Transcript 0508.gba.ppt

Extensive-Form Argumentation
Games
A. D. Procaccia & J. S. Rosenschein
Lecture Outline
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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: 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
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 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.
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): v2v1}
• 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
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 conflictfree 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
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.