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Equilibria in Social Belief Removal
Equilibria in Social Belief Removal
Thomas MeyerMeraka Institute
PretoriaSouth Africa
Richard BoothMahasarakham
UniversityThailand
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IntroductionIntroduction
• Multi-agent belief merging
• In multi-agent interaction, often have notions of equilibria
• Equilibria notions in belief merging?
• Guiding principle:“Each agent simultaneously makes the
appropriate response to what every other agent does”
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The Belief Merging ProblemThe Belief Merging Problem
•Set A = {1,…,n} of agents
•Each has beliefs
•Want to merge into single belief
•Problem: initial beliefs might be jointly inconsistent
¢(µ1,µ2,µ3,µ4)
µ4
µ3
µ2
µ1
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2-Stage Approach to Merging2-Stage Approach to Merging
•1st Stage: Agents remove beliefs to be jointly consistent
•Call this Social Belief Removal
•2nd Stage: Conjoin resulting beliefs
Á1ÆÁ2ÆÁ3ÆÁ4
µ4
µ3
µ2
µ1
Á1
Á2Á3
Á4
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Social Belief RemovalSocial Belief Removal
• Each agent has individual removal function >i
• >i(¸) = result of removing ¸
•Initial beliefs = >i(?)
•Call (>i)i 2 A a removal profile
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Social Belief RemovalSocial Belief Removal
• Definition: A social belief removal function takes a removal profile as input and outputs a consistent belief profile (Ái)i 2 A s.t. for each i there is ¸i s.t. Ái ≡ >i(¸i).
• Question: When is an outcome of SBR in equilibrium?
• Properties of >i?– Assumption: Each >i is a basic removal
function [BCMG 04]
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Basic Removal: PropertiesBasic Removal: Properties
Definition:> is a basic removal function iff it satisfies:
(>1) >(¸) 0 ¸
(>2) If ¸1 ≡ ¸2 then >(¸1) ≡ >(¸2)
(>3) If >(ÂƸ) `  then >(ÂƸÆÃ) ` Â
(>4) If >(ÂƸ) `  then >(ÂƸ) ` >(¸)
(>5) >(ÂƸ) ` >(Â) Ç >(¸)
(>6) If >(ÂƸ) 0 ¸ then >(¸) ` >(ÂƸ)
Definition:> is a basic removal function iff it satisfies:
(>1) >(¸) 0 ¸
(>2) If ¸1 ≡ ¸2 then >(¸1) ≡ >(¸2)
(>3) If >(ÂƸ) `  then >(ÂƸÆÃ) ` Â
(>4) If >(ÂƸ) `  then >(ÂƸ) ` >(¸)
(>5) >(ÂƸ) ` >(Â) Ç >(¸)
(>6) If >(ÂƸ) 0 ¸ then >(¸) ` >(ÂƸ)
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Basic Removal: Example 1Basic Removal: Example 1
Prioritised Removal: • Let Σ be a finite set of consistent
sentences, totally preordered by relation v.
• Σ(¸) = { ® 2 Σ | ® 0 ¸ }• >hΣ ,vi(¸) = Ç minv Σ(¸) if ÇΣ 0 ¸
> otherwise
• >hΣ ,vi satisfies (>1)- (>6)
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Prioritised Removal: Example 1Prioritised Removal: Example 1
hΣ ,vi:hΣ ,vi:
pp
pÇqpÇq
pÆ:q pÆ:q
pÇrpÇr
pÆrÆqpÆrÆq
:q:q
>(?)>(?) ≡ pÇq
≡ pÇq
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Prioritised Removal: Example 2Prioritised Removal: Example 2
hΣ ,vi:hΣ ,vi:
pp
pÇqpÇq
pÆ:q pÆ:q
pÇrpÇr
pÆrÆqpÆrÆq
:q:q
>(pÇq)>(pÇq) ≡ pÇr≡ pÇr
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Basic Removal: Example 2Basic Removal: Example 2
Severe Withdrawal [Rott+Pagnucco 99]:
•Sequence of sentences ½ = ¯1 ` ¯2 ` … ` ¯n
• >½(¸) = ¯i where i least such that ¯i 0 ¸
> if no such i exists
•>½ satisfies (>1)- (>6)
Severe Withdrawal [Rott+Pagnucco 99]:
•Sequence of sentences ½ = ¯1 ` ¯2 ` … ` ¯n
• >½(¸) = ¯i where i least such that ¯i 0 ¸
> if no such i exists
•>½ satisfies (>1)- (>6)
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Severe Withdrawal: Example 1Severe Withdrawal: Example 1
½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q
>(pÆq)>(pÆq)
≡ pÆ(qÇr) ≡ pÆ(qÇr)
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Severe Withdrawal: Example 2Severe Withdrawal: Example 2
½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q
>(p) >(p) ≡ pÇ:q ≡ pÇ:q
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1st Equilibrium Notion: Removal Equilibria
1st Equilibrium Notion: Removal Equilibria
µ4
µ3
µ2
µ1
Á1
Á2Á3
Á4
•For each agent i:
Ái ≡ >i(:ÆÁj)j≠ i
•Theorem Always exist for basic removal
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Removal Equilibria: ExampleRemoval Equilibria: Example
Assume 2 agents, using severe withdrawal:Assume 2 agents, using severe withdrawal:
pÆq ` q (:pÆ:q) ` (:pÇ:q)
pÆq >> :pÆ:qq :pÇ:q
3 removal equilibria:3 removal equilibria:
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2nd Equilibrium Notion: Entrenchment Equilibrium
2nd Equilibrium Notion: Entrenchment Equilibrium
Basic idea:
1. Convert (>i)i 2 A into strategic game G((>i)i 2 A )
2. Use Nash equilibria of G((>i)i 2 A )
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Strategic GamesStrategic Games
•Set A = {1,…,n} of players
•Each does an action
•Tuple of actions is an action profile
•Each player has preferences over action profiles
(a1,a2,a3,a4)
a4
a3
a2
a1
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Players’ Preferences in Strategic Games
Players’ Preferences in Strategic Games
(aj)j 2 A ¹i (bj)j 2 A
Means player i prefers (outcome from) (bj)j 2 A at least as much as (outcome from) (aj)j 2 A
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Nash EquilibriaNash Equilibria
•Definition: An action profile (a*i)i 2 A is a Nash equilibrium iff for every player j and every action aj for player j:
•Definition: An action profile (a*i)i 2 A is a Nash equilibrium iff for every player j and every action aj for player j:
(ai)i 2 A ¹j (a*i)i 2 A(ai)i 2 A ¹j (a*i)i 2 A
where a*i = ai for i jwhere a*i = ai for i j
•Each player makes best response to others•Each player makes best response to others
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Nash Equilibrium: ExampleNash Equilibrium: Example
Prisoners’ dilemma:Prisoners’ dilemma:
C D
C (3,3) (1,4)
D (4,1) (2,2)
Unique Nash EquilibriumUnique Nash Equilibrium
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Description of G((>i)i 2 A )Description of G((>i)i 2 A )
• Players = set A of agents
• Agent i’s actions = set of sentences
(agent chooses which sentence to remove)
• Agent i’s preference over action profiles:1. Prefers any consistent outcome to any inconsistent
one
2. Among consistent outcomes, prefers those in which i removes less entrenched sentences
¸ ¹i  iff >i(¸ÆÂ) 0 ¸
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2nd Idea: Entrenchment Equilibria2nd Idea: Entrenchment Equilibria
µ4
µ3
µ2
µ1
Á1
Á2Á3
Á4
•For each agent i:
Ái ≡ >i(¸i*)
where (¸i*)i 2 A is
a Nash equilibrium of G((>i)i 2 A )
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Connections Between EquilibriaConnections Between Equilibria
(Assuming agents use basic removal)
• Every removal equilibrium for (>i)i 2 A is an
entrenchment equilibrium for
(>i)i 2 A
• Converse holds only for a subclass of basic removal (which includes severe withdrawal, but not prioritised removal)
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ConclusionConclusion
• Defined several notions of equilibria in framework of social belief removal
• Proved existence, assuming agents use basic removal
• Future work:– Equilibria in social removal under integrity
constraints– (im)possibility theorems in social belief
removal