[EM] Fwd: agenda landau winner

[Susan Simmons]

Let's call this latest version ALW' and our longstanding version ALW (Agenda Landau Winner).

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.



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-------- Mensaje original --------
De: Susan Simmons <suzerainsimmons at outlook.com>
Fecha: 31/7/21 7:27 a. m. (GMT-08:0sion0)
A: Kristofer Munsterhjelm <km_elmet at t-online.de>, election-methods at lists.electorama.com
Asunto: Fwd: [EM] Fwd: agenda landau winner

Elect the most promising uncovered agenda item that covers each more promising item.

Proof of monotonicity:

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.

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.

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.

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.

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.

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."





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-------- Mensaje original --------
De: Susan Simmons <suzerainsimmons at outlook.com>
Fecha: 31/7/21 6:18 a. m. (GMT-08:00)
A: Kristofer Munsterhjelm <km_elmet at t-online.de>, election-methods at lists.electorama.com
Asunto: Re: [EM] Fwd: agenda landau winner

Kristofer,

Your comment below* has inspired an improved version of Agenda Based Landau:

Elect the most promising uncovered agenda item that covers each more promising item.

Proof of monotonicity coming in next message....

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-------- Mensaje original --------
De: Kristofer Munsterhjelm <km_elmet at t-online.de>
Fecha: 28/7/21 1:13 p. m. (GMT-08:00)
A: Susan Simmons <suzerainsimmons at outlook.com>, election-methods at lists.electorama.com
Asunto: Re: [EM] Fwd: agenda landau winner

On 28.07.2021 21:28, Susan Simmons wrote:
>
> We work from an agenda of alternatives listed in order of “promise.” The
> agenda is “monotone” if increasing ballot support for an alternative
> moves it towards the promising end of the agenda without altering the
> relative order of the other candidates in the list.

Of note here is that this monotone agenda criterion (strong mono-raise?)
is much stronger than ordinary mono-raise. For instance, Plurality fails
it but passes ordinary mono-raise.

Also: I don't think the relation above is iff; there are looser criteria
that, if met, guarantee monotonicity of the agenda method. For instance,
if the winner W is raised, then the property only has to hold for
candidates ranked below the winner on the agenda, because (by assumption
that W was the original winner),
* W covers everybody ranked higher.

When the agenda method is Smith rather than Landau, I think an even
looser criterion can be phrased in the term of beatpaths, or in that
raising A should not swap the position of any pair of candidates B and C
ranked below A where B beats A pairwise and C is beaten pairwise by A.

These looser criteria are much harder to reason about, though.

And finding a burial immune strong mono-raise compliant method would be
very nice :-)

-km
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