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

Ted Stern dodecatheon at gmail.com
Thu Sep 28 13:46:10 PDT 2017

```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.

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
- 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.

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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