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

[Michael Ossipoff]

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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