[EM] EM equilibrium dfn. NEO properties-examples.
Kevin Venzke
stepjak at yahoo.fr
Fri Sep 16 11:13:41 PDT 2016
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 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 > 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
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