Equilibrium: Difference between revisions
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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.
===Smith set equilibrium ===
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
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2: C>A>B
5: D</blockquote>A, B, and C are in the "majority Smith set" (every candidate in the Smith set [[majority-beat]]<nowiki/>s every candidate not in the 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.
Another example:
<br />
{| class="wikitable"
|+Losses and ties are bolded, with every win being a majority-win
!
!A
!B
!C
!D
!E
!F
!G
|-
|A
| ---
|Win
|'''Lose'''
|Win
|Win
|Win
|Win
|-
|B
|'''Lose'''
| ---
|Win
|Win
|Win
|Win
|Win
|-
|C
|Win
|'''Lose'''
| ---
|'''Lose'''
|Win
| Win
|Win
|-
|D
|'''Lose'''
|'''Lose'''
|Win
| ---
|'''Tie'''
|Win
|Win
|-
| E
|'''Lose'''
|'''Lose'''
|'''Lose'''
|'''Tie'''
| ---
| Win
|Win
|-
|F
|'''Lose'''
|'''Lose'''
|'''Lose'''
|'''Lose'''
|'''Lose'''
| ---
|Win
|-
|G
|'''Lose'''
|'''Lose'''
|'''Lose'''
|'''Lose'''
|'''Lose'''
|'''Lose'''
| ---
|}
A through E are in the Smith set (there are [[beat-or-tie path]]<nowiki/>s of A<C<B and C<D=E). Suppose F wins in [[Score voting]]; then the majority that prefers any of the Smith set members can set their [[approval threshold]] between the Smith set member and F, giving the Smith set member a majority of points and F a minority of points.
==Strong Nash equilibrium==
Applied to voting theory, a '''strong Nash equilibrium''' 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 ==
Although a particular type of candidate may be elected in equilibrium for a particular election method and election, that does not necessarily imply that this candidate type is in equilibrium for every election for that method. For example, [[Approval voting]] may elect the [[Condorcet winner]] in equilibrium in some elections, but not all: <blockquote>Here's my reasoning: consider a standard chicken dilemma:
{| class="wikitable"
!Number
! Ballots
|-
|1
| A
|-
|34
Line 43 ⟶ 124:
|B>A>C
|-
|
|C
|}
Line 72 ⟶ 153:
|'''Honesty (B)'''
|A wins, minor B utility loss
| A wins, minor B utility loss
|-
|'''Burial (B)'''
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|'''Honesty (B)'''
| -1, 0
| -1, 0
|-
|'''Burial (B)'''
| |||