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 Abstract Argumentation Frameworks • Argumentation framework =def (AR,). AR = set of arguments, is attack relation. • a is acceptable w.r.t. S iff: ba Sb. • S is conflict-free iff ~a,b in S s.t. ab. • 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 Abstract AF: Example c b a e d f g h • {b} is admissible •{d,e,g} is a stable extension 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. GBA Frameworks • Game-Based Argumentation Framework =def (AR,,AR1,AR2,U). ARi are finite • Dialogue is (a1,...,ar) s.t. ai in AR1 for odd i, in AR2 for even i, and aiai-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. GBA Frameworks as Game Trees I a b t1 a c b AR2 AR1 U(t1)=0.1, U(a,t2)=0.8, U(b,t2)=0.7, U(b,c,t1)=0.6 II 0.1 II t2 0.8 c t2 0.7 I t1 0.6 The interaction graph • Given (AR,,AR1,AR2,U), the associated interaction graph is the bipartite graph: V1=AR1, V2=AR2, E={(v1,v2): v2v1} • Proposition: Associated game tree is infinite iff interaction Graph contains a cycle. a c b AR1 d AR2 a c b d c 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 AR1AR2: ba Sb. • 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 conflictfree and S attacks all arguments in AR1AR2\S. • Proposition: Framework is normal and ARi are locally-stable Every node has infinite subtree. Subgame-infinite 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. 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 Future Research • Argumentation games of incomplete information. • Possible scenario: mediator. • 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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