<div dir="ltr">Continuing my search for a summable voting method that discourages burial and defection, I've come across this hybrid of Condorcet and median ratings that acts like Smith/Approval with an automatic approval cutoff. I'm calling it Pairwise Median Rating (PMR), but it could also be described as Smith//MR//Pairwise//MRScore:<br><ol><li>Equal Ranking and ranking gap allowed (essentially a ratings method with rank inferred). For purposes of this discussion, assume 6 slots (5 ranks above rejection).</li><li>In rank notation for this method, '>>' refers to a gap. So 'A >> B' means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap of two slots: 'A>>>B' means A is top ranked while B is in 4th place.</li><li>[Smith] </li><ol><li>Compute the pairwise preference array</li><li>The winner is the candidate who defeats each other candidate pairwise.</li><li>Otherwise, drop ballots that don't contain ranks above last for any member of the Smith Set.</li></ol><li>[Median Rating] </li><ol><li>Set the MR threshold to top rank.</li><li>While no Smith candidate has a majority of undropped ballots at or above the threshold, set the threshold to the next lower rank, until there is no lower rank. </li><li>The winner is the single candidate that has a majority of undropped ballots at or above the threshold.</li></ol><li>[Pairwise]<br></li><ol><li>Otherwise, if more than one candidate passes the threshold, look for a pairwise beats-all candidate among candidates meeting the MR threshold. (i.e. Condorcet on just the MR threshold set).</li><li> If there is one, you have a winner.</li></ol><li>[MR Score]<br></li><ol><li>Otherwise, the winner is the Smith set candidate with the largest number of ballots at or above the Median Rating threshold (their MRscore).</li></ol></ol><div>This method is essentially Smith//Approval(explicit) with the approval cutoff automatically inferred via median ratings<br><br>Step Smith.3, dropping non-Smith-candidate-voting ballots, could be considered optional, but by doing that, you ensure Immunity from Irrelevant Ballots (IIB), aka the zero ballot problem that affects other Median Rating / Majority Judgment methods. In other words, the majority threshold is unaffected by ballots that do not rank a viable candidate. It is possible to do this summably if need be.</div><div><br></div><div>PMR either passes the Chicken Dilemma criterion without adjustment, or there is a downranking strategy for defending against defection.<br><br>Consider the following examples from Chris Benham's post re MinLV(erw) Sorted Margins (<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html">http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html</a>):<br><pre style="color:rgb(0,0,0)">><i> 46 A>B
</i>><i> 44 B>C (sincere is B or B>A)
</i>><i> 05 C>A
</i>><i> 05 C>B
</i>><i>
</i>><i> A>B 51-49, B>C 90-10, C>A 54-46.
</i></pre>With sincere ballots, A is the Condorcet Winner (CW). With B's defection, there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR threshold is 2nd place, and A and B both pass the threshold. A defeats B, so A is the winner and B's defection/burial fails.<br><pre style="color:rgb(0,0,0)">><i> 25 A>B
</i>><i> 26 B>C
</i>><i> 23 C>A
</i>><i> 26 C
</i>><i>
</i>><i> C>A 75-25, A>B 48-26, B>C 51-49</i></pre><pre style="color:rgb(0,0,0)"><font face="arial, sans-serif">C wins with PMR (MR threshold is first place). B would win with most other Condorcet methods.</font></pre><pre style="color:rgb(0,0,0)"><pre style="text-wrap: wrap;">><i> 35 A
</i>><i> 10 A=B
</i>><i> 30 B>C (sincere B > A)
</i>><i> 25 C
</i>><i>
</i>><i> C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
</i><font face="arial, sans-serif">A wins with sincere voting. When B defects to try to win, which it would do with most other Condorcet methods, B wins. With PMR, C wins, an undesirable outcome for B.</font></pre><pre style="text-wrap: wrap;"><font face="arial, sans-serif">Here is another example from Rob LeGrand (<a href="https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html">https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html</a>). It's not a good example for chicken dilemma resistance, but it does demonstrate differences from Schulze, MMPO, RP and Bucklin:</font></pre><pre style="text-wrap: wrap;"><table border="0" cellpadding="0" cellspacing="0" style="font-family:Tinos;font-size:medium"><tbody><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"># example from method description page</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 98:Abby>Cora>Erin>Dave>Brad</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 64:Brad>Abby>Erin>Cora>Dave</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 12:Brad>Abby>Erin>Dave>Cora</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 98:Brad>Erin>Abby>Cora>Dave</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 13:Brad>Erin>Abby>Dave>Cora</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace">125:Brad>Erin>Dave>Abby>Cora</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace">124:Cora>Abby>Erin>Dave>Brad</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 76:Cora>Erin>Abby>Dave>Brad</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 21:Dave>Abby>Brad>Erin>Cora</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 30:Dave>Brad>Abby>Erin>Cora</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 98:Dave>Brad>Erin>Cora>Abby</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace">139:Dave>Cora>Abby>Brad>Erin</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 23:Dave>Cora>Brad>Abby>Erin</span></td></tr></tbody></table><p style="font-family:Tinos;font-size:medium">The pairwise matrix:</p><p style="font-family:Tinos;font-size:medium"></p><table border="" cellpadding="3"><tbody><tr align="center"><td colspan="2" rowspan="2"></td><th colspan="5">against</th></tr><tr align="center"><td class="gmail-against" style="background-color:rgb(208,192,200)"><span class="gmail-cand">Abby</span></td><td class="gmail-against" style="background-color:rgb(208,192,200)"><span class="gmail-cand">Brad</span></td><td class="gmail-against" style="background-color:rgb(208,192,200)"><span class="gmail-cand">Cora</span></td><td class="gmail-against" style="background-color:rgb(208,192,200)"><span class="gmail-cand">Dave</span></td><td class="gmail-against" style="background-color:rgb(208,192,200)"><span class="gmail-cand">Erin</span></td></tr><tr align="center"><th rowspan="5">for</th><td class="gmail-for" style="background-color:rgb(200,208,192)"><span class="gmail-cand">Abby</span></td><td></td><td class="gmail-loss" style="background-color:rgb(208,192,200)">458</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">461</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">485</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">511</td></tr><tr align="center"><td class="gmail-for" style="background-color:rgb(200,208,192)"><span class="gmail-cand">Brad</span></td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">463</td><td></td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">461</td><td class="gmail-loss" style="background-color:rgb(208,192,200)">312</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">623</td></tr><tr align="center"><td class="gmail-for" style="background-color:rgb(200,208,192)"><span class="gmail-cand">Cora</span></td><td class="gmail-loss" style="background-color:rgb(208,192,200)">460</td><td class="gmail-loss" style="background-color:rgb(208,192,200)">460</td><td></td><td class="gmail-loss" style="background-color:rgb(208,192,200)">460</td><td class="gmail-loss" style="background-color:rgb(208,192,200)">460</td></tr><tr align="center"><td class="gmail-for" style="background-color:rgb(200,208,192)"><span class="gmail-cand">Dave</span></td><td class="gmail-loss" style="background-color:rgb(208,192,200)">436</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">609</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">461</td><td></td><td class="gmail-loss" style="background-color:rgb(208,192,200)">311</td></tr><tr align="center"><td class="gmail-for" style="background-color:rgb(200,208,192)"><span class="gmail-cand">Erin</span></td><td class="gmail-loss" style="background-color:rgb(208,192,200)">410</td><td class="gmail-loss" style="background-color:rgb(208,192,200)">298</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">461</td><td class="gmail-win" style="background-color:rgb(200,208,192);font-weight:bold">610</td><td></td></tr></tbody></table><p></p><p style="font-family:Tinos;font-size:medium">There is no Condorcet winner. The Smith set is {<span class="gmail-cand" style="font-family:monospace">Abby</span>, <span class="gmail-cand" style="font-family:monospace">Brad</span>, <span class="gmail-cand" style="font-family:monospace">Dave</span>, <span class="gmail-cand" style="font-family:monospace">Erin</span>}.<br><br>Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin wins with Bucklin.<br><br>In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass the threshold. Brad defeats Abby and Erin to win. But Brad's threshold score of 484 is only slightly over the 50% mark of 460.5, so the Dave voters hold the balance of power. Dave defeats Brad pairwise, so Dave voters might not be as happy with a Brad victory, and Abby might be able to persuade Dave voters to downrank Brad but not Abby. If successful, Brad drops 44 points in MRScore and is no longer in the MR threshold set. Abby defeats Erin, so Abby wins.</p><pre style="text-wrap: wrap;"><table border="0" cellpadding="0" cellspacing="0" style="font-family:Tinos;font-size:medium"><tbody><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"><b> 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)</b></span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 30:Dave>Brad>Abby>Erin>Cora</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"> 98:Dave>Brad>Erin>Cora>Abby</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace">139:Dave>Cora>Abby>Brad>Erin</span></td></tr><tr align="left"><td><span class="gmail-cand" style="font-family:monospace"><b> 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)</b></span></td></tr></tbody></table></pre></pre><pre style="text-wrap: wrap;"><font face="arial, sans-serif">PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof. I believe it passes LNHelp. It probably fails Participation and IIA. There are probably weird examples where changing one vote changes the MR threshold. But overall, I think it has a good balance of incentive to deter burial and deliberate cycles.
Has Smith//Median Rating been proposed before? It seems like a simple modification to MR on its own.</font></pre></pre></div></div>