatest value of min(p1, p2) - p0.
>>
>> This actually gives two methods, since there are two natural ways of
>> selecting a candidate by random approval ballots.
>>
>> The first way is to select ballots at random until the approval set for
>> one of them has non-empty intersection with the set from which we are to
>> select a winner.  The names of the candidates are drawn randomly from a
>> hat.  The first name drawn of a candidate in the intersection set is the
>> name of the winner.
>>
>> The second way starts out as above, but once the first non-empty
>> intersection set is determined, additional ballots are drawn as needed to
>> narrow down the intersection to one candidate, the winner.
>>
>> More later ...
>>
>> From: Forest Simmons <fsimmons at pcc.edu>
>>> To: EM <election-methods at lists.electorama.com>,         Andy Jennings
>>>         <elections at jenningsstory.com>
>>>
>>>
>>> How about thie following ideas?
>>>
>>> Elect the pair that covers the most voters (i.e. that leaves the fewest
>>> voters with nobody that they approved elected).  In case of ties, among
>>> tied pairs elect the one whose weaker member has the most approval.
>>>
>>> Or this variant:  If no pair covers more than 70 percent of the voters,
>>> elect the pair that covers the greatest number of voters.  Otherwise
>>> consider all pairs that cover at least 70 percent of the voters to be
>>> tied.  Then among tied pairs, elect the one whose weaker member has the
>>> greatest approval.
>>>
>>>
>>> From: Andy Jennings <elections at jenningsstory.com>
>>> > To: Election Methods <election-methods at electorama.com>
>>> > Subject: [EM] Approval ballots. Two to elect. Best method?
>>> >
>>> > SPAV?
>>> > 1. Candidate with most approvals wins.
>>> > 2. That candidate's voters have their voting weight halved (or
>>> multiplied
>>> > by 1/3).
>>> > 3. Remaining candidate with most points wins.
>>> >
>>> > STV-like?
>>> > 1. Choose quota Q = one-third (or one half) of voters.
>>> > 2. Candidate with most approvals wins.  (T = # of approvals)
>>> > 3. That candidate's voters have their voting weight multiplied by
>>> > max(1-(Q/T), 0)
>>> > 4. Remaining candidate with most points wins.
>>> >
>>> > Monroe-like?
>>> > 1. For each pair of candidates, find the voter-assignment which
>>> maximizes
>>> > the number of voters assigned to a candidate they approved, such that
>>> no
>>> > more than half the voters are assigned to one candidate.
>>> > 2. Elect the pair which satisfies the most voters.
>>> >
>>> > Others?  Toby, what are your favorite PR methods at the moment?  Can
>>> you
>>> > give a short explanation of how Phragmen/Ebert would work with only
>>> two to
>>> > elect?
>>> >
>>> >
>>> >
>>> > Specifically, I'm worried that in practically every approval-ballot PR
>>> > method, if there is a candidate you really like, but are sure that she
>>> can
>>> > get elected without your vote, you gain an advantage by not approving
>>> > them.  Is there any method that minimizes that incentive?
>>> >
>>> > ~ Andy
>>> >
>>>
>>
>
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