Dynamic Unawareness and Rationalizable Behavior

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Dynamic Unawareness and Rationalizable Behavior

• Aviad Heifetz
• The Open University of Israel
• Martin Meier
• Institute of Economic Analysis, Barcelona
• Burkhard C. Schipper
• University of California, Davis

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Questions

• How to model dynamic interactions between players
  involving mutual unawareness?

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Questions

• How to model dynamic interactions between players
  involving mutual unawareness?
• Whats a good solution concept? (How to
  generalize the notion of extensive-form
  rationalizability (Pearce 1984) to this setting?)

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Example Battle of the Sexes
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Example Battle of the Sexes
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Example Battle of the Sexes
Unique extensive form rationalizable outcome.
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Example Battle of the Sexes
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Example Battle of the Sexes
The strategic implications of unawareness of an
action differ from the unavailability of the
action.
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Extensive Form Games with Unawareness
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Generalized Game
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Partially Ordered Set of Trees
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Partially Ordered Set of Trees
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Partially Ordered Set of Trees
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Partially Ordered Set of Trees
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Generalized Game
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Generalized Game
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a b c a b c
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a b c a b c
a b c a b c
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a b c a b c
a b c a b
a b c d e
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Generalized Game
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Information sets in a generalized game
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Standard
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Standard Generalized
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Standard Generalized New
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Information set across subtrees
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a b c
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a b
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a b c
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a b a b
a b a b
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Generalized Extensive-Form Games
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Strategies
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Extensive-form Rationalizability
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Belief Systems
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Rationality and Would-be Rationality
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Rationality ? Would-be Rationality
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Rationality ? Would-be Rationality
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Everywhere rationality ? Everywhere Would-be
Rationality
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Plan of action
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Plan of action
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Generalizing Extensive-form Rationalizability
(Pearce 1984, Battigalli 1997) to Dynamic
Unawareness
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Generalizing Extensive-form Rationalizability
(Pearce 1984, Battigalli 1997) to Dynamic
Unawareness
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Would-be rationalizability
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Would-be rationalizability
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Would-be rationalizability
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Introductory Example Revisited
Rationalizable Strategies SI1 SI \ (dont
tell, M, B, B), (dont tell, M, B, S), (dont
tell, M, S, B), (dont tell, M, S, S) SII1
(B, M), (S, M) SI2 (dont tell, B, M, B),
(tell, B, M, B), (dont tell, B, M, S), (tell,
B, M, S) SIIkSII1, SIk SI2, k 2,
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Example Sender-Receiver Game
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c
d1 d2
d3
d4
Announcer
Announcer
Announcer
Announcer
?2, ?3
?2, ?3
?2, ?3
?2, ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
DM
DM
DM
DM
DM DM DM
DM DM DM
DM DM DM
DM DM DM
l m r
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
5/3 5/3 2
5/3 5/3 2
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c
d1 d2
d3
d4
Announcer
Announcer
Announcer
Announcer
?2, ?3
?2, ?3
?2, ?3
?2, ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
DM
DM
DM
DM
DM DM DM
DM DM DM
DM DM DM
DM DM DM
l m r
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
5/3 5/3 2
5/3 5/3 2
DM
l m r
3 0 2
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c
d1 d2
d3
d4
Announcer
Announcer
Announcer
Announcer
?2, ?3
?2, ?3
?2, ?3
?2, ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
DM
DM
DM
DM
DM DM DM
DM DM DM
DM DM DM
DM DM DM
l m r
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
5/3 5/3 2
5/3 5/3 2
c
c
d1?1, ?2 d2?1, ?2 d4?1,
?2
d1?1, ?3 d3?1, ?3 d4?1,
?3
Announcer
Announcer
Announcer
Announcer
Announcer
Announcer
Ø ?2
Ø ?2
Ø ?3
Ø ?2
Ø ?3
Ø ?3
DM
DM
DM
DM
DM
DM
DM
DM
DM
DM
DM
DM
2 0 2
3 0 2
3 0 2
0 5 2
0 5 2
3/2 5/2 2
3/2 5/2 2
3 0 2
3 0 2
2 0 2
5/2 0 2
5/2 0 2
DM
l m r
3 0 2
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c
d1 d2
d3
d4
Announcer
Announcer
Announcer
Announcer
?2, ?3
?2, ?3
?2, ?3
?2, ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
Ø ?2 ?3
DM
DM
DM
DM
DM DM DM
DM DM DM
DM DM DM
DM DM DM
l m r
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
3 0 2
3 0 2
0 5 2
0 5 2
2 0 2
2 0 2
5/3 5/3 2
5/3 5/3 2
5/3 5/3 2
c
c
d1?1, ?2 d2?1, ?2 d4?1,
?2
d1?1, ?3 d3?1, ?3 d4?1,
?3
Announcer
Announcer
Announcer
Announcer
Announcer
Announcer
Ø ?2
Ø ?2
Ø ?3
Ø ?2
Ø ?3
Ø ?3
DM
DM
DM
DM
DM
DM
DM
DM
DM
DM
DM
DM
2 0 2
3 0 2
3 0 2
0 5 2
0 5 2
3/2 5/2 2
3/2 5/2 2
3 0 2
3 0 2
2 0 2
5/2 0 2
5/2 0 2
DM
l m r
3 0 2
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Prudent Rationalizability
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Prudence vs. Rationalizability Divining the
opponent's past behavior
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Prudent Rationalizability Would-be
Rationalizability
b c
a
II
6, 6
d e f d
e f
0, 0
5, 5
6, 6
10, 0
3, 4
4, 3
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Related Refinements
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Related Refinements
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Related Refinements
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Prudence vs. Rationalizability Divining the
opponent's future behavior
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Related Work
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Conditional Dominance
Is it possible to capture reasoning about
rational play in extensive-form games with
unawareness in an associated normal form?
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Conditional Dominance
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Conditional Dominance
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Conditional Dominance
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Conditional Dominance
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Conditional Dominance
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Set of outcomes ((dont give, M, M), (S, M)),
((give, M, M), (S, M))
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Unique outcome
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Associated Normal Form
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Whats in the paper?

• Define generalized extensive form games with
  unawareness
• Discuss notions of strategies and rationality in
  these games
• Generalize extensive form rationalizability to
  games with unawareness, show existence
• Show that the unawareness of actions has
  strategic implications different from the
  unavailability of actions
• Introduce a prudent extensive form
  rationalizability, show existence and that it is
  a outcome refinement of would-be
  rationalizability
• Define the normal form associated to generalized
  extensive games with unawareness
• Characterize extensive form rationalizability and
  prudent rationalizability by iterated elimination
  of conditionally strictly (resp. weakly)
  dominated strategies

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Bonus Slides
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Prudent Rationalizability A Refinement?
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in out
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2, 2
in out
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1, 1
in out
0, 100
2, 0

• Would-be Rationalizablity
• Level
• Level
• Level

• Prudent Rationalizablity
• Level
• Level