<div dir="ltr">As an example of APPR-Approval, consider the following ballots:<div><br></div><div><pre style="font-family:monospace,Courier;color:black;background-color:rgb(249,249,249);border:1px solid rgb(221,221,221);padding:1em;white-space:pre-wrap;line-height:1.3em;font-size:14px"> 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</pre></div><div>(modified from an example due to Rob LeGrand that is cited here: <a href="http://wiki.electorama.com/wiki/Definite_Majority_Choice">http://wiki.electorama.com/wiki/Definite_Majority_Choice</a>)<br><br>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.<br><br>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.<br><br>Compare with <a href="http://wiki.electorama.com/wiki/Marginal_Ranked_Approval_Voting">http://wiki.electorama.com/wiki/Marginal_Ranked_Approval_Voting</a>, which also finds the same result.</div></div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Sep 28, 2017 at 1:46 PM, Ted Stern <span dir="ltr"><<a href="mailto:dodecatheon@gmail.com" target="_blank">dodecatheon@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div>I am interested single-winner methods that find the variance-minimizing candidate, with resistance to strategic voting.</div><div><br></div><div dir="auto">Top two approval, STAR (top two score), and 3-2-1 voting, while all very good at resisting strategic voting, all fail clone resistance.<div dir="auto"><br></div><div dir="auto">When I raised the topic of top two approval on the EM list last November (<a href="http://election-methods.electorama.narkive.com/Vlwq75Zy/em-top-two-approval-pairwise-runoff-ttapr" target="_blank">http://election-methods.<wbr>electorama.narkive.com/<wbr>Vlwq75Zy/em-top-two-approval-<wbr>pairwise-runoff-ttapr</a>), it was suggested that using the ballots to approximate a two seat Proportional Representation style "parliament" would avoid the crowding effect of cloned candidates.</div><div dir="auto"><br></div><div>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. </div><div><br></div><div>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 <i>class</i> 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.</div><div><br></div><div>We can start with APPR-Approval, as in the cited thread above, since that is the easiest place to start.<br><br>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.</div><div><br></div><div><ul><li><i>Round 1</i>: Find the top two approved candidates, A (top score) and B (second-highest score).</li><li>Then drop <b>every</b> ballot that approves of A, and determine the new approval ratings for each candidate.</li><li><i>Round 2</i>: 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.</li><li><i>Candidates A and C are the Automatic Primary winners. They are the candidates to beat.</i></li><li>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.</li><li>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.</li></ul>As an example, assume B > A and B > C, but one of A or C defeats D. Then B wins. If <i>both</i> B and D defeat both A and C, the pairwise winner of B vs. D is the APPR.<br></div><div><br></div><div>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.</div><div><br></div><div>To win, the APPR is either the beats-all candidate among the 4, or has a beatpath through an Automatic Primary winner.</div><div><br></div><div><div>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.</div><div> </div><div>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.<br><br>Properties:<br><ul><li>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 <i>not</i> 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.<br></li><li>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.</li><li>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.</li><li>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.</li><li>Chicken-dilemma problems can be addressed via having rankings below the approval cutoff (see thread cited above).</li><li>Within each round, Favorite Betrayal Criterion (FBC) is satisfied through use of an FBC-compliant ratings method.</li><li>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.</li></ul><div>As described above, the particular method is <b>APPR-Approval</b>. But the APPR process could also be implemented with either Score or Majority Judgment in each round.</div></div></div><div><br></div><div>After thinking about this for a while, I have come to prefer <b>APPR-Score</b> 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.<br><br>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.<br><ul><li>Accumulate total scores (not averaged) for each candidate, counting blanks as zero scores.</li><li>Round 1: Find the score winner and runner up, A and B.</li><li>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.<br></li><li>C and D are determined from the Round 2 totals.</li></ul><div><b>APPR-MJ</b> 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.</div></div></div>
</div>
</blockquote></div><br></div>