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

[Ted Stern]

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.

[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.

[Toby Pereira]

I'm not sure that passing the Condorcet criterion is a primary aim of these methods, otherwise we might as well propose using one of the many Condorcet methods on offer. I think the main aim of methods like STAR and 3-2-1 voting is to maximise utility, without so much concern about strictly passing particular criteria in all cases.

In any case, there are situations where I think passing the Condorcet criterion is not a good thing. The following example has 3 candidates, 100 voters and score ballots with a maximum score of 10.

49 voters: A=10, B=0, C=1

49 voters: A=0, B=10, C=1

2 voters: A=1, B=0, C=10

C is the Condorcet winner, but a very weak winner. There are two main candidates (A and B) that polarise support, and a non-entity (C). C wins because the 98 A/B supporters decide to put C (marginally) above the main candidate they dislike. They might not even know anything about C.

This is a fairly extreme example, but it shows what can happen when you have two main candidates and a non-entity.

I think using score voting and then finding the top two candidates proportionally is a good idea. I would do this sequentially though, since we are ultimately looking for a single winner, not a two-person committee, and it would be strange to eliminate the top scoring candidate, which is what could happen with non-sequential election.

This method would still not be cloneproof as it stands, but I would solve this not by trying to eliminate the effect of clones, but by embracing clones. I would effectively clone each candidate. If the top two candidates in the sequential proportional election are a candidate and their clone, then that candidate would automatically be elected without the need for a top two head-to-head. If a candidate has that much of a lead over the others in a score ballot, then I don't think the head-to-head would be necessary, or even desirable. It would help eliminate the possibility of very weak winners, as could happen in my example above under some methods.

Toby

[Ted Stern]

This is an interesting example.

In what I'm calling the first round (equivalent to STAR), the top two scores are for A and B.

In the second round after deweighting top scorer A, the round 2 top two are B and C.

B is both the 2nd place round 1 score, and the round 2 score winner.

So maybe what this shows is that in such a case, in which we see that we don't need cloneproofing, we should just go with the top two.

In other words, only use full APPR if the round 1 runner up is not the round 2 top score.

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[Ted Stern]

Toby, you bring up a valid point.  There are indeed pathological cases in which the Condorcet winner does not have broad support.

My intention in introducing Automatic Primary as an adjunct method was to address the main problem possessed by top two or top N methods (such as SRV/STAR or 3-2-1), that the results are distorted by crowding.

For example, in 3-2-1, there might be a strong minority faction united behind a crowded slate of candidates, while the majority faction has a contentious chicken dilemma, effectively splitting its first place votes beyond the top three.  The same could be true in top two methods using approval, score or majority judgment.

I was hoping that some form of Condorcet compliance would be a side benefit of my earlier proposed automatic primary technique, but you have pointed out a situation in which that compliance could actually lead to a result with lower social utility or higher variance.

So I have thought of a further simplification that focuses solely on the crowding/splitting problem:

For any top two or top N method, run two different versions:  one with the original method, and another with automatic primary.  Each method will have a pairwise winner.  If the original top N method winner differs from the winner of the automatic primary pairwise runoff, the overall APPR winner is the pairwise preferred between those two candidates.

Automatic primary means:  for a particular ratings method, find the top scoring candidate  Then deweight ballots according to the degree that each ballot supports the ratings winner, and calculate the total rating winner using the remaining ballots or fraction of a ballot.  If using a top-3 method, apply another round of deweighting on those remaining ballots and find the next ratings winner.

In top two approval:

• Use a rating ballot with an approval cutoff, e.g. 5,4,3 = approved; 2,1,0 = disapproved; infer rankings from ratings.  Equal rating implies no pairwise preference vote.
• Find candidates A0 (highest approved) and A1 (approval runner-up).
• Drop all ballots that approve of A0
• Find the candidate with highest approval among ballots that don't approve of A0.  Call this candidate B.
• Do pairwise comparisons A0 vs. A1 (top two) and A0 vs. B (auto primary).  If the top two pairwise winner is different from the auto primary pairwise winner, the APPR winner is the most preferred the two.

In top two score [Score Runoff Voting (SRV) AKA Score + Automatic Runoff (STAR)]:

• Use a ratings ballot, infer rankings from ratings.  Equal rating implies no pairwise preference vote.
• Find candidates A0 (highest total score) and A1 (runner-up total score).
• For each ballot that gives A0 a score less than max-score, give each candidate their original score times (max_score - A0_score).  With scaling, this is the equivalent of removing the total score fraction for A0 from the original ballots.
• The automatic primary runner-up is the candidate with highest total score among these remaining candidates.  Call this candidate B.
• Proceed as before for APPR-Approval

One can proceed similarly for APPR-Majority Judgment.

It is interesting that the same idea APPR idea could also be applied to 3-2-1:

• Use a 3-2-1 3-slot ballot:  Good, OK, Reject.
• Find the total votes for each score level for each candidate.
• Find the top three candidates by total Good rating, A0, A1, A2.
• Drop the candidate with the most Reject scores.
• The 3-2-1 winner is the pairwise preferred between the remaining 2 candidates.  Call that winner A*.
• Find the automatic primary runner-up and second runner-up by first counting Good ratings on only those ballots that Reject A0.  The Good vote winner on those ballots is B1.
• Then count Good ratings on only those ballots that reject both A0 and B1.  The Good vote winner on those ballots is B2.
• Apply 3-2-1 on A0, B1, and B2.  Call that winner B*.
• If A* is not the same candidate as B*, the APPR-3-2-1 winner is the the pairwise preferred between A* and B*.

As with top-N methods, APPR applies another level of stabilization to the original ratings method.

 

On Tue, Oct 10, 2017 at 5:36 AM, 'Toby Pereira' via The Center for Election Science <electio...@googlegroups.com> wrote:

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