Equilibria in Social Belief Removal

Equilibria in Social Belief Removal

Thomas MeyerMeraka Institute

PretoriaSouth Africa

Richard BoothMahasarakham

UniversityThailand

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”

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

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

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

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]

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)

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)

Prioritised Removal: Example 1Prioritised Removal: Example 1

hΣ ,vi:hΣ ,vi:

pp

qq

pÇqpÇq

pÆ:q pÆ:q

pÇrpÇr

pÆrÆqpÆrÆq

:q:q

>(?)>(?) ≡ pÇq

≡ pÇq

Prioritised Removal: Example 2Prioritised Removal: Example 2

hΣ ,vi:hΣ ,vi:

pp

qq

pÇqpÇq

pÆ:q pÆ:q

pÇrpÇr

pÆrÆqpÆrÆq

:q:q

>(pÇq)>(pÇq) ≡ pÇr≡ pÇr

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)

Severe Withdrawal: Example 1Severe Withdrawal: Example 1

½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q

>(pÆq)>(pÆq)

≡ pÆ(qÇr) ≡ pÆ(qÇr)

Severe Withdrawal: Example 2Severe Withdrawal: Example 2

½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q½ = pÆqÆr ` pÆ(qÇr) ` pÇ:q

>(p) >(p) ≡ pÇ:q ≡ pÇ:q

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

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:

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 )

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

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

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

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

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 ¸

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 )

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)

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