A Cooperative-game Approach to Share Acceptability and Rank Arguments
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The document presents a cooperative-game approach to sharing acceptability and ranking arguments in the context of argumentation theory. It discusses various semantics frameworks, including Shapley value-based semantics and ranking-based semantics, highlighting their properties, criteria, and implications for argument acceptability. The conclusion emphasizes the necessity for refining the ranking function and comparing it with existing semantics.
A Cooperative-game Approach to Share Acceptability and Rank Arguments
- 1. A COOPERATIVE-GAME APPROACH TO SHARE ACCEPTABILITY AND RANK ARGUMENTS Stefano Bistarelli, Paolo Giuliodori, Francesco Santini and Carlo Taticchi
- 2. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 INDEX ▸ Argumentation Theory Argumentation Semantics ▸ Shapley Value Definition ▸ SV-based semantics Description + Example ▸ Conclusion 2
- 3. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 ABSTRACT ARGUMENTATION FRAMEWORKS1 3 a b c d e 1Phan Minh Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2):321–357.
- 4. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 EXTENSION-BASED SEMANTICS 4 a b c d e ADM = {{}, {a}, {c}, {d}, {a, c}, {a, d}} CF = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}} STA = {{a,d}} COM = {{a}, {a, c}, {a, d}}
- 5. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 LABELLING-BASED SEMANTICS2 5 a b c d e 2Martin Caminada. On the Issue of Reinstatement in Argumentation. JELIA 2006: 111-123.
- 6. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 LABELLING-BASED SEMANTICS2 6 a b c d e IN if it is attacked only by OUT arguments OUT if it is attacked by at least an IN argument UNDEC otherwise 2Martin Caminada. On the Issue of Reinstatement in Argumentation. JELIA 2006: 111-123.
- 7. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 7 a b c d e WHICH IS THE BEST? LABELLING-BASED SEMANTICS2
- 8. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 RANKING-BASED SEMANTICS3 ▸ Transforms an Argumentation Framework into a ranking 8 3Leila Amgoud, Jonathan Ben-Naim. Ranking-Based Semantics for Argumentation Frameworks. SUM 2013: 134-147. ≻a d ≻c ≻e ≻ b
- 9. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 RANKING-BASED SEMANTICS3 ▸ Transforms an Argumentation Framework into a ranking ▸ Criteria: direct attacks, lengths of the incoming paths, rewards 9 3Leila Amgoud, Jonathan Ben-Naim. Ranking-Based Semantics for Argumentation Frameworks. SUM 2013: 134-147. ≻a d ≻c ≻e ≻ b
- 10. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 RANKING-BASED SEMANTICS3 ▸ Transforms an Argumentation Framework into a ranking ▸ Criteria: direct attacks, lengths of the incoming paths, rewards ▸ Good properties 10 3Leila Amgoud, Jonathan Ben-Naim. Ranking-Based Semantics for Argumentation Frameworks. SUM 2013: 134-147. ≻a d ≻c ≻e ≻ b
- 11. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 CATEGORIZER4 11 Cat(x) = 1 if R− 1 (x) = 0 1 1 + ∑y∈R− 1 (x) Cat(y) otherwise 4P. Besnard and A. Hunter. A logic-based theory of deductive arguments. Artificial Intelligence 128(1-2):203–235. 2001. a b c d e
- 12. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 CATEGORIZER4 12 Cat(x) = 1 if R− 1 (x) = 0 1 1 + ∑y∈R− 1 (x) Cat(y) otherwise 4P. Besnard and A. Hunter. A logic-based theory of deductive arguments. Artificial Intelligence 128(1-2):203–235. 2001. a b c d e 1 0.5 0.38 0.65 0.53
- 13. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 CATEGORIZER4 13 Cat(x) = 1 if R− 1 (x) = 0 1 1 + ∑y∈R− 1 (x) Cat(y) otherwise 4P. Besnard and A. Hunter. A logic-based theory of deductive arguments. Artificial Intelligence 128(1-2):203–235. 2001. b ≻Cat d ≻Cat e ≻Cat c ≻Cat a a b c d e 1 0.5 0.38 0.65 0.53
- 15. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 THE SHAPLEY VALUE’S FORMULA5 ▸ i is a player ▸ n is the number of players ▸ S-i is any set of agents which does not contain i ▸ s is the cardinality of S-i ▸ v is a ranking function 15 ϕi(v) = ∑ S−i ⊆ G ∖ {i} s!(n − 1 − s)! n! (v(S−i ∪ {i}) − v(S−i)) 5Lloyd Stowell Shapley. Contributions to the Theory of Games. AM-28, Volume II - Princeton University Press, 1953.
- 16. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 RANKING FUNCTIONS 16 vI σ,F(S) = { 1, if S ∈ in(Lσ) 0, if otherwise a b c d e
- 17. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 RANKING FUNCTIONS 17 vI σ,F(S) = { 1, if S ∈ in(Lσ) 0, if otherwise vO σ,F(S) = { 1, if S ∈ out(Lσ) 0, if otherwise a b c d e
- 18. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 RANKING FUNCTIONS 18 vI σ,F(S) = { 1, if S ∈ in(Lσ) 0, if otherwise vO σ,F(S) = { 1, if S ∈ out(Lσ) 0, if otherwise Depends on the semantics σ a b c d e
- 19. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 SV-BASED SEMANTICS 19 ∀a, b ∈ A, a ≻ b iff ∙ ϕa(vI σ,F) > ϕb(vI σ,F), or ∙ ϕa(vI σ,F) = ϕb(vI σ,F) and ϕa(vO σ,F) < ϕb(vO σ,F)
- 20. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 SV-BASED SEMANTICS 20 ∀a, b ∈ A, a ≻ b iff ∙ ϕa(vI σ,F) > ϕb(vI σ,F), or ∙ ϕa(vI σ,F) = ϕb(vI σ,F) and ϕa(vO σ,F) < ϕb(vO σ,F) ∀a, b ∈ A, a ≃ b iff ∙ ϕa(vI σ,F) = ϕb(vI σ,F) and ϕa(vO σ,F) = ϕb(vO σ,F)
- 21. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 EXAMPLE 21 in(LCF) = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}} ϕi(v) = ∑ S−i ⊆ G ∖ {i} s!(n − 1 − s)! n! (v(S−i ∪ {i}) − v(S−i)) a b c d e
- 22. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 EXAMPLE 22 in(LCF) = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}} ϕa(v) = 0! ⋅ 4! 5! ⋅ (v({a}) − v({})) + ϕi(v) = ∑ S−i ⊆ G ∖ {i} s!(n − 1 − s)! n! (v(S−i ∪ {i}) − v(S−i)) a b c d e
- 23. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 EXAMPLE 23 in(LCF) = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}} ϕa(v) = 0! ⋅ 4! 5! ⋅ (v({a}) − v({})) + + 1! ⋅ 3! 5! ⋅ (v({a, b}) − v({b})) + … + ϕi(v) = ∑ S−i ⊆ G ∖ {i} s!(n − 1 − s)! n! (v(S−i ∪ {i}) − v(S−i)) a b c d e
- 24. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 EXAMPLE 24 in(LCF) = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}} ϕa(v) = 0! ⋅ 4! 5! ⋅ (v({a}) − v({})) + + 1! ⋅ 3! 5! ⋅ (v({a, b}) − v({b})) + … + ϕi(v) = ∑ S−i ⊆ G ∖ {i} s!(n − 1 − s)! n! (v(S−i ∪ {i}) − v(S−i)) a b c d e + 2! ⋅ 2! 5! ⋅ (v({a, b, d}) − v({b, d})) =
- 25. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 EXAMPLE 25 in(LCF) = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}} ϕa(v) = 0! ⋅ 4! 5! ⋅ (v({a}) − v({})) + + 1! ⋅ 3! 5! ⋅ (v({a, b}) − v({b})) + … + ϕi(v) = ∑ S−i ⊆ G ∖ {i} s!(n − 1 − s)! n! (v(S−i ∪ {i}) − v(S−i)) a b c d e + 2! ⋅ 2! 5! ⋅ (v({a, b, d}) − v({b, d})) = = 0 − 0.05 − 0.033 = − 0.084
- 26. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 26 EXAMPLE - RESULTS a b c d e
- 27. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 ▸ Abstraction ▸ Independence ▸ Non-attacked Equivalence ▸ Argument Equivalence SV-BASED SEMANTICS - PROPERTIES6 27 ▸ Total order ▸ Self-contradiction (only for the conflict-free semantics) 6Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, Nicolas Maudet. A Comparative Study of Ranking-Based Semantics for Abstract Argumentation. AAAI 2016: 914-920.
- 28. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 CONCLUSION 28 ▸ Ranking-based semantics • Uses the Shapley Value • Is parametric to σ • Satisfies good properties
- 29. A Cooperative-game Approach to Share Acceptability and Rank ArgumentsCarlo Taticchi — November 21, 2018 CONCLUSION 29 ▸ Ranking-based semantics • Uses the Shapley Value • Is parametric to σ • Satisfies good properties ▸ Next • Refine the ranking function • Check all the properties • Compare with existing semantics
- 30. A COOPERATIVE-GAME APPROACH TO SHARE ACCEPTABILITY AND RANK ARGUMENTS Thanks for your attention!