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ExtensiveForm Argumentation Games
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Argumentation framework =def (AR, ). AR = set of arguments, is attack relation. ... Game-Based Argumentation Framework =def (AR, ,AR1,AR2,U) ... – PowerPoint PPT presentation
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Title: ExtensiveForm Argumentation Games
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
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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
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The interaction graph
- Given (AR,?,AR1,AR2,U), the associated
interaction graph is the bipartite graph V1AR1,
V2AR2, 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
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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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