[EM] Automatic Primary + Pairwise Runoff (APPR), a class of cloneproof top-two-style methods

[Ted Stern]

As an example of APPR-Approval, consider the following ballots:

 98: Abby >  Cora >  Erin >> Dave > Brad
 64: Brad >  Abby >  Erin >> Cora > Dave
 12: Brad >  Abby >  Erin >> Dave > Cora
 98: Brad >  Erin >  Abby >> Cora > Dave
 13: Brad >  Erin >  Abby >> Dave > Cora
125: Brad >  Erin >> Dave >  Abby > Cora
124: Cora >  Abby >  Erin >> Dave > Brad
 76: Cora >  Erin >  Abby >> Dave > Brad
 21: Dave >  Abby >> Brad >  Erin > Cora
 30: Dave >> Brad >  Abby >  Erin > Cora
 98: Dave >  Brad >  Erin >> Cora > Abby
139: Dave >  Cora >  Abby >> Brad > Erin
 23: Dave >  Cora >> Brad >  Abby > Erin

(modified from an example due to Rob LeGrand that is cited here:
http://wiki.electorama.com/wiki/Definite_Majority_Choice)

In this case, the Round 1 winners are Erin and Abby, and the Round 2
winners are Dave and Abby.  The Automatic Primary winners are Erin and Dave.

Though Abby is never a round 1 or round 2 approval winner, she pairwise
defeats both automatic primary winners Erin and Dave, and is therefore the
APPR-Approval winner.  It can also be seen that Abby, though not the
highest approved candidate, is nevertheless rated highly by Erin and
non-Erin voters alike, and could be seen as the best compromise candidate
by both factions.  This should yield a high degree of satisfaction with the
results.

Compare with http://wiki.electorama.com/wiki/Marginal_Ranked_Approval_Voting,
which also finds the same result.

On Thu, Sep 28, 2017 at 1:46 PM, Ted Stern <dodecatheon at gmail.com> wrote:

> I am interested single-winner methods that find the variance-minimizing
> candidate, with resistance to strategic voting.
>
> Top two approval, STAR (top two score), and 3-2-1 voting, while all very
> good at resisting strategic voting, all fail clone resistance.
>
> When I raised the topic of top two approval on the EM list last November (
> http://election-methods.electorama.narkive.com/
> Vlwq75Zy/em-top-two-approval-pairwise-runoff-ttapr), it was suggested
> that using the ballots to approximate a two seat Proportional
> Representation style "parliament" would avoid the crowding effect of cloned
> candidates.
>
> There are several problems with this idea ... to start with, in a 3 person
> election, it fails the Condorcet criterion, which would be a minimal
> threshold for centrist approximating methods.  Another problem is that
> while picking the top two approved candidates is vulnerable to crowding,
> replacing the second-place winner with the second-seat parliament member
> means that there is no incentive for factions to cooperate, because doing
> so would lead to elimination from the second round.
>
> After playing around with this idea for a while, I think I've come up with
> a fairly straightforward modification.  I'm calling APPR a *class* of
> methods, since the initial candidate ranking can be based on any of several
> FBC-satisfying voter alignment metrics, such as Approval, Score, or
> Majority Judgment.
>
> We can start with APPR-Approval, as in the cited thread above, since that
> is the easiest place to start.
>
> Voters use a ratings ballot that is interpreted with ranking during
> tabulation.  I prefer a zero through 5 score rating, with scores 5, 4, 3,
> approved and 2, 1, 0, disapproved, but the actual implementation could vary
> as desired.
>
>
>    - *Round 1*: Find the top two approved candidates, A (top score) and B
>    (second-highest score).
>    - Then drop *every* ballot that approves of A, and determine the new
>    approval ratings for each candidate.
>    - *Round 2*: The top two approved candidates among these reweighted
>    ballots are C (top reweighted approval) and D (second-highest reweighted
>    approval).  NB: the reweighted approval totals can be accumulated summably
>    during the round 1 count.
>    - *Candidates A and C are the Automatic Primary winners.  They are the
>    candidates to beat.*
>    - From the original, non-reweighted, ballots, determine the pairwise
>    votes between candidates A, B, C, and D.  NB: the pairwise totals can be
>    accumulated summably during the round 1 count.
>    - To win, one of the four top-two winners from both rounds must defeat
>    all other Automatic Primary winners (i.e., A & C) pairwise.  If more than
>    one candidate satisfies this property, the pairwise preferred candidate is
>    the winner.
>
> As an example, assume B > A and B > C, but one of A or C defeats D.  Then
> B wins.  If *both* B and D defeat both A and C, the pairwise winner of B
> vs. D is the APPR.
>
> If B is defeated by either A or C, and D is defeated by either A or C, the
> APPR winner is the winner of A vs. C.
>
> To win, the APPR is either the beats-all candidate among the 4, or has a
> beatpath through an Automatic Primary winner.
>
> For 3 candidate elections, this is Condorcet compliant:  either B or D
> must be a repeat of one of A or C.  The winner is either the pairwise
> winner between A and C, and one of them defeats B/D, or B/D defeats both A
> and C pairwise.  In case of a Condorcet cycle, one of A or C must defeat
> B/D, so B/D is dropped, and the APPR winner is one of the two Automatic
> Primary winners, the victor of A vs. C.
>
> For 4 candidate elections, APPR is not strictly Condorcet, because it
> might be possible for B or D to overlap with A or C as before, and the
> fourth candidate left out might be preferred pairwise to the other three.
> But this is extremely unlikely except in highly fragmented elections with
> extremely low winning approvals.  When A through D are all distinct
> candidates, APPR is Condorcet compliant.
>
> Properties:
>
>    - The automatic primary avoids the both the splitting and the
>    clone/crowding problem, since the second round winner is chosen from only
>    those ballots that do *not* approve of A.  So the second round is
>    clearly from a different set of voters than those who would be crowded
>    around A.  Therefore, there is no advantage to be gained from crowding, but
>    no disadvantage either.
>    - Pushover is avoided because the automatic primary is based solely on
>    the highest approval winner, and it is not possible to engineer second
>    round top two placement for your favorite by approving one's weakest
>    opponent.
>    - By including the second-place approved candidates for each round, we
>    avoid the problem of eliminating the best representatives of strongly
>    aligned factions.  Consider a 2016-type situation:  Clinton wins round 1,
>    but after eliminating all Clinton-approving ballots, Trump wins round 2.
>    This is not a great choice for voters.  By including the runners-up, we get
>    to choose the most preferred of the candidates in each faction who defeat
>    both Clinton and Trump pairwise.  That is, if the Greens and Independents
>    partially aligned with Democrats, they are not penalized for that
>    alignment, and may in fact be rewarded for cooperation.
>    - Including the second-place candidates in each round adds a bit of
>    the flavor of 3-2-1 voting --- more than 2 factions can thus be considered.
>    - Chicken-dilemma problems can be addressed via having rankings below
>    the approval cutoff (see thread cited above).
>    - Within each round, Favorite Betrayal Criterion (FBC) is satisfied
>    through use of an FBC-compliant ratings method.
>    - While not Condorcet compliant for 4 or more candidates, APPR tends
>    to find the most preferred representatives of the two most preferred
>    disparate factional groups, and therefore should find the
>    variance-minimizing candidate most of the time.  I will be doing Yee-metric
>    tests on APPR to see just how well it performs in this respect.
>
> As described above, the particular method is *APPR-Approval*.  But the
> APPR process could also be implemented with either Score or Majority
> Judgment in each round.
>
> After thinking about this for a while, I have come to prefer *APPR-Score*
> due to its combination of expressiveness and its natural summable extension
> to the Automatic Primary part of the process.  I think that APPR-Score is
> the simplest way and most natural extension of STAR voting, without losing
> too much of STAR's simplicity.  Score based on total scores, instead of
> averages, also satisfies Participation and Immunity from irrelevant
> alternatives, in each round.
>
> I've described the Automatic Primary for score voting in other posts, but
> for clarity, I'm adding again here.  Assume a ratings ballot with range 0
> to 5.
>
>    - Accumulate total scores (not averaged) for each candidate, counting
>    blanks as zero scores.
>    - Round 1: Find the score winner and runner up, A and B.
>    - For each ballot that scores A above 0, accumulate scores of 5 minus
>    the ballot's A-score times the non-A score, for every other candidate on
>    the ballot.  So, for example, if the ballot scores A at 3, and candidate X
>    at 4, accumulate (5-3) * 4 = 8 points for X, and similarly for all other
>    non-zero scored candidates on the ballot.  Computationally, this preserves
>    exact integer arithmetic in the totals.  These totals are the Round 2
>    scores.  They can be converted into averages for reporting, if desired, by
>    dividing by the maximum score squared and the total number of ballots.
>    - C and D are determined from the Round 2 totals.
>
> *APPR-MJ* is similar to APPR-Approval in terms of dropping A-approving
> ballots to find the round 2 scores, but Majority Judgment is used in each
> round.  In the second round, the 50% level is determined by the number of
> remaining ballots instead of the original number of ballots.  There are
> some attractive aspects to this method, but they come at the cost of more
> complexity and unpractical summability.   Nevertheless, I would happily use
> this method if summabilty were not desirable.
>
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