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<div dir="ltr" data-setdir="false">As an example of a cycle inside a set of ballots where there is a Condorcet winner:</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">3: A>B>C</div><div dir="ltr" data-setdir="false">2: B>C>A</div><div dir="ltr" data-setdir="false">2: C>A>B</div><div dir="ltr" data-setdir="false">2: B>A>C</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">A is the Condorcet winner (beating both others 5-4), so there's no cycle in that sense. But you can remove the following cycle within the ballots:</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">2: A>B>C</div><div dir="ltr" data-setdir="false">2: B>C>A</div><div dir="ltr" data-setdir="false">2: C>A>B</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">This leaves:</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">1: A>B>C</div><div dir="ltr" data-setdir="false">2: B>A>C</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">B is now the Condorcet winner (and the Borda winner in the original set of ballots).</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Toby</div><div><br></div>
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On Monday, 18 March 2024 at 06:42:46 GMT, Closed Limelike Curves <closed.limelike.curves@gmail.com> wrote:
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<div><div id="ydp7a499decyiv8382822640"><div><div><div><div><div><div><i><span style="font-size: 16px; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; word-spacing: 1px; text-decoration: none; float: none; color: rgb(49, 49, 49); display: inline !important;">Only if there is a cycle involved. Either the election was in a cycle in the first place or if the burial strategy is to push the election into a cycle.</span><br clear="none" style="color: rgb(49, 49, 49); font-size: 16px; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; word-spacing: 1px; text-decoration: none;"><br clear="none" style="color: rgb(49, 49, 49); font-size: 16px; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; word-spacing: 1px; text-decoration: none;"><span style="font-size: 16px; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; word-spacing: 1px; text-decoration: none; float: none; color: rgb(49, 49, 49); display: inline !important;">If there never ever were any cycles, then Condorcet does not fail.</span></i></div><br clear="none"></div></div></div></div><div><div><div><div>This is like saying that if there never were any odd numbers, 3 would be an even number. If Condorcet cycles were mathematically impossible, that would be great. Sadly, they aren't, so we have to deal with them. </div><div><br clear="none"></div><div>For strategy, the mathematical *possibility* of a cycle--not its actual occurrence in a set of results--is what matters, because threatening to create a cycle is what forces a Condorcet winner's supporters to vote strategically.</div><div><br clear="none"></div><div>The system you're proposing--where we make cycles impossible--is Borda, not Condorcet. Saari shows that if you delete every subset of ballots that forms a cycle, then elect the Condorcet winner, you get Borda. If Condorcet is "strategyproof unless cycles are involved," then Borda is *always* strategy proof, because it's got none of them--but clearly it's not, and the reason why is because what matters is whether you *could*, hypothetically, have a cycle, not whether you actually *do*.</div><div><br clear="none"></div><div>In reality, cycles are always involved. Sometimes it's a real cycle. Sometimes it's a theoretical cycle that someone *could* create, and that threat gives them leverage.<br clear="none"></div><div><br clear="none"></div><div id="ydp7a499decyiv8382822640yqtfd98809" class="ydp7a499decyiv8382822640yqt7385802001"><div><br></div></div></div></div></div></div></div></div>
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