1. Equilibria in Social Belief Removal Thomas Meyer Meraka Institute Pretoria South Africa Richard Booth Mahasarakham University Thailand

2. Introduction 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”

3. The 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)¢(µ1,µ2,µ3,µ4) µ4µ4 µ3µ3 µ2µ2 µ1µ1

4. 2-Stage Approach to Merging 1 st Stage: Agents remove beliefs to be jointly consistent Call this Social Belief Removal 2 nd Stage: Conjoin resulting beliefs Á1ÆÁ2ÆÁ3ÆÁ4Á1ÆÁ2ÆÁ3ÆÁ4 µ4µ4 µ3µ3 µ2µ2 µ1µ1 Á1Á1 Á2Á2 Á3Á3 Á4Á4

5. Social Belief Removal Each agent has individual removal function > i > i ( ¸ ) = result of removing ¸ Initial beliefs = > i ( ? ) Call ( > i ) i 2 A a removal profile

6. Social Belief Removal Definition: A social belief removal function takes a removal profile as input and outputs a consistent belief profile ( Á i ) i 2 A s.t. for each i there is ¸ i s.t. Á i ≡ > i ( ¸ i ). Question: When is an outcome of SBR in equilibrium? Properties of > i ? –Assumption: Each > i is a basic removal function [BCMG 04]

7. Basic Removal: Properties Definition: > is a basic removal function iff it satisfies: ( > 1) > ( ¸ ) 0 ¸ ( > 2) If ¸ 1 ≡ ¸ 2 then > ( ¸ 1 ) ≡ > ( ¸ 2 ) ( > 3) If > ( Â Æ ¸ ) ` Â then > ( Â Æ ¸ Æ Ã ) ` Â ( > 4) If > ( Â Æ ¸ ) ` Â then > ( Â Æ ¸ ) ` > ( ¸ ) ( > 5) > ( Â Æ ¸ ) ` > ( Â ) Ç > ( ¸ ) ( > 6) If > ( Â Æ ¸ ) 0 ¸ then > ( ¸ ) ` > ( Â Æ ¸ )

8. Basic Removal: Example 1 Prioritised Removal: Let Σ be a finite set of consistent sentences, totally preordered by relation v. Σ( ¸ ) = { ® 2 Σ | ® 0 ¸ } > h Σ, vi ( ¸ ) = Ç min v Σ( ¸ ) if Ç Σ 0 ¸ > otherwise > h Σ, vi satisfies ( > 1)- ( > 6)

9. Prioritised Removal: Example 1 h Σ, vi : p q pÇqpÇq pÆ:qpÆ:q pÇrpÇr pÆrÆqpÆrÆq :q:q >(?)>(?) ≡ p Ç q

10. Prioritised Removal: Example 2 h Σ, vi : p q pÇqpÇq pÆ:qpÆ:q pÇrpÇr pÆrÆqpÆrÆq :q:q > (p Ç q) ≡ p Ç r

11. Basic 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)

12. Severe Withdrawal: Example 1 ½ = p Æ q Æ r ` p Æ (q Ç r) ` p Ç: q > (p Æ q) ≡ p Æ (q Ç r)

13. Severe Withdrawal: Example 2 ½ = p Æ q Æ r ` p Æ (q Ç r) ` p Ç: q > (p) ≡ p Ç: q

14. 1 st Equilibrium Notion: Removal Equilibria µ4µ4 µ3µ3 µ2µ2 µ1µ1 Á1Á1 Á2Á2 Á3Á3 Á4Á4 For each agent i : Á i ≡ > i ( : Æ Á j ) j ≠ i Theorem Always exist for basic removal

15. Removal Equilibria: Example Assume 2 agents, using severe withdrawal: p Æ q ` q ( : p Æ: q) ` ( : p Ç: q) pÆqpÆq > > : p Æ: q q : p Ç: q 3 removal equilibria:

16. 2 nd 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 )

17. Strategic 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 ( a 1, a 2, a 3, a 4 ) a4a4 a3a3 a2a2 a1a1

18. Players’ Preferences in Strategic Games ( a j ) j 2 A ¹ i ( b j ) j 2 A Means player i prefers (outcome from) ( b j ) j 2 A at least as much as (outcome from) ( a j ) j 2 A

19. Nash Equilibria Definition: An action profile ( a * i ) i 2 A is a Nash equilibrium iff for every player j and every action a j for player j : ( a i ) i 2 A ¹ j ( a * i ) i 2 A where a * i = a i for i  j Each player makes best response to others

20. Nash Equilibrium: Example Prisoners’ dilemma: CD C(3,3)(1,4) D(4,1)(2,2) Unique Nash Equilibrium

21. 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 ¸ ¹ i  iff > i ( ¸ Æ Â ) 0 ¸

22. 2 nd Idea: Entrenchment Equilibria µ4µ4 µ3µ3 µ2µ2 µ1µ1 Á1Á1 Á2Á2 Á3Á3 Á4Á4 For each agent i : Á i ≡ > i ( ¸ i *) where ( ¸ i *) i 2 A is a Nash equilibrium of G (( > i ) i 2 A )

23. Connections 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)

24. Conclusion 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