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
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
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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
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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.