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<div>Let's call this latest version ALW' and our longstanding version ALW (Agenda Landau Winner).</div>
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<div>If there is an ALW' winner, it will also be the ALW winner, but sadly, like the Condorcet Winner the ALW' winner does not always exist.</div>
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<div>De: Susan Simmons <suzerainsimmons@outlook.com> </div>
<div>Fecha: 31/7/21 7:27 a. m. (GMT-08:0sion0) </div>
<div>A: Kristofer Munsterhjelm <km_elmet@t-online.de>, election-methods@lists.electorama.com
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<div>Asunto: Fwd: [EM] Fwd: agenda landau winner </div>
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<div dir="auto"><font face="sans-serif">Elect</font> the most promising uncovered agenda item that covers each more promising item.</div>
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<div dir="auto">Proof of monotonicity:</div>
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<div dir="auto">Suppose that the winner W is uniquely given increased ballot support, but (by way of contradiction) after this change a different alternative W' emerges as the new winner.</div>
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<div dir="auto">We remark that (1) if W covers X before being uniquely raised, it continues covering X after the change ...and (2) unique raising cannot change W from its uncovered status to a covered status.</div>
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<div dir="auto">Since (by the first remark) W still covers all of the agenda items of greater promise, and W' is uncovered, this new winner must be an agenda item of lesser promise.</div>
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<div dir="auto">So W must be an item of greater promise than the new winner W', which entails that W' must cover W. But this contradicts the second remark.</div>
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<div dir="auto">So W' cannot be an item of greater or lesser promise than W, which contradicts the assumption that W' was an alternative different from W.</div>
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<div dir="auto">There is however a subtle opening for W' ... what if raising ballot support for W changed the relative agenda order of W and W' ? That would destroy the above argument, which presumes the preservation of their relative agenda order. This is
why we need an additional monotonicity assumption like... "uniquely raising ballot support for a candidate cannot move it to a less favorable agenda position relative to some other alternative."</div>
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<div>-------- Mensaje original --------</div>
<div>De: Susan Simmons <suzerainsimmons@outlook.com> </div>
<div>Fecha: 31/7/21 6:18 a. m. (GMT-08:00) </div>
<div>A: Kristofer Munsterhjelm <km_elmet@t-online.de>, election-methods@lists.electorama.com
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<div>Asunto: Re: [EM] Fwd: agenda landau winner </div>
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<div>Kristofer,</div>
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<div>Your comment below* has inspired an improved version of Agenda Based Landau:</div>
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<div><font face="sans-serif">Elect</font> the most promising uncovered agenda item that covers each more promising item.</div>
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<div>Proof of monotonicity coming in next message....</div>
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<div>-------- Mensaje original --------</div>
<div>De: Kristofer Munsterhjelm <km_elmet@t-online.de> </div>
<div>Fecha: 28/7/21 1:13 p. m. (GMT-08:00) </div>
<div>A: Susan Simmons <suzerainsimmons@outlook.com>, election-methods@lists.electorama.com
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<div>Asunto: Re: [EM] Fwd: agenda landau winner </div>
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<div class="PlainText">On 28.07.2021 21:28, Susan Simmons wrote:<br>
> <br>
> We work from an agenda of alternatives listed in order of “promise.” The<br>
> agenda is “monotone” if increasing ballot support for an alternative<br>
> moves it towards the promising end of the agenda without altering the<br>
> relative order of the other candidates in the list.<br>
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Of note here is that this monotone agenda criterion (strong mono-raise?)<br>
is much stronger than ordinary mono-raise. For instance, Plurality fails<br>
it but passes ordinary mono-raise.<br>
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Also: I don't think the relation above is iff; there are looser criteria<br>
that, if met, guarantee monotonicity of the agenda method. For instance,<br>
if the winner W is raised, then the property only has to hold for<br>
candidates ranked below the winner on the agenda, because (by assumption<br>
that W was the original winner), </div>
<div class="PlainText">* W covers everybody ranked higher.<br>
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When the agenda method is Smith rather than Landau, I think an even<br>
looser criterion can be phrased in the term of beatpaths, or in that<br>
raising A should not swap the position of any pair of candidates B and C<br>
ranked below A where B beats A pairwise and C is beaten pairwise by A.<br>
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These looser criteria are much harder to reason about, though.<br>
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And finding a burial immune strong mono-raise compliant method would be<br>
very nice :-)<br>
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-km<br>
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