Equilibrium: Difference between revisions
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All cabal equilibria are Nash equilibria but not vice versa.
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A strong Nash equilibrium is a set of ballots such that candidate X wins, and no set of voters can change their ballots such that a candidate Y whom all of them strictly prefer to X will win.
A slightly stronger and more restrictive concept is that of a strictly semi-honest strong Nash equilibrium; that is, one in which no voter puts any A above some B despite actually preferring B over A or being indifferent between the two.
If there is a majority Condorcet winner, there is almost certain to be a strong Nash equilibrium that favors that winner, in almost any reasonable deterministic voting system; but in some voting systems, that equilibrium may not be strictly semi-honest.
If there is a Condorcet winner but not a majority Condorcet winner (in other words, if enough voters are indifferent between the CW X and some other candidate Y, so that the social preference for X over Y is not a majority), it may not be possible to have a strictly semi-honest strong Nash equilibrium in a candidate-blind, non-dictatorial voting system.
Many voting methods that have an equilibrium around the Condorcet winner likely more generally have an equilibrium around any candidate in the [[Smith set]], particularly if every candidate in the Smith set majority-beats all candidates not in the Smith set. For example, with [[Approval voting]]:<blockquote>2: A>B>C
2: B>C>A
2: C>A>B
5: D</blockquote>A, B, and C are in the "majority Smith set". Every voter in the (A, B, C) [[solid coalition]] has an incentive to approve all of (A, B, C) to ensure that one of them wins, rather than D; if any of them approve fewer candidates, then D wins or at least ties, which is strictly worse from the solid coalition voters' perspectives.
==Strong Nash equilibrium==
A <nowiki>'''Strong Nash equilibrium'''</nowiki> is a concept from game theory. Applied to voting theory, it means a set of votes, where no coalition of voters can change their votes to get a result they all prefer. This is one of the strongest, most elusive kinds of equilibria in voting theory. The only ways to make it stronger are if it is known (through some reliable aspect of the system, not just through polling) and/or unique. It has also been called a "Coalition Proof Social Equilibrium" or CPSE.
==Notes==
Note that just because a voting method can have an equilibrium on a particular candidate, doesn't mean it will always. For example, consider [[Approval voting]] having an equilibrium on the [[Condorcet winner]]:<blockquote>Here's my reasoning: consider a standard chicken dilemma:
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|B>A>C
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|C
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==External links==
*[http://en.wikipedia.org/wiki/Nash_equilibrium Wikipedia: Nash equilibrium]
*[http://www.spaceandgames.com/?p=46 Peter de Blanc: A Stronger Type of Equilibrium]
*[http://www.spaceandgames.com/?p=134 Peter de Blanc: When is an honest vote a cabal equilibrium?]
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<references />
[[Category:Voting theory]]
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