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
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?
Kevin
De : Michael Ossipoff <email9648742@gmail.com>
À : election-methods@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 > BThe C voters are indifferent between A & B, & dislike bothRankings:
A voters: A > B
B voters: B
C voters: CTwo Approval Nash equilibria:A,B
B
CElecting B.andA
B,A
CElecting A.So, find the equilibria in an election with just A & B:A
BandA
B, AEither 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>MApproval Nash Equilibrium:F voters: F, M
M voters: M
W voters: WIf 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>FApproval Nash Equilibria;F voters: F, M
M voters: M
W voters: W,FThat'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, FThat'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
Election-Methods mailing list - see http://electorama.com/em for list info
On Sep 16, 2016 11:13 AM, "Kevin Venzke" stepjak@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@gmail.com
À : election-methods@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
Election-Methods mailing list - see http://electorama.com/em for list info