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Extensive-Form Argumentation Games

Extensive-Form Argumentation Games. A. D. Procaccia & J. S. Rosenschein. Lecture Outline. Abstract argumentation Motivation and related work Game-based argumentation frameworks Structure of the game tree The interaction graph Local semantics Algorithmic issues Future work.

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Extensive-Form Argumentation Games

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  1. Extensive-Form Argumentation Games A. D. Procaccia & J. S. Rosenschein

  2. Lecture Outline • Abstract argumentation • Motivation and related work • Game-based argumentation frameworks • Structure of the game tree • The interaction graph • Local semantics • Algorithmic issues • Future work

  3. Abstract Argumentation Frameworks • Argumentation framework =def (AR,). AR = set of arguments,  is attack relation. • a is acceptable w.r.t. S iff: ba  Sb. • S is conflict-free iff ~a,b in S s.t. ab. • S is admissible iff S is conflict-free and a in S is acceptable w.r.t. S. • S is a stable extension iff S is conflict-free and S attacks all arguments in AR\S. stable  admissible

  4. Abstract AF: Example c b a e d f g h • {b} is admissible • {d,e,g} is a stable extension

  5. Motivation and Related Work • Abstract Argumentation is static in nature. • Wish to model interaction between several players (but keep abstraction!). • A body of work on dialectic argumentation addresses these issues (independently). • Two advantages of our approach: • Flexible rewards. • Algorithmic game theory.

  6. GBA Frameworks • Game-Based Argumentation Framework =def (AR,,AR1,AR2,U). • Dialogue is (a1,...,ar) s.t. ai in AR1 for odd i, in AR2 for even i, and aiai-1. • U assigns utility to every valid dialogue. • Terminates with t1 or t2. • Real values in [0,1] which sum to 1: useful for divisible goods. • Normal Framework: ARi disjoint and nonempty. Two players, ARi are finite

  7. GBA Frameworks as Game Trees I a b t1 AR1 AR2 a II II 0.1 c t2 t2 b c 0.8 I 0.7 U(t1)=0.1, U(a,t2)=0.8, U(b,t2)=0.7, U(b,c,t1)=0.6 t1 0.6

  8. The interaction graph • Given (AR,,AR1,AR2,U), the associated interaction graph is the bipartite graph: V1=AR1, V2=AR2, E={(v1,v2): v2v1} • Proposition: Associated game tree is infinite iff interaction Graph contains a cycle. a AR1 a c b d c BFS

  9. Local Semantics and the Game Tree • How do properties of argument sets affect the size of the game tree? • a is locally-acceptable w.r.t. S iff b in AR1AR2: ba  Sb. • S is locally-admissible iff S is conflict-free and a in S is locally-acceptable w.r.t. S. • S is a locally-stable extension iff S is conflict-free and S attacks all arguments in AR1AR2\S. • Proposition: Framework is normal and ARi are locally-stable  Every node has infinite subtree. Subgame-infinite

  10. Algorithmic issues: Simplifying • Several ways to insure tree is finite: • Each argument can be used once. • k-bounded: restricting length of arguments. • Finite game trees can be solved by backward induction. • Complexity is linear in size of game tree. • Solution is subgame-perfect Nash equilibrium. Alpha-beta pruning

  11. Algorithmic Issues: Concise Utility • Tree may be very large, although framework can be concisely represented. • Pure framework: • Utility 0 to player who terminates the dialogue. • Can be concisely represented. • Proposition: In a k-bounded pure argumentation framework, the winner can be identified in time poly(|AR1|,|AR2|, k). Proof: dynamic programming

  12. Future Research • Argumentation games of incomplete information. • U is zero-sum. • Two-player zero-sum extensive-form game of incomplete information but with perfect recall: equilibria are solutions of LP.

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