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<div>Gustav and Kristofer - I might have not thought that through properly. I was only considering removing candidates from one end of the pecking and not the other.</div><div><br></div><div>I also agree it can be of theoretical interest and that there may be even some cases where it is of practical use. But when I see it listed alongside other criteria as if it's an equal, I wonder how it made the list.</div><div><br></div><div>But regardless of practicalities, and just from a theoretical "good candidate" standpoint, if you have an A>B>C>A cycle and A is the winner, I don't have any intuition that tells me B should automatically be considered better than C, which LIIA would suggest is the case. (Obviously B beats C pairwise but we have a cycle.)</div><div><br></div><div>Toby</div><div><br></div>
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On Friday, 1 May 2026 at 19:09:49 BST, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:
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<div><div dir="ltr">On 2026-05-01 18:28, Toby Pereira via Election-Methods wrote:<br></div><div dir="ltr">> I'm not sure if this relates to your question at all, but any method can <br></div><div dir="ltr">> easily be converted to an LIIA-passing method, without changing the <br></div><div dir="ltr">> winner. Instead of using the method's "natural" finishing order, declare <br></div><div dir="ltr">> just the winner initially and then for 2nd place, remove the winner from <br></div><div dir="ltr">> the process and find the new winner and declare them to be 2nd, and so on.<br></div><div dir="ltr"><br></div><div dir="ltr">Is that true? Consider the definition as stated on Electowiki:<br></div><div dir="ltr"><br></div><div dir="ltr">>> LIIA requires that both of the following conditions always hold:<br></div><div dir="ltr"><br></div><div dir="ltr">>> If the option that finished in last place is deleted from all the<br></div><div dir="ltr">>> votes, then the order of finish of the remaining options must not<br></div><div dir="ltr">>> change. (The winner must not change.)<br></div><div dir="ltr">>> If the winning option is deleted from all the votes, the order of<br></div><div dir="ltr">>> finish of the remaining options must not change. (The option that<br></div><div dir="ltr">>> finished in second place must become the winner.)<br></div><div dir="ltr"><br></div><div dir="ltr">Suppose that you construct a method like the above, where the winner is <br></div><div dir="ltr">the winner of the original method, then the second-place candidate is <br></div><div dir="ltr">the winner with the original winner removed, etc. It is then not at all <br></div><div dir="ltr">clear that removing the loser (the candidate ranked last) will preserve <br></div><div dir="ltr">the ranking of the other candidates.<br></div><div dir="ltr"><br></div><div dir="ltr">Another reason that this seems wrong is, a method that satisfies <br></div><div dir="ltr">majority and LIIA must also satisfy Condorcet, and then Smith, and then <br></div><div dir="ltr">ISDA.<br></div><div dir="ltr"><br></div><div dir="ltr">It must satisfy the property that if A is ranked immediately ahead of B, <br></div><div dir="ltr">then A beats B pairwise; suppose otherwise, then eliminate all <br></div><div dir="ltr">candidates ranked below B. This shouldn't change anything. Then <br></div><div dir="ltr">eliminate all candidates ranked above A. This shouldn't change anything, <br></div><div dir="ltr">either. Then majority requires that A win.<br></div><div dir="ltr"><br></div><div dir="ltr"> From this, it satisfies Condorcet because suppose A is the CW but not <br></div><div dir="ltr">ranked first, then the candidate ranked above A must beat A pairwise, <br></div><div dir="ltr">which is a contradiction.<br></div><div dir="ltr"><br></div><div dir="ltr">Similar reasoning leads to Smith (since if not Smith, then someone in <br></div><div dir="ltr">the Smith set is ranked below someone not in it) and ISDA (because you <br></div><div dir="ltr">can eliminate everybody outside the Smith set as we've established the <br></div><div dir="ltr">Smith set must be ranked before any non-Smith candidate).<br></div><div dir="ltr"><br></div><div dir="ltr">But the given construction would let you make a "LIIA" method that ranks <br></div><div dir="ltr">any candidate first, even a Condorcet loser. Which doesn't seem right.<br></div><div dir="ltr"><br></div><div dir="ltr">> Going off on a tangent, I've always felt that LIIA has somehow found its <br></div><div dir="ltr">> way into the "standard list" of election method criteria without any <br></div><div dir="ltr">> proper scrutiny of its utility. It's not clear what purpose it serves. <br></div><div dir="ltr">> It sounds good because it has "IIA" in it, but it doesn't really have <br></div><div dir="ltr">> much, if anything, to do with the IIA criterion. It's certainly not a <br></div><div dir="ltr">> stepping stone towards it.<br></div><div dir="ltr">> <br></div><div dir="ltr">> I think when I mentioned this before, Kristofer said that if the winner <br></div><div dir="ltr">> drops out for some reason, then you can just elect 2nd place as the <br></div><div dir="ltr">> order wouldn't change if you ran the election again without the original <br></div><div dir="ltr">> winner. But the flipside of this is that after the election, 2nd place <br></div><div dir="ltr">> might be found to be ineligible for some reason, and there would be some <br></div><div dir="ltr">> elections where an LIIA-failing method would save you from the <br></div><div dir="ltr">> embarrassment of the original 3rd placed candidate becoming the new winner.<br></div><div dir="ltr"><br></div><div dir="ltr">It might also have some uses in Condorcet STV methods. Suppose a class <br></div><div dir="ltr">of STV-like methods is constructed like this:<br></div><div dir="ltr"><br></div><div dir="ltr"> 1. If we've filled every seat, exit.<br></div><div dir="ltr"> 2. If anybody has more than a Droop quota of the first preferences:<br></div><div dir="ltr"> 2.1. Elect and eliminate that candidate.<br></div><div dir="ltr"> 2.2. Redistribute surpluses based on first preferences.<br></div><div dir="ltr"> 2.3. Go to 1.<br></div><div dir="ltr"> 3. Otherwise:<br></div><div dir="ltr"> 3.1. Determine a winning order by some base method X<br></div><div dir="ltr"> 3.2. Eliminate the loser according to X<br></div><div dir="ltr"> 3.3. Go to 1.<br></div><div dir="ltr"><br></div><div dir="ltr">There are two cases where small initial differences may be amplified to <br></div><div dir="ltr">cause widely diverging outcomes: election (where electing A instead of B <br></div><div dir="ltr">may change who gets elected next) and elimination (similar to IRV's <br></div><div dir="ltr">chaos). If you use a LIIA method, then the second source vanishes, <br></div><div dir="ltr">because eliminating the loser of X doesn't change the order of victory <br></div><div dir="ltr">of the other candidates.<br></div><div dir="ltr"><br></div><div dir="ltr">This might lead to a more orderly method, possibly fewer monotonicity <br></div><div dir="ltr">violations, etc. I don't know this for sure: it's just an intuitive <br></div><div dir="ltr">argument.<br></div><div dir="ltr"><br></div><div dir="ltr">-km<br></div></div>
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