[EM] Summary (and tweaking) of recent PR ideas inspired by Andy Jennings the two winner approval posting

[Forest Simmons]

I have tweaked things slightly in order to make things slightly simpler and
to make sure that when Range style ballots are used the method reduces to
standard Range in the one winner case.

The general method takes any proportional lottery based on score/range
(including approval) style ballots and converts it into a PR election
method.

Recall that a lottery method is a system of assigning probabilities to
candidates.  A lottery method is "proportional" if it assigns probabilities
in proportion to the respective faction sizes when all faction members vote
with single-minded and exclusive loyalty in favor of their favorite.

Let's suppose that we are in the context of an election where w>0
candidates are to be elected, and that there is at least one subset of
candidates W of size w, such that when our lottery method is applied to W,
all members of W are assigned positive probabilities.

If there is only one such subset, then that is the subset selected by our
method.

Otherwise, for each subset W of candidates of the requisite size w, we do
the following steps:

(1) Partition the ballots into sets S and S' that do and don't,
respectively, give at least one candidate of W a positive rating.

(2) Let p be the probability that a randomly drawn ballot would be a member
of the set S.  Let q = 1 - p. [Note this is a change; the definitions of p
and q have been switched, for esthetic reasons.]

(3) Let p1, p2, ... be the respective probabilities assigned to the members
of W by our lottery method, when that method is restricted to the ballot
set S.

(4) Let a1, a2, ... be the respective averages of the candidate ballot
scores over the respective ballots that rate the respective candidates
positively. [In case of approval, all of these averages will be ones]

Having completed these four steps for each candidate set W of size w, elect
the set W that maximizes the value of the expression

min(a1*p1, a2*p2, ...)*(p),

Now you I will show you the reason for the tweaks:  suppose that w=1.  Then
the value of p1 is 1, and the value of a1 is the average rating of W's only
candidate over the ballots in the set S,  The average rating of W's member
over S' is zero, so the average score over the union of S and S' is the
weighted average  a1*p1*p + 0*q, which simplifies to a1*p1*p  .  Therefore
in the case w=1, the standard range winner wins!

We have mentioned several of the various random ballot lotteries.  More
variations are possible.  Ordinal ballots can be used via the "Implicit
approval cutoff," or with the help of an explicit one.  If the lottery is
random nallot the only step needing an approval measure is step (4) above.
There are other possible ways to adapt to ordinal ballots (ranked
preference ballots).

There are other proportional lotteries besides the random ballot ones.  One
is the so called Ultimate Lottery, a restricted version of which is called
the Nash Lottery.  The Nash Lottery Method picks the lottery that maximizes
the product of the ballot expectations based on the lottery.  It turns out
that the crucial factor that makes this lottery proportional is the
"homogeneity" of the ballot expectations in the probabilities.  So if we
widen the admissible ballots to include any homogeneous functions of the
probabilities (along with the natural requirement that such functions not
be decreasing in any of their arguments), the we get the Ultimate Lottery
Method.

Jobst Heitzig has come up with many lotteries with special attention to
those with low entropy, which makes those lotteries useful in single winner
elections.  Why would we want to use a lottery in single winner elections?
It turns out that the element of chance, when skillfully incorporated, can
(more or less) remove incentives for insincere voting.  The best known
example of this is the "benchmark" standard random ballot lottery.  In that
method there is no incentive for insincere voting, but the resulting
lotteries tend to be high entropy lotteries, hence not good for single
winner elections.

One of Jobst's best methods has two stages.

The first stage generates a set of approval ballots from "thresh-hold"
information supplied by the voters on their ballots.  I won't go into the
details of this stage, but the voters give tentative approvals that allow
the method to automatically build approval ballots that would back-fire on
defectors.

The second stage calculates the total approvals for the respective
candidates and assigns each ballot B to the candidate with the greatest
total approval of any candidate approved on ballot B.  The lottery
probabilities are proportional to the number of ballots assigned to each
candidate.

For our multi-winner purposes we can replace the first stage with direct
use of approval ballots or by conversion of range ballots into approval
ballots via Toby's idea or by other ideas, including the "sincere approval
strategy" technique, for example.

Where else can we find suitable proportional lotteries?

Andy Jennings found that he could take any sequential PR method and convert
it into a proportional lottery by cloning all of the candidates as many
times as needed, and allowing a huge number of winners.  Then the candidate
lottery probabilities are proportional to the number of clones they have in
the winning circle.

In fact it turned out that Andy;s first application of that technique
generated the Nash Lottery by use of sequential PAV adapted to range
ballots.

More recently, Andy and I have seen that there are many ways to convert
practlcally any single winner method into a sequential PR method.

Putting all of this together we have the following diagram:

single winner -> sequential PR -> Lottery -> multi-winner PR

So in a standard way we can convert any single winner method into a
multi-winner PR method that retains some of the flavor of the single winner
method.  Why not just stop at the sequential PR stage?  That's always a
possibility, but sequential methods tend to generate an hierarchy among the
winners in order of election.  It is sometimes better philosophically to
have a method that compares (many if not all) possible winning sets
(including the winning sets generated by various sequential PR methods)
against each other. In this regard, Jameson has just invented (and posted
the EM list) a method that compares methods on the basis of sum of squared
envy.


That's it for now ...

On Thu, Dec 3, 2015 at 2:55 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

> Now I just have time to point out that in place of the random ballot
> lottery we could use any proportional lottery, such as PAV or The Ultimate
> Lottery.
>
> More on next time ...
>
> 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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