Unawareness
- Minicourse -
Burkhard C. Schipper
University of California, Davis
Outline
1.
2.
3.
4.
5.
6.
7.
8.
Informal introduction
Epistemic models of unawareness
Type spaces with unawareness
Speculation
Bayesian games with unawareness
Revealed unawareness
Dynamic games with unawareness
…(???)
1. An informal introduction
“Awareness” in Natural Language
“I was aware of the red traffic light.” (Just knowledge?)
“Be of aware of sexually transmitted diseases!"
(“generally taking into account", “being present in mind", “paying
attention to“)
Etymology: “aware” ← “wary” ← “gewӕr” (old English) ← “gewahr”
(German)
Psychiatry: Lack of self-awareness means that a patient is oblivious to
aspects of an illness that is obvious to his/her social contacts. (Failure of
negative introspection.)
“Reports that say that something hasn't happened are
always interesting to me, because as we know, there are
known knowns; there are things we know we know. We
also know there are known unknowns; that is to say we
know there are some things we do not know. But there are
also unknown unknowns – the ones we don't know we
don't know. And if one looks throughout the history of our
country and other free countries, it is the latter category
that tend to be the difficult ones.”
(Former) United States Secretary of Defense, Donald Rumsfeld,
February 12, 2002
Unawareness as Lack of Conception
In most formal approaches:
Unawareness means the lack of conception.
(“not being present in the mind”)
Lack of information versus lack of conception:
We all know the homo oeconomicus.
But how did he look as a baby?
!1 !2 !3
!4 !5 !6 !7 !8 !9 !10 !11
!12 !13 !14 !15 !16 !17 !18 !19
!20 !21 !22 !23 !24 !25 !26 …
!1 !2 !3
!4 !5 !6 !7 !8 !9 !10 !11
!12 !13 !14 !15 !16 !17 !18 !19
!20 !21 !22 !23 !24 !25 !26 …
!1 !2 !3
!4 !5 !6 !7 !8 !9 !10 !11
!12 !13 !14 !15 !16 !17 !18 !19
!20 !21 !22 !23 !24 !25 !26 …
!
Standard models allow for the lack of
information but not for the lack of conception.
In standard models, learning means shrinking
the relevant state space but never discovering
of new possibilities.
An agent may have lack of conception of an event
because:
• never thought about it (i.e., novelties)
• does not pay attention to it at the very moment
we model (different from rational inattention)
Relevance of unawareness
•
•
•
•
•
•
•
It is real phenomena
Incomplete contracting, incomplete markets
Speculation in financial markets
Disclosure of information
Strategic negotiations and bargaining
Modelling discoveries and innovations
Modeling games where the perception of the strategic
context is not necessarily “common” among players
• Exploring robustness of decision theory to small changes
in assumptions on the primitives
• …
Non-robustness of Decision Theory
In decision theory, the decision maker’s perception of the
problem are captured in the primitives (state space, set of
consequences, acts etc.)
These primitives are assumed by the modeler and are not
revealed.
Preference are revealed given primitives.
How do we know that the decision maker views the
problem the same as the modeler?
Example: Non-robustness of the
Ellsberg Paradox
1
2
3
Bets Exp. Val. Red Green Blue
A
33.33 $100
$0
$0
B
33.33
$0 $100
$0
C
D
66.67 $100
$0 $100
66.67
$0 $100 $100
Balls #
30
60
30
30
30
Example: Non-robustness of the
Ellsberg Paradox
1
Bets
A
B
C
D
2
3
3
4 Red
5
6 Green
7
8 Blue
9
10
11
12
13
14
15
16
Bets 1Exp.2 Val.
Exp. Val. Red Green Blue
A $100 $0 33.33
$100
33.33
$0 $0 $100
$0
$0 $0
$0 $100 $0$0 $100 $100
$0 $100 $100 $100
33.33
$0 $100
$0 $0
$0 $100
$0
$0 $100 $100
$0 $100 $100
$0 $100 $100
B
33.33
$0 $100
$0
66.67 $100
$0 $100 $0
66.67
$0 $100 $100 $0
Balls #
30
60
30
30
30
0
C
D
$0
$0
$0 $100
$0
$0
$0 $100
$0 $100
$0 $100 $100 $100
$0 $100
$0
$0 $100
$0 $100 $100 $100 $100
66.67 $100
$0 0 $100
0
0
0
0
0
66.67
$0 $100 $100
Balls #
30
60
30
30
30
0
0
0
0
0
0
Example: Non-robustness of the
Ellsberg Paradox
1
2
3
4
5
6
Bets Exp. Val. Red Green Blue
A
34.07 $100
$0
$0 $0 $100
$0
B
32.97
$0 $100
$0 $0
$0 $100
C
D
65.93 $100
$0 $100 $0
67.03
$0 $100 $100 $0
Balls #
30
60
30
30
30
0
$0
$0
0
7
$0
$0
8
0
10
11
12
13
14
15
16
$0 $100
$0 $100 $100
$0 $100 $100 $100
$0 $100 $100
$0 $100 $100
$0 $100 $100
$0 $100
$0
$0
$0 $100
0
9
0
$0 $100
$0 $100 $100 $100
$0 $100
$0
$0 $100
$0 $100 $100 $100 $100
0
0
1
0
0
0
0
Rationalizes Ellsberg behavior with expected utility
on a slightly different state-space.
Unawareness models allow us to analyze decisions
under varying primitives.
0
How to model unawareness?
2. Epistemic models of
unawareness
What’s the problem with modeling unawareness?
Why not take a state-space model a la Aumann or a Kripke
frame?
What’s the problem with modeling unawareness?
Digression: Necessitation of Belief
Possibility correspondence:
¦
i
: S ¡ ! 2S n f ; g
K nowledge operat or de¯ned on event s E 2 2S :
K i (E ) := f ! 2 S : ¦ i (! ) µ E g
Sat is¯es Necessit at ion by de¯nit ion.
Digression: Necessitation of Belief
Type mapping
t i : S ¡ ! ¢ (S; F S )
Probability-at -least -p-belief operat or de¯ned on event s E 2 F S :
B ip (E ) := f ! 2 S : t i (! )(E ) ¸ pg
Sat is¯es Necessit at ion by de¯nit ion of probability measures
Digression: Necessitation of Belief
\ Ambiguity" type mapping (mult iple probability measures model)
t i : S ¡ ! F ¢ ( S)
Ambiguous probability-at -least -p-belief operat or de¯ned
on event s E 2 F S :
B~ip (E ) := f ! 2 S : 9¹ 2 t i (! ); ¹ (E ) ¸ pg
Sat is¯es Necessit at ion by de¯nit ion of probability measures
What’s the problem with modeling unawareness?
Modeling Unawareness
Various approaches, interdisciplinary
Computer science
Economics
• Modeling logical nonomniscience and limited
reasoning
• Inspired by Kripke
structures
• Analyst’s description of
agents’ reasoning
• Seminal work: Fagin and
Halpern (AI 1988)
• Focused on modeling
lack of conception while
keeping everything else
standard
• Inspired by Aumann
structures and Harsanyi
type spaces
• Players’ descriptions of
players’ reasoning
Unawareness Structures
Goals:
1. Define a structure consistent with non-trivial
unawareness in the multi-agent case.
2. Prove all properties of awareness of Dekel,
Lipman, Rustichini (1998), Modica and
Rustichini (1999), Halpern (2001)
3. Prove unawareness is consistent with strong
notions of knowledge (like S4 or stronger).
4. Clear separation between syntax and
semantics to facilitate applications.
hS; ¹ i nonempty complete lattice of nonempty disjoint spaces
Digression: Lattices
Let X is a set and ¹ be a binary relat ion de¯ned on X .
¹ is partial order if it is
re° exive: x ¹ x for all x 2 X
ant isymmet ric: for any x; y 2 X , x ¹ y and y ¹ x implies x = y
t ransit ive: for any x; y; z 2 X , x ¹ y and y ¹ z implies x ¹ z
Digression: Lattices
Let Y µ X .
u is an upper bound of Y if x ¹ u for all x 2 Y
` is a lower bound of Y if ` ¹ x for all x 2 Y
u is a least upper bound, join, or surpremum of Y if u ¹ x
for all upper bounds x of Y
` is a greatest lower bound, meet, or in¯mum of Y if x ¹ `
for all lower bounds x of Y
Digression: Lattices
X is a lattice if for any x; y 2 X has a join and meet .
X is a complete lattice if any Y µ X has a join and a meet .
Every nonempty ¯nit e lat t ice is complet e.
For any set , t he set of all subset s is a complet e lat t ice ordered
by set inclusion.
hS; ¹ i nonempty complet e lat t ice of nonempty disjoint spaces
For S; S0 2 S, S0 º S st ands for \ S0 is more expressive t han S"
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For S; S0 2 S, S0 º S st ands for \ S0 is more expressive t han S"
- :=
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S2 S
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hS; ¹ i nonempty complet e lat t ice of nonempty disjoint spaces
For S; S0 2 S, S0 º S st ands for \ S0 is more expressive t han S"
- :=
S
S2 S
S
0
0
For S; S 2 S wit h S º S,
S0
rS
: S0 ¡ !
S surject ive project ion.
For any S 2 S, r SS = i dS .
0
00
00
0
For any S; S ; S 2 S, S º S º S,
S 00
rS
=
S0
rS
S 00
± r S0 .
Digression: Surjection
A funct ion f : X ¡ ! Y is surject ive or ont o if for
every y 2 Y t here exist s an x 2 X such t hat f (x) = y.
Student 1
Student 2
Student 3
Student 4
Thomas
Anna
京
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For S; S0 2 S, S0 º S st ands for \ S0 is more expressive t han S"
- :=
S
S2 S
S
0
0
For S; S 2 S wit h S º S,
S0
rS
: S0 ¡ !
S surject ive project ion.
For any S 2 S, r SS = i dS .
0
00
00
0
For any S; S ; S 2 S, S º S º S,
S 00
rS
=
S0
rS
S 00
± r S0 .
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hS; ¹ i nonempty complet e lat t ice of nonempty disjoint spaces
For S; S0 2 S, S0 º S st ands for \ S0 is more expressive t han S"
- :=
S
S2 S
S
0
0
For S; S 2 S wit h S º S,
S0
rS
: S0 ¡ !
S surject ive project ion.
For any S 2 S, r SS = i dS .
0
00
00
0
For any S; S ; S 2 S, S º S º S,
0
For ! 2 S , !
S
=
S0
r S (!
S 00
rS
=
S0
rS
). For D ½ S0, D S = f !
S 00
± r S0 .
S
: ! 2 D g.
"
For D µ S, D :=
³
S
S 02 S :S 0º S
r SS
0
´¡
1
(D ).
E µ - is an event if E = D " for some base D µ S
in some base space S 2 S. (S(E ))
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For D µ S, D :=
³
S
S 02 S :S 0º S
r SS
0
´¡
1
(D ).
E µ - is an event if E = D " for some base D µ S
in some base space S 2 S. (S(E ))
We writ e ; S for t he vacuous event wit h base space S.
Not every subset of - is an event .
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For D µ S, D :=
³
S
S 02 S :S 0º S
r SS
0
´¡
1
(D ).
E µ - is an event if E = D " for some base D µ S
in some base space S 2 S. (S(E ))
We writ e ; S for t he vacuous event wit h base space S.
Not every subset of - is an event .
Negat ion: For D " an event ,
³ 0´ ¡ 1
S
: D " := S 02 S :S 0º S r SS
(S n D ).
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"
For D µ S, D :=
³
S
S 02 S :S 0º S
r SS
0
´¡
1
(D ).
E µ - is an event if E = D " for some base D µ S
in some base space S 2 S. (S(E ))
We writ e ; S for t he vacuous event wit h base space S.
Not every subset of - is an event .
Negat ion: For D " an event ,
³ 0´ ¡ 1
S
: D " := S 02 S :S 0º S r SS
(S n D ).
"
For D µ S, D :=
³
S
S 02 S :S 0º S
r SS
0
´¡
1
(D ).
E µ - is an event if E = D " for some base D µ S
in some base space S 2 S. (S(E ))
We writ e ; S for t he vacuous event wit h base space S.
Not every subset of - is an event .
Negat ion: For D " an event ,
³ 0´ ¡ 1
S
: D " := S 02 S :S 0º S r SS
(S n D ).
Typcially : E $ - n E .
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f E j g collect ion of event s
Conjunct ion:
V
j
E j :=
T
j
Ej
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f E j g collect ion of event s
Conjunct ion:
Disjunct ion:
Typically
W
j
V
j
W
j
E j :=
T
E j := :
Ej $
S
j
Ej
j
Ej
³V
´
j
: Ej
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f E j g collect ion of event s
Conjunct ion:
Disjunct ion:
Typically
W
j
V
j
W
j
E j :=
T
E j := :
Ej $
S
j
j
Ej
³V
´
j
: Ej
Ej
E µ F if and olny if E µ F and S(E ) º S(F ).
Set of all event s is not necessarily an algebra
(many vacuous event s).
For each i 2 I , t here is a possibility correspondence ¦
sat isfying t he following propert ies:
i
: - ¡ ! 2- n f ; g
0. Con¯nedment : If ! 2 S t hen ¦ i (! ) µ S0 for some S0 ¹ S.
!
S
¦i(!)
S’
!
¦i(!‘)
¦i(!) !’
 !’’
S
S
¦i(!)
!
S’
S“
¦i(!‘‘)
S’
S’’’
For each i 2 I , t here is a possibility correspondence ¦
sat isfying t he following propert ies:
i
: - ¡ ! 2- n f ; g
0. Con¯nedment : If ! 2 S t hen ¦ i (! ) µ S0 for some S0 ¹ S.
1. Generalized Re° exivity: ! 2 ¦ "i (! ) for every ! 2 - .
2. St at ionarity: ! 0 2 ¦ i (! ) implies ¦ i (! 0) = ¦ i (! ).
3. Project ions Preserve Awareness: If ! 2 S0, ! 2 ¦ i (! ) and S ¹ S0
t hen ! S 2 ¦ i (! S ).
4. Project ions Preserve Ignorance: If ! 2 S0 and S ¹ S0
t hen ¦ "i (! ) µ ¦ "i (! S ).
5. Project ions Preserve K nowledge: If S ¹ S0 ¹ S00, ! 2 S00
and ¦ i (! ) µ S0 t hen (¦ i (! )) S = ¦ i (! S ).
¡
¢
S ¡ 1
r S0
(¦ i (!
!
))
S
¦i(!)
S’
¡
¢
S ¡ 1
r S0
(¦ i (!
!
))

S
¦i(!)
¡
¢
S ¡ 1
r S0
(¦ i (!
))
S
¦i(!)
S’
S’
For each i 2 I , t here is a possibility correspondence ¦
sat isfying t he following propert ies:
i
: - ¡ ! 2- n f ; g
0. Con¯nedment : If ! 2 S t hen ¦ i (! ) µ S0 for some S0 ¹ S.
1. Generalized Re° exivity: ! 2 ¦ "i (! ) for every ! 2 - .
2. St at ionarity: ! 0 2 ¦ i (! ) implies ¦ i (! 0) = ¦ i (! ).
¦i(!) = ¦i(!‘)
!
 !’
S
¦i(!)
¦i(!‘)
!
 !’
 !’’
¦i(!) = ¦i(!‘)
!
 !’
S
S
¦i(!‘‘)
S’
For each i 2 I , t here is a possibility correspondence ¦
sat isfying t he following propert ies:
i
: - ¡ ! 2- n f ; g
0. Con¯nedment : If ! 2 S t hen ¦ i (! ) µ S0 for some S0 ¹ S.
1. Generalized Re° exivity: ! 2 ¦ "i (! ) for every ! 2 - .
2. St at ionarity: ! 0 2 ¦ i (! ) implies ¦ i (! 0) = ¦ i (! ).
3. Project ions Preserve Awareness: If ! 2 S0, ! 2 ¦ i (! ) and S ¹ S0
t hen ! S 2 ¦ i (! S ).
!
¦i(!)
S’
 !S
¦i(!S)
S
!
!
¦i(!)
¦i(!)
S’
S’
 !S
S
 !S
¦i(!S)
¦i(!S)
S
S‘‘
For each i 2 I , t here is a possibility correspondence ¦
sat isfying t he following propert ies:
i
: - ¡ ! 2- n f ; g
0. Con¯nedment : If ! 2 S t hen ¦ i (! ) µ S0 for some S0 ¹ S.
1. Generalized Re° exivity: ! 2 ¦ "i (! ) for every ! 2 - .
2. St at ionarity: ! 0 2 ¦ i (! ) implies ¦ i (! 0) = ¦ i (! ).
3. Project ions Preserve Awareness: If ! 2 S0, ! 2 ¦ i (! ) and S ¹ S0
t hen ! S 2 ¦ i (! S ).
4. Project ions Preserve Ignorance: If ! 2 S0 and S ¹ S0
t hen ¦ "i (! ) µ ¦ "i (! S ).
!
S’
¦i(!)
 !S
S
¦i(!S)
!
!
S’
S’
¦i(!)
¦i(!)
 !S
 !S
S
S
¦i(!S)
¦i(!S)
For each i 2 I , t here is a possibility correspondence ¦
sat isfying t he following propert ies:
i
: - ¡ ! 2- n f ; g
0. Con¯nedment : If ! 2 S t hen ¦ i (! ) µ S0 for some S0 ¹ S.
1. Generalized Re° exivity: ! 2 ¦ "i (! ) for every ! 2 - .
2. St at ionarity: ! 0 2 ¦ i (! ) implies ¦ i (! 0) = ¦ i (! ).
3. Project ions Preserve Awareness: If ! 2 S0, ! 2 ¦ i (! ) and S ¹ S0
t hen ! S 2 ¦ i (! S ).
4. Project ions Preserve Ignorance: If ! 2 S0 and S ¹ S0
t hen ¦ "i (! ) µ ¦ "i (! S ).
5. Project ions Preserve K nowledge: If S ¹ S0 ¹ S00, ! 2 S00
and ¦ i (! ) µ S0 t hen (¦ i (! )) S = ¦ i (! S ).
!
S“
¦i(!)
S’
 !S
¦i(!S) = (¦i(!))S
S
!
!
S“
¦i(!)
S“
¦i(!)
S’
 !S
¦i(!S) = (¦i(!))S
S
S’
 !S
¦i(!S) ¾ (¦i(!))S
S
For each i 2 I , t here is a possibility correspondence ¦
sat isfying t he following propert ies:
i
: - ¡ ! 2- n f ; g
0. Con¯nedment : If ! 2 S t hen ¦ i (! ) µ S0 for some S0 ¹ S.
1. Generalized Re° exivity: ! 2 ¦ "i (! ) for every ! 2 - .
2. St at ionarity: ! 0 2 ¦ i (! ) implies ¦ i (! 0) = ¦ i (! ).
3. Project ions Preserve Awareness: If ! 2 S0, ! 2 ¦ i (! ) and S ¹ S0
t hen ! S 2 ¦ i (! S ).
4. Project ions Preserve Ignorance: If ! 2 S0 and S ¹ S0
t hen ¦ "i (! ) µ ¦ "i (! S ).
5. Project ions Preserve K nowledge: If S ¹ S0 ¹ S00, ! 2 S00
and ¦ i (! ) µ S0 t hen (¦ i (! )) S = ¦ i (! S ).
R em ar k Generalized Re° exivity implies t hat if S0 ¹ S, ! 2 S
and ¦ i (! ) µ S0, t hen r SS0 (! ) 2 ¦ i (! ). In part icular, we have
¦ i (! ) 6
= ; , for all ! 2 - .
R em ar k Property 5 and Con¯nement (Property 0) imply Property 3.
R em ar k Generalized Re° exivity implies t hat if S0 ¹ S, ! 2 S
and ¦ i (! ) µ S0, t hen r SS 0 (! ) 2 ¦ i (! ). In part icular, we have
¦ i (! ) 6
= ; , for all ! 2 - .
R em ar k Property 5 and Con¯nement (Property 0) imply Property 3.
D
S0
hS; ¹ i ; (r S ) S¹
E
S 0 ; (¦ i ) i 2 I
is an unawareness structure.
T he knowledge operator of individual i on event s E is de¯ned,
as usual, by
K i (E ) := f ! 2 - : ¦ i (! ) µ E g ;
if t here is a st at e ! such t hat ¦ i (! ) µ E , and by
K i (E ) := ; S( E )
ot herwise.
T he knowledge operator of individual i on event s E is de¯ned,
as usual, by
K i (E ) := f ! 2 - : ¦ i (! ) µ E g ;
if t here is a st at e ! such t hat ¦ i (! ) µ E , and by
K i (E ) := ; S( E )
ot herwise.
P r op osit ion If E is an event , t hen K i (E ) is an S(E )-based event .
P r op osit ion T he knowledge operat or K i has t he following propert ies:
(i) Necessit at ion: K i (- ) = - ,
³T
´
T
(ii) Conjunct ion: K i
= j K i (E j ),
j Ej
(iii) Trut h: K i (E ) µ E ,
(iv) Posit ive Int rospect ion: K i (E ) µ K i K i (E ),
(v) Monot onicity: E µ F implies K i (E ) µ K i (F ),
(vi) : K i (E ) \ : K i : K i (E ) µ : K i : K i : K i (E ).
T he awareness operator of individual i is de¯ned on event s E by
A i (E ) = f ! 2 - : ¦ i (! ) µ S(E )g;
if t here is a st at e ! 2 - such t hat ¦ i (! ) µ S(E ), and by
A i (E ) = ; S( E )
ot herwise.
T he awareness operator of individual i is de¯ned on event s E by
©
ª
"
A i (E ) = ! 2 - : ¦ i (! ) µ S(E ) ;
if t here is a st at e ! 2 - such t hat ¦ i (! ) µ S(E ), and by
A i (E ) = ; S( E )
ot herwise.
T he unawareness operator is t hen nat urally de¯ned by
Ui (E ) = : A i (E ):
P r op osit i on T he following propert ies obt ain:
1. K U Int rospect ion: K i Ui (E ) = ; S( E ) ,
2. AU Int rospect ion: Ui (E ) = Ui Ui (E ),
3. Plausibility: Ui (E ) = : K i (E ) \ : K i : K i (E ),
T1
n
4. St rong Plausibility: Ui (E ) = n = 1 (: K i ) (E ),
³
´
"
5. Weak Necessit at ion: A i (E ) = K i S (E ) ,
6. Weak Negat ive Int rospect ion: : K i (E ) \ A i : K i (E ) = K i : K i (E ),
7. Symmet ry: A i (E ) = A i (: E ),
¡T
¢
T
8. A-Conjunct ion: ¸ 2 L A i (E ¸ ) = A i
¸ 2 L E¸ ,
9. AK - Re° ect ion: A i (E ) = A i K i (E ),
10. AA- Re° ect ion: A i (E ) = A i A i (E ),
11. A-Int rospect ion: A i (E ) = K i A i (E ).
Illustration
E = {!2}
!2
!3
!1
Ab(E)={!2}
Ub(E) = {!3}
:KaUb(E) = {!2, !3}
:KaAb(E) = {!2, !3}
Kb:KaUb(E) = {!2}
AaKb:KaUb(E) = {!2, !3}
…
T he mut ual knowledge operat or on event s is de¯ned by
\
K (E ) =
K i (E ):
i2I
T he mut ual knowledge operat or on event s is de¯ned by
\
K (E ) =
K i (E ):
i2I
T he common knowledge operat or is de¯ned by
\1
CK (E ) =
n= 1
K n (E ):
T he mut ual knowledge operat or on event s is de¯ned by
\
K (E ) =
K i (E ):
i2I
T he common knowledge operat or is de¯ned by
\1
K n (E ):
CK (E ) =
n= 1
T he mut ual awareness operat or on event s is de¯ned by
\
A(E ) =
A i (E );
i2I
and t he common awareness operat or by
\1
CA(E ) =
n
(A) (E ):
n= 1
P r op osit i on T he following mult i-agent propert ies obt ain:
1. A i (E ) = A i A j (E ),
2. A i (E ) = A i K j (E ),
3. K i (E ) µ A i K j (E ),
4. A(E ) = K (S(E ) " ),
5. A(E ) = CA(E ),
6. K (E ) µ A(E ),
7. CK (E ) µ CA(E ),
8. CK (S(E ) " ) µ CA(E ).
Logic of unawareness
• Syntax versus semantics: What’s the internal structure of
states? What’s the interpretation of the lattice?
• How comprehensive are unawareness structures? Can
the model itself be subject to agents’ uncertainties? Can
any minute detail of awareness, beliefs, mutual beliefs,
etc. of all individuals be described in some state of the
model?
• Soundness: Is any theorem valid in all states?
• Completeness: Is every formula that is valid in all states
also provable?
Given a nonempty set of agent s i 2 I , a nonempty set of
at omic formulae p 2 At as well as t he special formula > ,
t he formulae ' of t he language L kI ;a (At ) are de¯ned
by t he grammar
' ::= p j : ' j ' ^ Ã j ki ' j ai ' :
at omic formula: \ penicillium rubens has ant ibiot ic propert ies"
Given a nonempty set of agent s i 2 I , a nonempty set of
at omic formulae p 2 At as well as t he special formula > ,
t he formulae ' of t he language L kI ;a (At ) are de¯ned
by t he grammar
' ::= p j : ' j ' ^ Ã j ki ' j ai ' :
at omic formula: \ penicillium rubens has ant ibiot ic propert ies"
' _ Ã := : (: ' ^ : Ã)
(' ) Ã) := (: ' _ Ã)
(' , Ã) := (' ) Ã) ^ (Ã ) ' )
De¯ne induct ively t he set of primit ive proposit ions At (' )
t hat appear in '
² At (> ) := ; ,
² At (p) := p, for p 2 At ,
² At (: ' ) := At (' ),
² At (' ^ Ã) := At (' ) [ At (Ã),
² At (ki ' ) := At (' ) = : At (ai ' ).
De¯ne induct ively t he set of primit ive proposit ions At (' )
t hat appear in '
² At (> ) := ; ,
² At (p) := p, for p 2 At ,
² At (: ' ) := At (' ),
² At (' ^ Ã) := At (' ) [ At (Ã),
² At (ki ' ) := At (' ) = : At (ai ' ).
Modica-Rust ichini de¯nit ion of awareness: ai ' := ki ' _ ki : ki ' .
An axiom is a formula assumed.
An inference rule infers a formula (i.e., a conclusion) from
a collect ion of formulae (i.e., t he hypot hesis).
An axiom system consist s of a collect ion of axioms and
inferences rules.
Axiom syst em S5kI ;a :
Prop. All subst it ut ions inst ances of t aut ologies of proposit ional logic,
including t he formula > .
AS. ai : ' ,
ai ' (Symmet ry)
AC. ai (' ^ Ã) ,
A i K j R. ai ' ,
ai ' ^ ai à (Awareness Conjunct ion)
ai kj ' , for all j 2 I (Awareness K nowledge Re° ect ion)
T . ki ' ) ' (Axiom of Trut h)
4. ki ' ) ki ki ' (Posit ive Int rospect ion Axiom)
MP. From ' and ' ) à infer à (modus ponens)
RK . For all nat uralSnumbers n ¸ 1, if ' 1 ; ' 2 ; :::; ' n and ' are such
t hat At (' ) µ n`= 1 At (' ` ), t hen ' 1 ^ ' 2 ^ ¢¢¢^ ' n ) '
implies ki ' 1 ^ ki ' 2 ^ ¢¢¢^ ki ' n ) ki ' . (RK -Inference)
A proof in an axiom syst em consist s of a sequence of formulae,
where each formula is eit her an axiom in t he axiom syst em or follows
by an applicat ion of an inference rule.
A proof is a proof of a formula ' if t he last formula in t he proof is ' .
A formula ' is provable in an axiom syst em if t here is a proof of '
in t he axiom syst em.
T he set of theorems of an axiom syst em is t he smallest set of
formulae t hat cont ain all axioms and t hat is closed under inference
rules of t he axiom syst em.
De¯ne ui ' := : ai ' .
R em ar k T he Modica and Rust ichini de¯nit ion of awareness
and axiom syst em S5kI ;a implies:
K . ki ' ^ ki (' ) Ã) ) ki '
ki ' ^ ki à ) ki (' ^ Ã)
NNI. ui ' ) : ki : ki : ki '
AI. ai ' ) ki ai '
V
AGPP. ai ' ,
p2 A t ( ' ) ai p
GenA . If ' is a t heorem, t hen ai ' ) ki ' is a t heorem.
Given a language L (At ), a set of formulae ¡ is consistent
wit h respect t o an axiom syst em if and only if t here is no
formula ' such t hat bot h ' and : ' are provable from ¡ .
! is maximally consistent of formulae if it is consist ent and
for any formula ' 2 L (At ) n ! , t he set ! [ f ' g is not
consist ent .
Every consist ent subset of L (At ) can be ext ended t o a
maximally consist ent subset ! of L (At ). Moreover,
¡ µ L (At ) is a maximally consist ent subset of L (At )
if and only if ¡ is consist ent and for every ' 2 L(At),
' 2 ¡ or : ' 2 ¡ .
Canonical const r uct ion:
For every At 0 µ At, let SA t 0 be t he set of maximally consist ent
set s ! A t 0 of formulae in t he sublanguage L kI ;a (At 0).
C anonical const r uct ion:
For every At 0 µ At, let SA t 0 be t he set of maximally consist ent
set s ! A t 0 of formulae in t he sublanguage L kI ;a (At 0).
hf SA t 0 gA t 0µ At ; ¹ i is a complet e lat t ice of disjoint spaces wit h t he
order de¯ned by SA t 00 º SAt 0 if and only if At 00 ¶ At 0.
- :=
S
At 0µ At
SAt 0 .
C anonical const r uct ion:
For every At 0 µ At, let SA t 0 be t he set of maximally consist ent
set s ! A t 0 of formulae in t he sublanguage L kI ;a (At 0).
hf SA t 0 gA t 0µ At ; ¹ i is a complet e lat t ice of disjoint spaces wit h t he
order de¯ned by SA t 00 º SAt 0 if and only if At 00 ¶ At 0.
- :=
S
At 0µ At
SAt 0 .
00
For any SA t 00 º SA t 0 , surject ive project ions r AA tt 0 : SA t 00 ¡ !
A t 00
are de¯ned by r A t 0 (! ) := ! \ L kI ;a (At 0).
SA t 0
T heor em For every ! and i 2 I , t he possibility correspondence de¯ned
by ¦ i (! )
½
¾
0
(i) ki ' implies ' 2 ! ; and
:= ! 0 2 - : For every formula ' ;
(ii) ai ' 2 ! i® ' 2 ! 0 or : ' 2 ! 0
sat is¯es Con¯nement , Generalized Re° exivity, Project ions Preserve
Ignorance, and Project ions Preserve K nowledge. Moreover, for every
formula ' , t he set of st at es [' ] := f ! 2 - : ' 2 ! g is a SA t ( ' ) -based
event , and [: ' ] = : [' ], [' ^ Ã] = [' ] \ [Ã], [ki ' ] = K i [' ], [ai ' ] = A i [' ],
and [ui ' ] = Ui [' ].
Set of event s in unawareness st ruct ures §
Valuat ion V : At ¡ ! §
Set of event s in unawareness st ruct ures §
Valuat ion V : At ¡ ! §
0
Denot e M = hhS; ¹ i ; (r SS ) S 0¹ S ; (¦ i ) i 2 I ; V i .
Set of event s in unawareness st ruct ures §
Valuat ion V : At ¡ !
§
0
Denot e M = hhS; ¹ i ; (r SS ) S 0¹ S ; (¦ i ) i 2 I ; V i .
De¯ne induct ively t he sat isfact ion relat ion on t he st ruct ure of
formulae in L kI ;a (At )
M ; ! j= > , for all ! 2 - ,
M ; ! j= p if and only if ! 2 V (p),
M ; ! j= ' ^ Ã if and only if ! 2 [' ] \ [Ã],
M ; ! j= : ' if and only if ! 2 [: ' ],
M ; ! j= ki ' if and only if ! 2 K i [' ],
where [' ] := f ! 0 2 - : M ; ! 0 j= ' g, for every formula ' .
In a Kripke st ruct ure, a formula is valid if it is t rue in every st at e.
In an unawareness st ruct ure, a formula may not be de¯ned in
every st at e!
In a K ripke st ruct ure, a formula is valid if it is t rue in every st at e.
In an unawareness st ruct ure, a formula may not be de¯ned in
every st at e!
' is de¯ned in state ! in M if
A formula
T
! 2 p2 At ( ' ) (V (p) [ : V (p)).
A formula ' is valid in M if M ; ! j= ' for all ! in which ' is
de¯ned.
A formula ' is valid if it is valid in all M .
An axiom syst em is sound for a language L wit h respect t o a
class of st ruct ures if every formula in L t hat is provable in t he
axiom syst em is valid wit h respect t o every st ruct ure M .
An axiom syst em is complete for a language L wit h respect t o a
class of st ruct ures if every formula in L t hat is valid in every
st ruct ure M is provable in t he axiom syst em.
An axiom syst em is sound for a language L wit h respect t o a
class of st ruct ures if every formula in L t hat is provable in t he
axiom syst em is valid wit h respect t o every st ruct ure M .
An axiom syst em is complete for a language L wit h respect t o a
class of st ruct ures if every formula in L t hat is valid in every
st ruct ure M is provable in t he axiom syst em.
T heor em For t he language L kI ;a (At ), t he axiom syst em S5kI ;a
is a sound and complet e axiomat izat ion wit h respect t o
unawareness st ruct ures.
An Example: Speculative Trade
How to model the following example?
Two agents: An owner o of a firm and a potential buyer b.
Status quo value of the firm is $100.
The owner is aware of a potential lawsuit l that may cost the firm
$20. The buyer is unaware of it. The owner knows that the buyer is
unaware of it.
The buyer is aware of a potential innovation n that may enhance
the value of the firm by $20. The owner is unaware of it. The buyer
knows that the owner is unaware of it.
Suppose the buyer is offering to buy the firm for $100. Is the owner
going to sell to her?
An Example: Speculative Trade
●
n, l
●
●
n, :l
:n, l
●
:n,:l
Sn, l
●
n
●
●
:n
l
●
:l
Sl
Sn
●
>
S;
The First Approach: Awareness Structures
(Fagin and Halpern, Artificial Intelligence 1988)
L(n)
●
L(l)
●
●
L(l)
n, :l
n, l
L(n)
L(n)
L(l)
:n, l
L(n)
L(l)
●
:n,:l
L(n)
●
L(l)
●
L(l)
n, :l
n, l
L(n)
L(n)
L(l)
●
:n, l
L(n)
Fagin and Halpern (1988), Halpern
(2001), Halpern and Rego (2008)
provide sound and complete
axiomatizations of awareness
structures under various
assumptions on accessibility
relations and awareness
correspondences. (details)
L(l)
●
:n,:l
The example models our story.
Buyer:
L(n)
●
L(l)
●
L(l)
n, :l
n, l
L(n)
L(n)
L(l)
●
:n, l
L(n)
L(l)
●
:n,:l
“I (implicitly) know that the
owner is aware of the lawsuit
but unfortunately I am
unaware of the lawsuit.”
It models reasoning of an
outside observer about the
knowledge and awareness of
agents but not the agents’
subjective reasoning that
economists, decision
theorists, and game theorists
are interested in.
Semantics is not syntax-free. What would
be the analogue to a Kripke frame?
Awareness structures are very flexible. For instance,
Should be very useful to model various forms of framing.
The Second Approach: Generalized Standard Models
(Modica and Rustichini, GEB 1999)
●
n, l
●
n, :l
●
:n, l
●
Single-agent structure
:n,:l
Sn, l
●
n
Sn
●
:n
Modica and Rustichini (1999) prove a sound
and complete axiomatization.
Use “objective” validity.
Models reasoning of an outside observer
about knowledge and awareness of a single
agent.
A Fourth Approach: Product Models
(Li, 2008, JET 2009)
n
●
l
(1n,1l)
n
●
l
(0n,1l)
*
n
●
l
(1n,0l)
n
●
l
(0n,0l)
A Fourth Approach: Product Models
(Li, 2008, JET 2009)
n
●
l
n
●
l
(1n,1l) (1n,0l)
n
●
n
l
(0n,1l)
●
l
(0n,0l)
*
●
(1n)
●
●
(0n)
(1n)
b(*)
o(*)
●
o (b(*)) = b (o(*))
●
(0n)
A Fourth Approach: Product Models
(Li, 2008, JET 2009)
n
●
n
l
●
n
l
(1n,1l) (1n,0l)
●
n
l
(0n,1l)
l
●
(0n,0l)
*
n
●
;
(1n)
n
●
;
;
(0n)
●
l
(1n)
b(*)
o(*)
;
●
;
o (b(*)) = b (o(*))
;
●
l
(0n)
A Fourth Approach: Product Models
(Li, 2008, JET 2009)
n
●
n
l
●
n
l
(1n,1l) (1n,0l)
●
n
l
(0n,1l)
Example models our story
l
●
(0n,0l)
Possibility correspondence
models implicit knowledge.
*
n
●
;
(1n)
n
●
;
;
(0n)
●
l
(1n)
b(*)
o(*)
;
●
;
●
l
(0n)
Sublattices represent
subjective perceptions of
the situation, while the
objective state space
represents the outside
modeler’s view.
;
o (b(*)) = b (o(*))
Li (2009) proves properties
of unawareness and
knowledge
Further Epistemic Approaches
Awareness of Unawareness:
• Awareness structures with propositional quantifiers:
Halpern and Rego (MSS 2013), Halpern and Rego (GEB
2009)
• First-order modal logic with unawareness of objects:
Board and Chung (2011)
• First-order modal logic with unawareness of objects and
properties: Quantified awareness neighborhood
structures: Sillari (RSL 2008)
• Awareness of unawareness without quantifiers: Agotnes
and Alechina (2007), Walker (MSS 2014)
Dynamic awareness:
…
“Evolution” of the literature on
epistemic models of unawareness
Survey:
Schipper, B.C. “Awareness”, in: Handbook of Epistemic Logic,
Chapter 3, H. van Ditmarsch, J.Y. Halpern, W. van der Hoek
and B. Kooi (Eds.), College Publications, London, 2015,
77–146.
Li (2009)
Prod. models
Heinsalu (2012)
Halpern (2001)
Single-agent aw. struc.
Halpern (2001)
Modica Rustichini (1999)
Gen. stand. models
Li (2008) Multiagent prod. m.
Single-agent
Multi-agent
HMS (2006, 2008)
Unawareness struct.
Board Chung Schipper (2011)
Ewerhart
(2001)
Pires
(1994)
Halpern Rego (2008)
Board Chung (2011b)
Obj.-based unaw.
FH (1988) Prop. gen.
awareness structures
van Ditmarsch et al.
(2013)
Feinberg (2004,
2005, 2012)
Galanis
(2011, 2013)
Wansing (1990)
Fagin Halpern (1988)
Awarness struct.
Agotnes Alechina
(2007)
Walker
(2013)
Quantification
Sillari (2008b)
Sillari (2008a)
Aw. neighb.
Sillari (2008a) Imp.
worlds neighb.
Halpern Rego (2009)
Prop. quant. aw. struct.
Halpern Rego (2013)
Prop. quant. aw. struct.
Board Chung (2011a)
FO Obj.-based unaw.
Rantala (1982)
Impossible worlds
Sillari (2008b)
Sillari (2008a)
FO Aw. neighb.
Sillari (2008a) FO
Imposs. w. neighb.
Returning to the speculative trade example:
Say that at a state s an agent is willing to trade at the price $x if either
she strictly prefers to trade at $x or she is indifferent between trading or
not at $x.
• At $100, both agents are willing
to trade (in every state).
• This is common knowledge
among the agents at all states.
• Yet, each agent has a strict
preference to trade in all in all
states except the lowest space.
Both are indifferent between
trading and not trading in the
lowest space. The states in the
lowest space are the “states of
mind” of an agent as viewed by
the other agent.
Returning to the speculative trade example:
Say that at a state s an agent is willing to trade at the price $x if either
she strictly prefers to trade at $x or she is indifferent between trading or
not at $x.
• At $100, both agents are willing
to trade (in every state).
• This is common knowledge
among the agents at all states.
• Yet, each agent has a strict
preference to trade in all in all
states except the lowest space.
Both are indifferent between
trading and not trading in the
lowest space. The states in the
lowest space are the “states of
mind” of an agent as viewed by
the other agent.
This is in contrast to the “Nospeculative-trade-theorems” of
structures without unawareness
(e.g., Milgrom and Stokey, 1982):
If there is a common prior and
common knowledge of willingness to
trade, then both must be indifferent
to trade.
Need to talk about probabilities.
Outline
1.
2.
3.
4.
5.
6.
7.
8.
Informal introduction
Epistemic models of unawareness
Type spaces with unawareness
Speculation
Bayesian games with unawareness
Revealed unawareness
Dynamic games with unawareness
…(???)
3. Type spaces with
unawareness
Type Spaces with Unawareness
• Economists frequently make use of
probabilistic notions of belief
• Quantified notions of belief are often
useful for applications (optimization under
uncertainty)
• How to model unawareness and
probabilistic beliefs?
Each S 2 S is now a measurable space wit h sigma-¯eld F S .
If S0 º S, we require a measurable surject ive project ion
S0
r S : S0 ! S.
§ denot es now t he set of measurable event s in t he
unawareness st ruct ure.
Denot e by 4 (S) is a set of probability measures on (S; F S )
endowed wit h t he sigma-¯eld F 4 ( S) generat ed by
f ¹ 2 4 (S) : ¹ (D ) ¸ pg, D 2 F S , p 2 [0; 1].
For ¹ 2 4 (S0), t he marginal ¹ j S of ¹ on S ¹ S0 is de¯ned by
µ³
¶
´
¡
1
S0
¹ j S (D ) := ¹
rS
(D ) ; D 2 F S :
Denot e by S¹ t he space on which ¹ is a probability measure.
Whenever S¹ º S(E ), we abuse not at ion ¹ (E ) = ¹ (E \ S¹ ).
If S¹ 6
º S(E ), t hen ¹ (E ) is unde¯ned.
D e¯nit ion
S For each individual i 2 I t here is a type mapping
ti : - !
®2 A ¢ (S® ), which is measurable in t he sense t hat for
every S 2 S and Q 2 F ¢ ( S) we have t ¡i 1 (Q) \ S 2 F S , for all
S 2 S.
We require t he type mapping t i t o sat isfy t he following
propert ies:
(0) Con¯nement: If ! 2 S0 t hen t i (! ) 2 ¢ (S) for some S ¹ S0.

S’
ti()
S

ti()
 ’
S’
S’
ti()

S
S
ti(’)
S“
D e¯nit ion
S For each individual i 2 I t here is a type mapping
ti : - !
®2 A ¢ (S® ), which is measurable in t he sense t hat for
every S 2 S and Q 2 F ¢ ( S) we have t ¡i 1 (Q) \ S 2 F S , for all
S 2 S.
We require t he type mapping t i t o sat isfy t he following
propert ies:
(0) Con¯nement: If ! 2 S0 t hen t i (! ) 2 ¢ (S) for some S ¹ S0.
(1) If S00 º S0 º S, ! 2 S00, and t i (! ) 2 4 (S) t hen
t i (! S 0 ) = t i (! ).

S’’
 S’
S’
ti() = ti(S’)
S


S“
S’’
 S’
 S’
S’
S’
ti()
ti() = ti(S’)
S
ti(S')
S
ti(S’)
S*
D e¯nit ion
S For each individual i 2 I t here is a type mapping
ti : - !
®2 A ¢ (S® ), which is measurable in t he sense t hat for
every S 2 S and Q 2 F ¢ ( S) we have t ¡i 1 (Q) \ S 2 F S , for all
S 2 S.
We require t he type mapping t i t o sat isfy t he following
propert ies:
(0) Con¯nement: If ! 2 S0 t hen t i (! ) 2 ¢ (S) for some S ¹ S0.
(1) If S00 º S0 º S, ! 2 S00, and t i (! ) 2 4 (S) t hen
t i (! S 0 ) = t i (! ).
(2) If S00 º S0 º S, ! 2 S00, and t i (! ) 2 4 (S0) t hen
t i (! S ) = t i (! ) j S .

S’’
ti()
S’
ti(S) = ti()|S
S


S’’
S’’
ti()
ti()
S’
S’
ti(S) = ti()|S
S
ti(S)
ti()|S
S
D e¯nit ion
S For each individual i 2 I t here is a type mapping
ti : - !
®2 A ¢ (S® ), which is measurable in t he sense t hat for
every S 2 S and Q 2 F ¢ ( S) we have t ¡i 1 (Q) \ S 2 F S , for all
S 2 S.
We require t he type mapping t i t o sat isfy t he following
propert ies:
(0) Con¯nement: If ! 2 S0 t hen t i (! ) 2 ¢ (S) for some S ¹ S0.
(1) If S00 º S0 º S, ! 2 S00, and t i (! ) 2 4 (S) t hen
t i (! S 0 ) = t i (! ).
(2) If S00 º S0 º S, ! 2 S00, and t i (! ) 2 4 (S0) t hen
t i (! S ) = t i (! ) j S .
(3) If S00 º S0 º S, ! 2 S00, and t i (!
S 0)
2 4 (S) t hen St i ( ! ) º S.

S’’
ti()
S’
S’
St ()
i
ti(S’)
S


S’’
S’’
ti()
S’
S’
St ()
i
 S’
ti(S’)
S
S’
ti()
ti(S’)
S
St ()
i
ti()
D e¯nit ion
S For each individual i 2 I t here is a type mapping
ti : - !
®2 A ¢ (S® ), which is measurable in t he sense t hat for
every S 2 S and Q 2 F ¢ ( S) we have t ¡i 1 (Q) \ S 2 F S , for all
S 2 S.
We require t he type mapping t i t o sat isfy t he following
propert ies:
(0) Con¯nement: If ! 2 S0 t hen t i (! ) 2 ¢ (S) for some S ¹ S0.
(1) If S00 º S0 º S, ! 2 S00, and t i (! ) 2 4 (S) t hen
t i (! S 0 ) = t i (! ).
(2) If S00 º S0 º S, ! 2 S00, and t i (! ) 2 4 (S0) t hen
t i (! S ) = t i (! ) j S .
(3) If S00 º S0 º S, ! 2 S00, and t i (!
S 0)
2 4 (S) t hen St i ( ! ) º S.
R em ar k Property 1 is implied by t he ot her propert ies.
A
n ssum pt ion ( I nt r osp ect ion)
o If
! 0 2 - : t i (! 0) j St i ( ! ) = t i (! ) µ E , E an event , t hen
t i (! )(E ) = 1.
D e¯nit i on For i 2 I and an event E , de¯ne t he awareness
operator
A i (E ) := f ! 2 - : t i (! ) 2 ¢ (S) ; S º S (E )g
if t here is a st at e ! such t hat t i (! ) 2 ¢ (S) wit h S º S(E ),
and by
A i (E ) := ; S( E )
ot herwise.
T he unawareness operator of individual i 2 I on event s is now
de¯ned by
Ui (E ) = : A i (E ):
L em m a If E is a (not necessarily measurable) event , t hen
A i (E ) (and t hus Ui (E )) is an S (E )-based event .
D e¯nit i on For i 2 I , p 2 [0; 1] and an event E , t he p-belief
operator is de¯ned, as usual, by
B ip (E ) := f ! 2 - : t i (! )(E ) ¸ pg;
if t here is a st at e ! such t hat t i (! )(E ) ¸ p, and by
B ip (E ) := ; S( E )
ot herwise.
P r op osit i on If E is an event , t hen B ip (E ) is an S (E )-based
event .
P r op osit i on ( St andar d P r op er t ies) Let E and F be event s,
f E l gl = 1;2;::: be an at most count able collect ion of event s, and
p; q 2 [0; 1]. T he following propert ies of belief obt ain:
(o) B ip (E ) µ B iq (E ), for q · p,
(i) Necessit at ion: B i1 (- ) = - ,
(ii) Addit ivity: B ip (E ) µ : B iq (: E ), for p + q > 1,
T1
p T1
(iiia) B i ( l = 1 E l ) µ l = 1 B ip (E l ),
1
(iiib) for any
decreasing
sequence
of
event
s
f
E
g
l
l = 1,
T
T
B ip ( 1l= 1 E l ) = 1l= 1 B ip (E l ),
T1
T1
1
(iiic) B i ( l = 1 E l ) = l = 1 B i1 (E l ),
(iv) Monot onicity: E µ F implies B ip (E ) µ B ip (F ),
(v) Int rospect ion: B ip (E ) µ B i1 B ip (E ).
P r op osit ion Let E be an event and p; q 2 [0; 1]. T he following propert ies of
awareness and belief obt ain:
1. Plausibility: Ui (E ) µ : B ip (E ) \ : B ip : B ip (E ),
T1
n
2. St rong Plausibility: Ui (E ) µ n = 1 (: B ip ) (E ),
3. B p U Int rospect ion: B ip Ui (E ) = ; S( E ) for p 2 (0; 1], B i0 Ui (E ) = A i (E ),
4. AU Int rospect ion: Ui (E ) = Ui Ui (E ),
¡
¢
1
"
5. Weak Necessit at ion: A i (E ) = B i S(E ) ,
6. B ip (E ) µ A i (E ), B i0 (E ) = A i (E ),
7. B ip (E ) µ A i B iq (E ),
8. Symmet ry: A i (E ) = A i (: E ),
¡T
¢
T
9. A Conjunct ion: ¸ 2 L A i (E ¸ ) = A i
¸ 2 L E¸ ,
10. AB p Self Re° ect ion: A i B ip (E ) = A i (E ),
11. AA Self Re° ect ion: A i A i (E ) = A i (E ),
12. B ip A i (E ) = A i (E ).
B elief
A war eness
mut ualT belief
B p (E ) := i 2 I B ip (E )
mut ual T
awareness
A(E ) := i 2 I A i (E )
commonTcert ainty
1
CB 1 (E ) := n = 1 (B 1 ) n (E )
commonT awareness
1
n
CA(E ) := n = 1 (A) (E )
P r op osit ion Let E be an event and p; q 2 [0; 1]. T he following mult i-person
propert ies obt ain:
1. A i (E ) = A i A j (E ),
2. A i (E ) = A i B jp (E ),
3. B ip (E ) µ A i B jq (E ),
4. B ip (E ) µ A i A j (E ),
5. CA(E ) = A(E ),
6. CB 1 (E ) µ CA(E ),
7. B p (E ) µ A(E ), B 0 (E ) = A(E ),
8. B p (E ) µ CA(E ), B 0 (E ) = CA(E ),
9. A(E ) = B 1 (S(E ) " ),
10. CA(E ) = B 1 (S(E ) " ),
11. CB 1 (S(E ) " ) µ A(E ),
12. CB 1 (S(E ) " ) µ CA(E ).
4. Speculation
Revisiting to the speculative trade example
• Status quo value of the firm depends on high
($100) or low ($80) sales, which is
equiprobable.
• Both whether or not there is a lawsuit and
whether or not there is a novelty are
equiprobably too.
• Moreover, all events are independent.
nls
n¬ls
nl¬s
n¬l¬s
¬nls
¬n¬ls
¬nl¬s
¬n¬l¬s
●
●
●
●
●
●
●
●
S{nls}
ns
n¬s
¬ns
¬n¬s
ls
l¬s
¬ls
¬l¬s
●
●
●
●
●
●
●
●
S{ns}
S{ls}
s
¬s
●
●
S{s}
nls
n¬ls
nl¬s
n¬l¬s
¬nls
¬n¬ls
¬nl¬s
¬n¬l¬s
●
●
●
●
●
●
●
●
S{nls}
ns
n¬s
¬ns
¬n¬s
ls
l¬s
¬ls
¬l¬s
●
●
●
●
●
●
●
●
¼
¼
¼
¼
¼
¼
¼
¼
S{ns}
S{ls}
s
¬s
●
●
½
½
S{s}
nls
n¬ls
nl¬s
n¬l¬s
¬nls
¬n¬ls
¬nl¬s
¬n¬l¬s
●
●
●
●
●
●
●
●
⅛
⅛
⅛
⅛
⅛
⅛
⅛
⅛
S{nls}
ns
n¬s
¬ns
¬n¬s
ls
l¬s
¬ls
¬l¬s
●
●
●
●
●
●
●
●
¼
¼
¼
¼
¼
¼
¼
¼
S{ns}
S{ls}
s
¬s
●
●
½
½
S{s}
Revisiting to the speculative trade example
• There is a common prior (i.e., a projective
system of probability measures with which
agents’ types are consistent) and common
certainty of willingness to trade but each
agent has a strict preference to trade.
• Counterexample to Milgrom and Stokey
(1982).
• Speculative trade is possible under
unawareness.
• Knife-edge case???
Prior in a st andard type space S (Samet , 1998):
For every event E 2 F S ,
PiS
(E ) =
R
S
t
(¢
)
(E
)
dP
i
i (¢).
S
² convex combinat ion of post eriors (types)
² a consist ency condit ion on types inst ead of a belief
at a \ prior st age"
D e¯nit i on ( P r ior ) A prior
¡ Sfor
¢ playerQi is a syst em of
probability measures Pi = Pi S2 S 2 S2 S ¢ (S) such t hat
1. T he syst em is project ive: If S0 ¹ S t hen t he marginal
S
0
S0
of Pi on S is Pi . (T hat is, if E 2 § is an event whose
base-space S (E ) is lower or equal t o S0, t hen
S
S0
Pi (E ) = Pi (E ).)
2. Each probability measure PiS is a convex combinat ion
of i 's beliefs in S: For every event E 2 § such t hat
S(E ) ¹ S,
Z
PiS (E \ S \ A i (E )) =
t i (¢) (E ) dPiS (¢) :
¡
S
¢
S\ A i ( E )
Q
P = P S2 S 2 S2 S ¢ (S) is a common prior if P is a prior
for every player i 2 I .
 3
1/9
 1
1/18
 4
2/9
 2
2/18
 5
1/18
 7
1/9
 6
2/18
 8
2/9
S’
 odd  even
1/3
2/3
S
[t i (! )] := f ! 0 2 - : t i (! 0) = t i (! )g
¡
S
¢
Q
D e¯nit ion ( P r ior ) A common prior P = P S2 S 2 S2 S ¢ (S)
is positive if and only if for all i³ 2 I and ! 2 - : If ´t i (! ) 2 4 (S0),
"
t hen [t i (! )] \ S0 2 F S 0 and P S ([t i (! )] \ S0) \ S > 0 for all
S º S0.
Let x 1 and x 2 be real numbers
and v a random variable on - .o
n
R
· x1
De¯ne t he set s E 1 := ! 2 - : St ( ! ) v (¢) d (t 1 (! )) (¢) · x 1
1
n
o
R
and E 2¸ x 2 := ! 2 - : St ( ! ) v (¢) d (t 2 (! )) (¢) ¸ x 2 .
2
We say t hat at ! , condit ional on his informat ion, player 1 (resp.
player 2) believes t hat t he expect at ion of v is weakly below x 1
(resp. weakly above x 2 ) if and only if ! 2 E 1· x 1 (resp. ! 2 E 2¸ x 2 ).
belief st ruct ure wit h
T heor em Consider a ¯nit e unawareness
¡ S¢
Q
a posit ive common prior P = P S2 S 2 S2 S ¢ (S). T hen
t here is no st at e !~ 2 - such t hat t here are a random variable
v : - ¡ ! R and x 1 ; x 2 2 R, x 1 < x 2 , wit h t he following
property: at !~ it is common cert ainty t hat condit ional on
her informat ion, player 1 believes t hat t he expect at ion of v
is weakly below x 1 and, condit ional on his informat ion,
player 2 believes t hat t he expect at ion of v is weakly above x 2 .
Arbit rary small t ransact ion cost s (like a Tobin t ax) rule out
speculat ive t rade under unawareness.
Converse t o t he t heorem is false.
5. Bayesian games with
unawareness
Bayesian games with unawareness
• Unawareness of (payoff-)relevant events,
actions, and players.
• Replace type space with unawareness
type space.
• Digression: How do we model uncertainty
over action sets in standard games?
D e¯nit ion A Bayesian game wit h unawareness
¿
À
³
´
¡ (S) = hS; ¹ i ; r SS¯®
; (t i ) i 2 I ; (M i ) i 2 I ; (M i ) i 2 I ; (ui ) i 2 I
S¯ ¹ S®
consist
s of a ¯nit e unawareness
¿
À type space
³
´
hS; ¹ i ; r SS¯®
; (t i ) i 2 I and
S¯ ¹ S®
(i) a nonempty ¯nit e set of act ions M i , for i 2 I , and a
correspondence M i : - ¡ ! 2M i n f ; g, for i 2 I , such t hat for
any nonempty subset of act ions1 M i0 µ M i ,
[M i0] := f ! 2 - : M i0 µ M i (! )g is an event (in t he
unawareness type space), and ! 0; ! 00 2 [t i (! )] \ St i ( ! )
implies M i (! 0) = M i (! 00), for all ! 2 - ,
(ii) for every i ³ 2³ I , a ut ility funct
ion ´
´
S
Q
ui : ! 2 j 2 I M j (! ) £ f ! g ¡ !
R.
A strategy of player i in a Bayesian game wit h unawareness is a
funct ion ¾i : - ¡ ! ¢ (M i ) such t hat for all ! 2 - ,
³
´
(i) ¾i (! ) 2 ¢ M i (! St i ( ! ) ) , and
(ii) t i (! 0) = t i (! ) implies ¾i (! 0) = ¾i (! ).
Example (Feinberg 2005)
Unique Nash equilibrium (a2, b1)
!1
!2
S
!3
S’
An equilibrium
!1
!2
S
!3
S’
An equilibrium
³
Denot e ¾St i ( !
:=
)
´
0
(¾j (! )) j 2 I
. T he expected utility of player-type
! 02 S t i ( !
)
(i ; t i (! )) from t he st rat egy pro¯le ¾St i ( !
)
U( i ;t i ( !
) ) (¾S t i ( !
)
) :=
X
! 02 S t i
0
X
(! )
m2
µ
Q
j 2I
M
j
¶
!
0
St
0
j (! )
is given by
1
Y
@ ¾j (! 0) (f m j g) A ¢ui ((m j ) j 2 I ; ! 0) t i (! )(f ! 0g):
j 2I
D e¯nit i on Given a Bayesian game wit h unawareness ¡ (S), de¯ne t he associat ed st rat egic game by
(i) f (i ; t i (! )) : ! 2 - and i 2 I g is t he set of players,
and for each player (i ; t i (! )),
(ii) t he set of mixed st rat egies is ¢ (M i (!
S t i ( ! ) ),
and
(iii) t he ut ility funct ion is given above.
A pro¯le (¾i ) i 2 I is an equilibrium of t he Bayesian game wit h unawareness if and
only if t he following is an equilibrium of t he associat ed st rat egic game: (i ; t i (! ))
plays ¾i (! ), for all i 2 I and ! 2 - .
P r op osit ion ( Ex ist ence) Every ¯nit e Bayesian game wit h unawareness has
an equilibrium.
Sublat t ice wit h least upper bound S is l (S) := f S0 2 S : S0 ¹ Sg
Given ¡ (S), t he S-part ial game is ¡ (l(S)).
At ! 2 - , player i views t he game as given by ¡ (l(St i ( ! ) )).
P r op osit i on Given a Bayesian game wit h unawareness ¡ (S), consider for
S0; S00 2 S wit h S0 ¹ S00 t he S0-part ial (resp. S00 -part ial) Bayesian game wit h
unawareness. If I , - , and (M i ) i 2 I are ¯nit e, t hen for every equilibrium of t he
S0-part ial Bayesian game, t here is an equilibrium of t he S00-part ial Bayesian
0
game
in
which
equilibrium
st
rat
egies
of
player-types
in
f
(i
;
t
(!
))
:
!
2
=
i
S
0
S2 l ( S 0) S and i 2 I g are ident ical wit h t he equilibrium st rat egies in t he S part ial Bayesian game.
R em ar k Consider for S0; S00 2 S wit h S0 ¹ S00 t he S0-part ial (resp. S00part ial) Bayesian game wit h unawareness of given game ¡ (S). T hen for every
equilibrium of t he S00-part ial Bayesian game t here is a unique equilibrium of
t he S0-part ial Bayesian game
in which t he equilibrium st rat egies of player-types
S
in f (i ; t i (! )) : ! 2 - 0 = S2 l ( S 0) S and i 2 I g are ident ical t o t he equilibrium
st rat egies of t he S00-part ial Bayesian game.
(Peleg and T ijs, 1996, Peleg, Pot t ers, and T ijs, 1996)
Unawareness perfection
Unawareness perfection
Unawareness perfection
Unawareness perfection
Unawareness perfection
Unawareness perfection
Unawareness perfection
Unawareness perfection
Unawareness perfection
• Opponents could have any awareness level
• Full support belief over types of opponent
• Consider limit belief over opponent’s types close to him being fully
aware.
• Limit Bayesian Nash equilibrium selects (U, L), the undominated Nash
equilibrium.
For each strategic game, construct a Bayesian game with
unawareness in which each player has a full support belief over
opponents’ having any awareness of action sets.
Each such Bayesian game with unawareness has a Bayesian Nash
equilibrium.
Definition An unawareness perfect equilibrium of a strategic game
is a Nash equilibrium for which there exists a sequence Bayesian
games with certainty of awareness at the limit and a corresponding
sequence of Bayesian Nash equilibria such that the limit of the
sequence at the true state converges to the Nash equilibrium.
Theorem For every finite strategic game, an unawareness perfect
equilibrium exist.
Theorem A Nash equilibrium is an unawareness perfect equilibrium
if and only if undominated (i.e., not weakly dominated).
Interpretation
Undominated Nash equilibrium: No opponents’ action may be
excluded from consideration.
Unawareness perfect equilibrium: Slight chance that any
opponents’ subset of action profiles may be excluded from
consideration.
(Increases the probability that other not excluded actions are
played.)
Comparison to Trembling Hand Perfection:
• Independence of opponents’ mistakes.
• Mutual belief in opponents’ mistakes.
• No assumption of irrational behavior / mistakes under
unawareness perfection. Instead beliefs are formed over
opponents’ perception of the game.
6. Revealed Unawareness
Revealed Unawareness
• What’s the difference between a zeroprobability event and unawareness?
• Can we reveal unawareness?
Savage confined subjective expected utility theory
(SEU) to the decision maker’s small world.
The decision maker chooses among acts, i.e.,
mappings from states to outcomes.
Can the decision maker choose among acts
without being made aware of events by the
description of acts?
Yes, in reality we do it all the times.
No, in standard SEU models.
Proposal for resolution:
• Replace the state space in Savage by lattice of
spaces.
• Define acts on the union of spaces.
• Let preferences depend on the space (i.e., the
awareness level).
• The decision maker chooses among actions that
are “names” for acts that are not necessarily
understood by the decision maker. His
understanding depends on her awareness level.
l
80
:l
100
;
l
:l
100 100
~
;
l
:l
100 80
~
;
100
100
100
Contract 1
Contract 2
Contract 3
Choice behavior is inconsistent with either “lawsuit” or
“not lawsuit” being a Savage null event.
But it is consistent with unawareness of “lawsuit”.
Outline
1.
2.
3.
4.
5.
6.
7.
8.
Informal introduction
Epistemic models of unawareness
Type spaces with unawareness
Speculation
Bayesian games with unawareness
Revealed unawareness
Dynamic games with unawareness
…(???)
7. Dynamic games with
unawareness
• How to model dynamic interactions between
players with asymmetric awareness?
What are the problems:
o Players may be aware of different subsets
players, moves of nature, or actions (i.e.,
different “representations” of the game)
o Players may become aware of further
players, moves of nature or actions during
the play of the game (i.e., endogenous
changes of their “representations” of the
game)
• What is a sensible solution concept?
What are the problems:
o Players couldn’t learn equilibrium through
repeated play (because of “novel”
situations)
o Subgame perfection, sequential equilibrium
etc. ignore surprises but with unawareness
players could be surprised frequently
o Players may make others strategically
aware. Thus players should be able to
reason carefully when being surprised by
others rather than discarding surprising
actions of others as a mistake.
Introductory Example
• A glimpse of how we model unawareness
in dynamic games
• what solution concept we suggest to these
models
• Relevance: Show that outcomes under
unawareness of an action may differ from
outcomes under the unavailability of an
action
Example: Battle of the Sexes
II
B
S
B
3, 1
0, 0
S
0, 0
1, 3
I
Example: Battle of the Sexes
B
II
S
M
B
3, 1
0, 0
0, 4
I S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
Example: A kind of “Battle of the Sexes”
I
not give car
to player II
give car
to player II
II
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
M
0, 0
0, 0
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
S1I = {any strategy except ones with “M” after not giving the car}
S1II = {(B, M), (S, M)}
S2I = {(not give, B, M), (give, B, M)}
S2II = {(B, M), (S, M)}
S3I = {not give, B, M), (give, B, M)}
S3II = {(B, M)}
S4I = {not give, B, M)}
S4II = {(B, M)}
Unique extensive-form rationalizable outcome
Example: A kind of “Battle of the Sexes”
I
not give car
to player II
give car
to player II
II
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
M
0, 0
0, 0
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
Example: A kind of “Battle of the Sexes”
I
not tell player II about
the Mozart concert
tell player II about
the Mozart concert
II
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
M
0, 0
0, 0
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
Two novel features:
- Forest of trees, partial order
- Information set at a history of a tree
may be located in a less expressive tree
Example: A kind of “Battle of the Sexes”
I
not tell player II about
the Mozart concert
tell player II about
the Mozart concert
II
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
M
0, 0
0, 0
II
I
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
Notion of strategy
B
S
B
3, 1
0, 0
S
0, 0
1, 3
Example: A kind of “Battle of the Sexes”
I
not tell player II about
the Mozart concert
tell player II about
the Mozart concert
II
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
M
0, 0
0, 0
II
I
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
Notion of strategy
B
S
B
3, 1
0, 0
S
0, 0
1, 3
Player I
Example: A kind of “Battle of the Sexes”
I
not tell player II about
the Mozart concert
tell player II about
the Mozart concert
II
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
M
0, 0
0, 0
II
I
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
Notion of strategy
B
S
B
3, 1
0, 0
S
0, 0
1, 3
Player I
Player II
Example: A kind of “Battle of the Sexes”
S1I = {any strategy except ones with “M” after not tell}
S1II = {(M, B), (M, S)}
S2I = {don’t tell, B, M, B), (don’t tell, B, M, S),
(tell, B, M, B), (tell, B, M, S)}
S2II = {(M, B), (M, S)}
S3I = {don’t tell, B, M, B), (don’t tell, B, M, S),
(tell, B, M, B), (tell, B, M, S)}
S3II = {(M, B), (M, S)}
Different extensive-form rationalizable outcomes
Extensive Form Games with Unawareness
Generalized Game
Partially Ordered Set of Trees
Partially Ordered Set of Trees
Partially Ordered Set of Trees
Partially Ordered Set of Trees
Generalized Game
T
●
●n
●
●
●
●
T’
n’
Generalized Game
●
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● n’
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●n
a
●
b
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●
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b
●
c
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a
c
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●
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●
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a
●
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
●
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●
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● n’
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a
c
●
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b
c
●
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a
●
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b
●
Information sets in a generalized game
Standard
Standard
Generalized
Standard
Generalized
New
●
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Information set
across subtrees
●
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●
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●
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● n’
a
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●
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●
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●
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●n
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● n’
a
b
●
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●
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T’
●
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●
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b
●
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●
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a
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b
a
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b
a
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ai
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ai
●
●
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●
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●
n’
i
●
●
n’’
●
Generalized Extensive-Form Games
Extensive-form Rationalizability
Strategies
Belief Systems
Generalizing Extensive-form Rationalizability
(Pearce 1984, Battigalli 1997)
to Dynamic Unawareness
Generalizing Extensive-form Rationalizability
(Pearce 1984, Battigalli 1997)
to Dynamic Unawareness
Prudent Rationalizability
Prudent Rationalizability
Prudent rationalizability is not a refinement of extensive-form
rationalizability in terms of strategies.
Prudence vs. Rationalizability: Divining the
opponent's past behavior
I
Prudent rationalizability
Extensive-form rationalizability
b
a
c
II
6, 6
d
3, 4
e
4, 3
f
10, 0
d
5, 5
e
6, 6
f
0, 0
Application to Verifiable Information
1. With Full Awareness: a la Milgrom and Roberts (1986)
• One seller, one buyer
• One merchandise with quality L or H
• For each quality, the seller is better off from selling a larger quantity.
• The buyer’s utility is strictly concave in quantity with single peak ¯(¢)
and ¯(L) < ¯(H).
• Before a sale takes place, the seller can provide a certified signal
about the quality: can be imprecise but must be truthful
• The buyer’s unique prudent rationalizable strategy is to buy ¯(L) if
presented with {L} or {L, H} and ¯(H) if presented with {H}, while the
seller’s unique prudent rationalizable strategy is to certify {L, H} when
his quality is L and {H} when is quality is H. Full unraveling.
Application to Verifiable Information
2. With Unawareness of a Dimension of Quality
• One merchandise with quality q 2 {L, H} £ {0, *}
• For instance, available certificates in state (L,0): {L, H} £ {0, *}, {L} £ {0, *},
{L, H} £ {0}, and {L} £ {0}
• ¯(L,*) < ¯(L,0) < ¯(H,0) < ¯(H,*).
• If the buyer is fully aware of both dimensions of quality, we still obtain full
unraveling by previous arguments.
• Assume now that the buyer is aware only of dimension {L, H} while being
unaware of dimension {0, *}. Assume ¯(L) = ¯(L,0) and ¯(H) = ¯(H,0).
c
L*
L0
Seller
Seller
Seller
{L,H}£{0,
{L,H}£{0, *}
●
●
{L,H}£{0}
…
●
{H}£{0}
●
●
…
●
{L,H}£{*}
●
{H}
…
{H}£{0, *}
…
{L, H}
…
…
{L,H}£{0, *}
{H}£{0, *}
{L, H}
{L}
*}
{L,H}£{0}
{L}£{0, *}
●
…
●
{L}£{*}
{L, H}
…
*}
{L,H}£{*}
●
{L}£{0}
Seller
{L,H}£{0,
●
…
{L}£{0, *}
{L}
H*
H0
…
…
{H}
…
c
H
Seller
Seller
{L}
…
{L,H}
{L,H}
●
●
…
…
…
…
L
●
{L, H}
…
…
{H}£{*}
{H}
…
Application to Verifiable Information
2. With Unawareness of a Dimension of Quality
• One merchandise with quality q 2 {L, H} £ {0, *}
• For instance, available certificates in state (L,0): {L, H} £ {0, *}, {L} £ {0, *},
{L, H} £ {0}, and {L} £ {0}
• ¯(L,*) < ¯(L,0) < ¯(H,0) < ¯(H,*).
• If the buyer is fully aware of both dimensions of quality, we still obtain full
unraveling by previous arguments.
• Assume now that the buyer is aware only of dimension {L, H} while being
unaware of dimension {0, *}. Assume ¯(L) = ¯(L,0) and ¯(H,0) = ¯(H).
• In the verifiable information model in which the buyer is unaware of some
dimension of the the good's quality, the seller does not fully reveal the
quality in any prudent rationalizable outcome. E.g., (L, *).
Conditional Dominance
Can we capture these solution concepts in
“normal-form“ games with unawareness?
How crucial is the “time-structure” for the
analysis of strategic interaction under
unawareness?
For standard extensive-form games, Shimoji
and Watson (JET 1998) show that
extensive-form rationalizability is
characterized by iterated elimination of
conditionally strictly dominated strategies.
Chen and Micali (TE 2012) show orderindependence of iterated elimination of
conditionally strictly dominated strategies.
How about dynamic games with
unawareness?
Example: A kind of “Battle of the Sexes”
I
n
t
II
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
M
0, 0
0, 0
II
I
B
S
B
3, 1
0, 0
S
0, 0
1, 3
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 0
1, 3
0, 4
M
0, 0
0, 0
2, 6
Example: A kind of “Battle of the Sexes”
I
T
n
t
II
II
I
B
S
B
3, 1
0, 0
S
0, 1
1, 0
M
4, 0
2, 3
II
I
T’
B
S
B
3, 1
0, 0
S
0, 1
1, 0
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 1
1, 0
0, 4
M
4, 0
2, 3
2, 6
I
T
n
t
II
II
I
B
S
B
3, 1
0, 0
S
0, 1
1, 0
M
4, 0
2, 3
II
I
I
B
S
M
B
3, 1
0, 0
0, 4
S
0, 1
1, 0
0, 4
M
4, 0
2, 3
2, 6
What is the associated normal-form game?
T’
B
S
B
3, 1
0, 0
S
0, 1
1, 0
II
I
T
BB
BS
SB
SS
MB
MS
nBBB
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nSBB
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nMBB
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
nBSB
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nSSB
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nMSB
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
nBMB
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nSMB
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nMMB
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
nBBS
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nSBS
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nMBS
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
nBSS
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nSSS
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nMSS
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
nBMS
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nSMS
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nMMS
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
tBBB
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
tSBB
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
tMBB
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
tBSB
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
tSSB
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
tMSB
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
tBMB
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
tSMB
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
tMMB
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
tBBS
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
tSBS
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
tMBS
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
tBSS
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
tSSS
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
tMSS
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
tBMS
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
tSMS
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
tMMS
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
T’
I
II
B
S
B
3, 1
0, 0
S
0, 1
1, 0
T
I
II
BB
BS
SB
SS
MB
MS
nB**
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nS**
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nM**
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
t*B*
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
t*S*
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
t*M*
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
1.
1.
2.
1.
1.
II
I
T’
B
S
B
3, 1
0, 0
S
0, 1
1, 0
2.
1.
Iterated conditional dominance coincides with extensive-form rationalizability.
Iterated conditional dominance requires
normal-form information sets and thus
structure from the extensive-form. Can we
dispense with any extensive-form structure
entirely and obtain forward-induction
outcomes in the associated normal-form
game?
=> Iterated admissibility?
Iterated admissibility:
T
I
II
BB
BS
SB
SS
MB
MS
nB**
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nS**
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nM**
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
t*B*
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
t*S*
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
t*M*
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
1.
II
I
T’
B
S
B
3, 1
0, 0
S
0, 1
1, 0
1.
1.
1.
A naïve approach would delete
MB in the second step, which is
the only extensive-form
rationalizable strategy (i.e., the
only strategy capturing forward
induction)!
How about iterated admissibility?
T
I
II
BB
BS
SB
SS
MB
MS
nB**
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nS**
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nM**
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
t*B*
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
t*S*
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
t*M*
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
1.
II
I
T’
B
S
B
3, 1
0, 0
S
0, 1
1, 0
1.
1.
A naïve approach would delete
MB in the second step, which is
the only extensive-form
rationalizable strategy (i.e., the
only strategy capturing forward
induction)!
1.
Iterated admissibility must rely on information sets (i.e., extensive-form structures).
How about iterated admissibility?
T
I
II
BB
BS
SB
SS
MB
MS
nB**
3, 1
0, 0
3, 1
0, 0
3, 1
0, 0
nS**
0, 1
1, 0
0, 1
1, 0
0, 1
1, 0
nM**
4, 0
2, 3
4, 0
2, 3
4, 0
2, 3
t*B*
3, 1
3, 1
0, 0
0, 0
0, 4
0, 4
t*S*
0, 1
0, 1
1, 0
1, 0
0, 4
0, 4
t*M*
4, 0
4, 0
2, 3
2, 3
2, 6
2, 6
1.
II
I
T’
B
S
B
3, 1
0, 0
S
0, 1
1, 0
1.
1.
2.
A naïve1.
approach would delete
MB in the second step, which is
the only extensive-form
rationalizable strategy (i.e., the
only strategy capturing forward
induction)!
2.
1.
Iterated admissibility must rely on information sets (i.e., extensive-form structures).
We defined the normal form associated to dynamic
games with unawareness.
We characterize extensive-form rationalizability by
iterative conditional strict dominance and prudent
rationalizability by iterative conditional weak
dominance.
We show that iterative admissibility in the associated
normal form must depend on information sets.
We show that iterative conditional weak dominance
coincides with iterated admissibility in dynamic games
with unawareness.
Other approaches to extensive-form
games with unawareness
An Example: Unforeseen Roadwork (Feinberg 2012)
Alice (baker), Bob (coffee-shop owner), Carol (Alice’s
worker)
Alice has to deliver pastries to Bob on Monday or with
penalties on Tuesdays or Wednesdays unless unforeseen
contingencies occur
Bob notices unforeseen roadwork on Monday morning
that is likely to delay delivery; Bob thinks that Alice is
unaware of roadwork
Bob can decide to call Alice and renegotiate later delivery
Carol notices upcoming roadwork on Sunday and emails
Alice; Bob is unaware of Carol
¡; = ¡Road Work noticed by Carol, e-mail, Not Call
= ¡Road Work Notice by Carol, e-mail, Call Alice
=…
Nature
Feinberg (2012)
Carol
Bob
Alice
M
T
M
W
Bob
Alice
T
W
M
T
W
Alice
M
T
W
Alice
M
T
W
Alice
M
T
W
Alice
M
Nature
¡ Road Work = ¡ Road Work noticed by Carol, No e-mail,
= ¡ Road Work noticed by Carol, E-mail = ¡ Road Work, Call Alice
= ¡ Road Work noticed by Carol, No e-mail, Call Alice
=…
Bob
Alice
M
T W
M
Alice
T
W
M
T
W
Alice
M
T
W
¡ No Road Work
= ¡ Road Work, Not Call
= ¡ Road Work noticed by Carol, No e-mail, Not Call
=…
T
W
Halpern and Rego (MSS 2014)
¡
Nature
Carol
Bob
Alice
M
T
W
M
Bob
Alice
T
W
M
Nature
T
W
Alice
Alice
M
T
W
M
T
W
Alice
M
T
F
¡1
Bob
Alice
M
T
W
M
Alice
T
W
M
T
W
W
Alice
M
T
W
¡2
Alice
M
T
W
Grant and Quiggin (ET 2013)
¡
Nature
Carol
Bob
Alice
M
T
W
M
Bob
Alice
T
W
M
Nature
T
W
Alice
Alice
M
T
W
M
T
W
Alice
M
T
e
Z
¡1
Bob
Alice
M
T
W
M
Alice
T
W
M
T
W
W
Alice
M
T
W
¡2
Alice
M
T
W
Heifetz, Meier, Schipper (GEB 2013)
¡
Nature
Carol
Bob
Alice
M
Bob
Alice
T
W
M
T
Alice
W
M
Nature
T
W
Bob
M
T
W
Alice
Alice
Alice
M
T
W
M
T
¡1
Bob
Alice
Alice
M
T
W
M
T
W
M
T
W
Alice
M
T
W
¡2
W
Alice
M
T
W
In a Nutshell:
Similarity: Every approach uses a forest of game trees.
Main differences: How is dynamic awareness represented in the
forest of game trees:
• Feinberg: infinite sequences of views
• Halpern-Rego: awareness correspondence
• Grant-Quggin: perception correspondence
• Heifetz, Meier, and Schipper: information sets
Solution concepts:
• Feinberg, Halpern-Rego, Grant-Quiggin: Sequential equilibrium
• Heifetz, Meier, and Schipper: Extensive-form rationalizability
Self-confirming games
Self-confirming games
Questions:
• How do we arrive at our perceptions of strategic
contexts?
• How to model the process of discovering novel
features of “repeated” strategic contexts?
• Is there something like a “steady-state” of
perceptions and if there is do they necessarily
involve a common perception among players?
• Is there a notion of equilibrium that makes sense in
dynamic strategic interaction under unawareness?
Equilibrium under unawareness?
Problems:
• Although mathematical definitions of standard equilibrium
concepts can be extended to games with unawareness
(Halpern and Rego, 2014, Rego and Halpern, 2012, Feinberg,
2012, Li 2008, Grant and Quiggin, 2013, Ozbay, 2007, Meier
and Schipper, 2013), they cannot be interpreted as steadystates of behavior.
• Players could not learn equilibrium through repeated play
because they become aware of novel features. The perception
of the game may “change” between “repetitions” and even
along the supposed equilibrium path of a given game with
unawareness.
• How to reconcile equilibrium notions and non-equilibrium
solutions (i.e., extensive-form rationalizability) in games with
unawareness?
Main ideas
• Formal model of rationalizable discoveries in games
with unawareness
• Explain endogenously how players perceive strategic
contexts.
• Existence of equilibrium of perceptions and behavior:
Show that for every game with unawareness there
exists a discovered version that can be viewed as a
steady-state of player’s perceptions and possess an
equilibrium that can be interpreted as a steady-state
of behavior.
Examples
1
T
l1
r1
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1
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1, 7
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1
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1
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1
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1
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1
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1
T
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1
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1
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1
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1
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1
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1
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1
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1
T
l1
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2
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1
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1
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1
T
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• Does discovery always lead to common
awareness?
• Rationalizable discoveries versus nonrationalizable discoveries: Does rational play
select among discoveries and thus
representations of the strategic context?
1
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• Consider extensive-form games with
unawarenes a la Heifetz, Meier, Schipper
(2013) with the stronger properties of
unawareness.
• Require additionally that the set of trees is a
lattice.
• Devise information sets also at terminal
nodes.
Self-confirming equilibrium
Profile of behavioral strategies satisfying: For each
player
(0) awareness is self-confirming (i.e. constant) along
the path(s)
(i) is rational along the path(s)
(ii) beliefs are self-confirming (constant and correct)
along the path(s)
Mutual knowledge of no changes of awareness?
Lemma: No change of awareness for each player
implies “mutual knowledge” of no change of
awareness.
Proof: Follows from complete lattice of trees
and conditions on info. sets.
Existence of self-confirming equilibrium
• exists in standard games
• often fails in games with unawareness (because of
changes of awareness implied by rational play)
A game is a self-confirming game if it possesses a selfconfirming equilibrium.
Remark: Every game with common constant awareness
is a self-confirming game. Converse is false.
Discovered version
Given an extensive-form game with
unawareness and a strategy profile, the
discovered version is defined by
• the same set of players, same lattice of trees,
the same player function, same payoffs
• “updated” information sets:
T5
:
:
T4
:
:
T3
:
:
T2
:
T1
:
T5
:
:
T4
:
:
T3
:
:
T2
:
T1
:
T5
:
:
T5
:
:
T4
:
:
T4
:
:
T3
:
:
T3
:
:
T2
:
T2
:
T2
:
T1
:
Discovered version
Discovered versions are well-defined:
Lemma: For any extensive-form game with
unawareness and strategy profile, the discovered
version is an extensive-form game with unawareness.
Lemma: Given an extensive-form game with
unawareness, any two strategy profiles that generate
the same path have identical discovered versions.
Discovery process
Finite state machine:
• Set of states: set of extensive-form games with
unawareness (with identical upmost tree)
• Initial state/game
• Output function assigns strategy profile to each
game
• Transition function assigns the discovered version to
each game and strategy profile
• Final states/games (absorbing)
Discovery process
Proposition: For every discovery process, the final state exists
and is unique.
Remark: Final states may not have common constant awareness.
Proposition: For every extensive-form game with unawareness,
there exists a discovery process that leads to a self-confirming
game.
Remark: Typically extensive-form games with unawareness have
several self-confirming versions.
Extensive-form rationalizability
Inductive definition
Level 0
• All belief systems
• Any strategy for which there exists a belief system such that the
strategy is optimal at every non-terminal info. set.
Level k
• All belief systems concentrated on k-1 level rationalizable strategies
of opponents if possible.
• Any strategy for which there exists a level-k belief system such that
the strategy is optimal at every non-terminal info. set.
Extensive-form rationalizable strategies: intersection of every level-k
rationalizable strategy sets.
Rationalizable discoveries
A discovery process is rationalizable if the
output function assigns to each game an
extensive-form rationalizable strategy.
Proposition: For every extensive-form game
with unawareness, there exists a rationalizable
discovery process whose final state is a selfconfirming version. This refines the set of selfconfirming versions.
Rationalizable self-confirming
equilibrium
A self-confirming equilibrium is rationalizable if
non-rationalizable paths of play have zero
probability. (???)
Conjecture: For every extensive-form game with
unawareness, there exists a rationalizable
discovery process whose final state is a game
with a rationalizable self-confirming equilibrium.
Applications of unawareness
•
•
•
•
•
•
•
•
•
•
Dynamic games: Halpern and Rego (MSS 2014), Rego and Halpern (IJGT 2012),
Heifetz, Meier, and Schipper (GEB 2013), Feinberg (2004, 2005, 2012), Grant
and Quiggin (ET 2013), Li (2006), Ozbay (2008), Nielsen and Sebald (2013)
Bayesian games with unawareness: Meier and Schipper (ET 2014), Sadzik
(2006)
Value of information, value of awareness: Galanis (2013), Quiggin (2013)
Disclosure of information: Heifetz, Meier, and Schipper (2012), Li, Peitz, and
Zhao (MSS 2014), Schipper and Woo (2015)
Contract theory: Auster (GEB 2013), Filiz-Ozbay (GEB 2012), Grant, Kline, and
Quiggin (JEBO 2012), Lee (2008), Zhao and van Thadden (RES 2013)
Decision theory: Karni and Viero (AER 2013), Schipper (IJGT 2013), Li (2006)
Electoral Campaigning: Schipper and Woo (2015)
General equilibrium: Exchange economies: Modica, Rustichini, and Thallon
(ET 1998), Production economies: Kawamura (JET 2005)
Finance: Siddiqi (2014)
… (need more)
I am looking forward to receiving your
paper on unawareness.