Fwd: agenda landau winner
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-------- Mensaje original --------
De: Susan Simmons suzerainsimmons@outlook.com
Fecha: 28/7/21 12:25 p. m. (GMT-08:00)
A: Kristofer Munsterhjelm km_elmet@t-online.de
Asunto: agenda landau winner
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.
We wish to show that the following method for choosing a winner from a monotone agenda satisfies the mono-raise property of election methods.
Agenda Landau Winner (ALW):
Let X be the least promising (remaining) alternative. Let Y be the winner of this ALW method applied recursively to the agenda restricted to the remaining alternatives with X (also) eliminated. If X covers Y, then elect X, else elect Y.
First suppose the winner W moves towards the more promising end of the agenda without changing the covering relation.Then, if anything, W attains an unassailable position at an earlier stage of the recursion.
Now suppose ballots are altered only by raising the ALW winner W relative to the other alternatives. We will show that this change cannot result in a new winner, by deriving a contradiction from the assumption that some W' other than W becomes the new winner.
First note that after the change W still covers every alternative it covered before the change, and W is still uncovered (its short beatpaths remain short).
Let A be the most promising agenda item. Could W' be A? No, because W still covers A, and the new winner W' has to be uncovered.
Does W' cover A? Yes, because an ALW winner (e.g. W') must cover the most promising agenda item (when not equal to it).
Let b1=A, b2, ... bN=W be the sequence of agenda items such that before the change (i.e. the raising of W) the agenda alternative b(k+1) was the most promising agenda item that covered b(k).
Let a1=A, a2, ... aN=W' be the sequence of agenda items such that a(k +1) is (after the change) the most promising item that covers a(k).
Let j be the smallest value of k, such that a(k) = b(k).
Then a(j+1) is a more promising alternative than b(j+1), so a(j+1) did not cover b(j) = a(j) before the change.
Let Z be the alternative that W beat pairwise after the change but not before. To simplify notation, let V = b(j) = a(j), and let U = a(j+1).
In summary, U covers V after the change but not before, and W beats Z after, but not before.
How can this be? U not covering V before entails V beating some alternative that U did not before, but now does. What alternative could that be?
The only pairwise beat change is W going from not beating Z to beating Z.
Evidently W = U and Z = V.
So W did not beat Z = V = b(j) before the change.
This contradicts the construction in which W covers every b(k) .... (by the transitivity of the covering relation).
This contradiction shows that the supposition W' not equal to W was untenable.
The ALW method satisfies mono-raise!
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
¿Am I correct in understanding that in this agenda contexto Min Pairwise Support would satisfy strong mono-raise, along with Range in general, Above MidRange Approval, and Chiastic Approval?.. (my favorite agenda setters)
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-------- Mensaje original --------
De: Kristofer Munsterhjelm km_elmet@t-online.de
Fecha: 28/7/21 1:13 p. m. (GMT-08:00)
A: Susan Simmons suzerainsimmons@outlook.com, election-methods@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
Chiastic approval works ... for each alternative it depends only on the ballot ratings for that alternative ... it is a cross between MJ and above midrange approval.
Sent from my MetroPCS 4G LTE Android Device
-------- Mensaje original --------
De: Kristofer Munsterhjelm km_elmet@t-online.de
Fecha: 28/7/21 1:13 p. m. (GMT-08:00)
A: Susan Simmons suzerainsimmons@outlook.com, election-methods@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