[EM] EM equilibrium dfn. NEO properties-examples.

[Michael Ossipoff]

On Sep 16, 2016 11:13 AM, "Kevin Venzke" <stepjak at yahoo.fr> wrote:
>
> Hi Mike,
>
> If the cast ballots (in your first example) were A>B, B, and C, how could
a possible equilibrium be A, B=A, and C? Only one faction showed any
willingness to vote for A, correct?

Yes. But, because it must be assumed that the B voters are indifferent
between A & C, then, if they _were_ approving A, resulting in an A win,
then they wouldn't improve their outcome by withdrawing that approval &
letting C win.

Since no cohort can improve their outcome by changing their vote, it's an
equilibrium.

But there's another problem:

...an equilibrium that I didn't notice:

A
B,C
C

The B voters obviously wouldn't improve their  outcome by withdrawing that
approval.

C wins in that equilibrium.

But having said all that, that isn't a CD failure. So it isn't a problem at
all. But it remains in this post because I don't have a way to delete.

Michael Ossipoff
>
> Kevin
>
>
> ________________________________
> De : Michael Ossipoff <email9648742 at gmail.com>
> À : election-methods at electorama.com
> Envoyé le : Vendredi 16 septembre 2016 10h47
> Objet : [EM] EM equilibrium dfn. NEO properties-examples.
>
> EM has discussed Nash equilibrium a number of times, & this, if seems to
me, is what EM agrees Nash equilibrium to mean:
> A "cohort" is a set of voters who prefer & vote the same as eachother.
> At EM, for voting-systems, a Nash equilibrium is an outcome that no
cohort can improve for itself by changing its vote.
> (end of dfn)
> NEO assumes that the voters' rankings are sincere, & indicate the voters'
actual preferences & indifferences.
> Chicken dilemma:
> The usual example:
> 3 candidates: A, B, & C.
> The A voters & B voters are a majority who greatly prefer A & B to C.
(though NEO of course doesn't recognize unexpressed preferences)
> Faction size relations:
> C > A > B
> The C voters are indifferent between A & B, & dislike both
> Rankings:
> A voters: A > B
> B voters: B
> C voters: C
> Two Approval Nash equilibria:
> A,B
> B
> C
> Electing B.
> and
> A
> B,A
> C
> Electing A.
> So, find the equilibria in an election with just A & B:
> A
> B
> and
> A
> B, A
> Either way A wins.
> CD's requirement, that B not win, is met.
> Truncation against CWs:
> Instead of A, B, & C, I prefer:
> Worst, Middle, & Favorite.
> W, M, & F.
> More expressive. Of course the W voters are the offensive strategizers.
> Rankings:
> W voters: W
> M voters: M>W
> F voters: F>M
> Approval Nash Equilibrium:
> F voters: F, M
> M voters: M
> W voters: W
> If the F voters don't approve M, that could only change the winner to W,
worsening the outcome for them.
> If the M voters approve W, that could only change the winner to W,
worsening the outcome for them.
> W voters gain nothing by approving M. That's another Nash equilibrium.
> M wins in both Equilibria.
> Burial & defensive truncation:
> Rankings:
> F voters: F>M
> M voters: M
> W voters: W>F
> Approval Nash Equilibria;
> F voters: F, M
> M voters: M
> W voters: W,F
> That's a disequilibrium, because the F voters' approval of M could change
the winner from F to M. They withdraw that Approval:
> F voters : F
> M voters: M
> W voters: W, F
> That's a Nash equilibrium.
> (W voters are assumed to prefer F to M, due to their ranking)
> F wins the NEO election. The burial is thwarted & penalized.
> Michael Ossipoff
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