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

[Forest Simmons]

I'll try to explain the Toby transformation in the five minutes remaining
on my account:

Let ballot B be a range style ballot with scores in the range between zero
and 100 inclusive.

For each k between zero and 99 (including zero and 99) we form an approval
ballot B_k that approves all of the candidates rated above k by ballot B.

This gives us 100 approval ballots to use in place of ballot B.

Do this for every ballot B in the set of ballots, and give them equal
weight, say 1/100 if you are worried about the "one man one vote" fanatics.

To be continued ...

On Thu, Dec 3, 2015 at 2:42 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

> 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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