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<div dir="ltr" data-setdir="false">I think it's a bit of a myth that one can always clone one's way to victory in Borda. It requires co-ordinating your voters to put the candidate you want to win of the clones at the top while at the same time hoping that the opposition don't do the reverse.</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 Wednesday 16 April 2025 at 10:44:06 BST, Kristofer Munsterhjelm via Election-Methods <election-methods@lists.electorama.com> wrote:
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<div><div dir="ltr">The other day, I was thinking about how to define a method having exit <br></div><div dir="ltr">or entry incentives, and thus being vulnerable to strategic nomination.<br></div><div dir="ltr"><br></div><div dir="ltr">Strategic nomination incentives are subsets of IIA failures. For <br></div><div dir="ltr">single-candidate incentives, it seems pretty straightforward to define:<br></div><div dir="ltr"><br></div><div dir="ltr">Suppose we have a c-candidate election in a spatial model. Candidates <br></div><div dir="ltr">prefer those who are closer to themselves in the spatial model to those <br></div><div dir="ltr">further away, just like the voters do.<br></div><div dir="ltr"><br></div><div dir="ltr">Then, if there exists some candidate X (at some position in issue space) <br></div><div dir="ltr">who, by entering, could change the winner from A to B in the resulting <br></div><div dir="ltr">(c+1)-candidate election, and B is closer to X than A is, then there is <br></div><div dir="ltr">an entry incentive.<br></div><div dir="ltr"><br></div><div dir="ltr">And if there exists some current non-winning candidate C who, were he <br></div><div dir="ltr">not to run, could change the winner from A to B in the resulting <br></div><div dir="ltr">(c-1)-candidate election, and B is closer to C than A is, then there is <br></div><div dir="ltr">an exit incentive.<br></div><div dir="ltr"><br></div><div dir="ltr">So far so good.<br></div><div dir="ltr"><br></div><div dir="ltr">But when discussing general IIA failures, it's usually a good idea to <br></div><div dir="ltr">consider coalitions (groups of voters). This is what makes Condorcet <br></div><div dir="ltr">methods have no more IIA failures than non-Condorcet methods. If there <br></div><div dir="ltr">is a CW, eliminating anybody will still let that CW be the CW. On the <br></div><div dir="ltr">other hand, if there is a cycle and the winner is X, or if the method <br></div><div dir="ltr">fails to elect the Condorcet winner, we can eliminate everybody but X <br></div><div dir="ltr">and some Y with Y>X to demonstrate IIA failure.<br></div><div dir="ltr"><br></div><div dir="ltr">For exit incentive, this is still quite easy: if there exist a group of <br></div><div dir="ltr">candidates, all of whom prefer B to the current winner A, and <br></div><div dir="ltr">eliminating them all makes B win, then there's an exit incentive.<br></div><div dir="ltr"><br></div><div dir="ltr">But for entry incentive, things get much harder, and kind of weird, too. <br></div><div dir="ltr">The coalitional version would be "if there exist k points in opinion <br></div><div dir="ltr">space closer to B than A, and inserting a candidate at each of these <br></div><div dir="ltr">points makes B win, then there's an entry incentive".<br></div><div dir="ltr"><br></div><div dir="ltr">The latter is harder to calculate, but worse is that it produces extreme <br></div><div dir="ltr">results. For instance, in Borda, one can always "clone one's way to <br></div><div dir="ltr">victory". Any loser B can win just by inserting enough clones of himself <br></div><div dir="ltr">into the election. So Borda's coalitional entry incentive would be 100%, <br></div><div dir="ltr">all the time no matter the pre-cloning number of candidates.<br></div><div dir="ltr"><br></div><div dir="ltr">What do you think? Does Borda's 100% coalitional entry incentive suggest <br></div><div dir="ltr">that extending entry incentive to coalitions doesn't really work, or <br></div><div dir="ltr">does it just show that Borda is that bad?<br></div><div dir="ltr"><br></div><div dir="ltr">-km<br></div><div dir="ltr">----<br></div><div dir="ltr">Election-Methods mailing list - see <a href="https://electorama.com/em" rel="nofollow" target="_blank">https://electorama.com/em</a> for list info<br></div></div>
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