Awareness in Games, Awareness in Logic - Cornell Computer Science
Awareness in Games, Awareness in Logic Joseph Halpern
ˆ Leandro Rego
Cornell University
Awareness in Games, Awareness in Logic – p. 1/37
Game Theory Standard game theory models assume that the structure of the game is common knowledge among the players. This includes the possible moves and the set of players Problem: Not always a reasonable assumption; for example: war settings one side may not be aware of weapons the other side has financial markets an investor may not be aware of new innovations auctions in large networks, you may not be aware of who the bidders are ... Awareness in Games, Awareness in Logic – p. 2/37
This talk: Discuss how awareness can be added in games Halpern and Rego: Extensive games with possibly unaware players Lots of other work in the game theory community Discuss how awareness can be added to epistemic logic Halpern and Rego: Reasoning About Knowledge of Unawareness Revisited Work goes back to Fagin and Halpern (1985) Now lots of work in the game theory community!
Awareness in Games, Awareness in Logic – p. 3/37
Nash Equilibrium Arguably, the major goal is to define appropriate solution concepts how a game is/should be played The standard solution concept in game theory is Nash equilibrium (NE) No player can gain by unilaterally changing his strategy But Nash equilibrium does not always make sense if players are not aware of all moves . . .
Awareness in Games, Awareness in Logic – p. 4/37
A Simple Game
One Nash equilibrium of this game (not unique).
plays down
plays across ,
can play down ,
will play down .
is not aware that
But if
Awareness in Games, Awareness in Logic – p. 5/37
We need a solution concept that takes awareness into account! First step: represent games where players may be unaware
Awareness in Games, Awareness in Logic – p. 6/37
Representing lack of awareness
: an underlying standard extensive game.
describes the moves actually available to players is essentially a standard game
An augmented game based on
that also determines for each history
an awareness level,
the set of runs in the underlying game that the player who moves at
is aware of
Intuition: an augmented game describes the game from the point of view of an omniscient modeler or one of the players.
Awareness in Games, Awareness in Logic – p. 7/37
Augmented Games
are aware of all histories of the game;
across , down
is aware of run
is uncertain as to whether player
player
and
players
Consider the earlier game. Suppose that
and believes that
is unaware of it with
that is aware of the run across , down
the type of player
probability ; and
is aware of all histories, and he knows
is
aware that player
is
’s awareness level and knows the probability .
uncertain about
To represent this, we need three augmented games.
Awareness in Games, Awareness in Logic – p. 8/37
are aware of all histories of the underlying game.
considers it possible that
is unaware.
But
and
Both
Modeler’s Game
To represent
’s viewpoint, we need another augmented game. Awareness in Games, Awareness in Logic – p. 9/37
,
is not aware of the run across , down
At node
’s View of the Game
.
We need yet another augmented game to represent this. Awareness in Games, Awareness in Logic – p. 10/37
’s view
neither is
is not aware of across , down
,
!
At node
;
!
( ’s view of)
at
.
Moral: to fully represent a game with awareness we need a set of augmented games. Like a set of possible worlds in Kripke structures Awareness in Games, Awareness in Logic – p. 11/37
"
$%" # ()' & " ( "* +,
Game with Awareness is a tuple
'
A game with awareness based on
, where
;
"
'
is a countable set of augmented games based on is an omniscient modeler’s view of the game
2,
( " ( . / 0, $ / ( "&
1
*-
of game
0 and he is in information set
0
0
2
is
, player believes that the actual game
2
"1
in history
then
".
+& If
"1
( ’s information set) describes the histories where thinks
"#
might be in There are some consistency conditions that
must satisfy
See paper Awareness in Games, Awareness in Logic – p. 12/37
Example Continued
Awareness in Games, Awareness in Logic – p. 13/37
Adding Awareness to Games 3
There are many games of awareness based on an underlying game
, that vary depending on
how players’ awareness levels change over time;
3
players’ beliefs about other players’ awareness level.
67
, we have considered so far, both
4
Example: If in the simple game
5
players were indeed unaware of run across , down
, and this
was common knowledge, then in the modeler’s game of this
36
example, players’ awareness level would not include this run, and we would only need game
to model players’ view of the game.
Awareness in Games, Awareness in Logic – p. 14/37
8
Canonical Representation D D@ for
and ,
M
B@
9A
AE G ;F
G
and
.
8
, where
M
@
K
; < 8 = ; ?@
can be identified with the game
9:
A standard extensive game
This is the canonical representation of
as a game with
awareness.
8
Intuition: as a game with awareness, all
8
In the canonical representation of
players understand the structure of the underlying game
and this
is common knowledge among players. A standard game can be viewed as a special case of a game with awareness, where the underlying game is common knowledge. Awareness in Games, Awareness in Logic – p. 15/37
N
Strategies
N
In a standard game, a behavioral strategy for player is a function
N
N
from -information sets to a distribution over moves.
N
must do the same thing at histories cannot tell apart. A strategy is a universal plan, describing what will do in every possible circumstance. In games with awareness, this does not make sense! A player cannot plan in advance what he will do when he becomes aware of new moves.
Awareness in Games, Awareness in Logic – p. 16/37
Q%O P T)S R T O U VW
Local Strategies In a game
with awareness, we consider a
collection of local strategies, one for each augmented game an
O]
agent may consider to be the true one in some situation. for game
is the strategy that
^
^
^
CZY X [\
Intuitively, a local strategy
O]
would use if were called upon to play and thought that the true game was
. for
O]
ZCY X [\
There may be no relationship between the strategies different games
.
Awareness in Games, Awareness in Logic – p. 17/37
is a generalized Nash equilibrium if for every player , if
believes he is playing game
, then his local strategy
is a
c d Cfe ` gh
b
cd
Intuition:
b
a`_
Generalized Nash Equilibrium
.
`_
best response to the local strategies of other players in .
c a`_
`_
The local strategies of the other players are part of iff
is a
c
is a Nash equilibrium of a standard game
(generalized) Nash equilibrium of the canonical representation of as a game with awareness. Theorem: Every game with awareness has at least one generalized Nash equilibrium.
Awareness in Games, Awareness in Logic – p. 18/37
Awareness of Unawareness Sometimes players may be aware that they are unaware of relevant moves: War settings: you know that an enemy may have new technologies of which you are not aware Delaying a decision you may become aware of new issues tomorrow Chess “inability to compute”
i
“lack of awareness”
Awareness in Games, Awareness in Logic – p. 19/37
can make a move at
that is not aware of,
then
can make a “virtual move” at
j
k
l
If is aware that
j
k
j
l
Modeling Awareness of Unawareness
in ’s subjective
j
representation of the game The payoffs after a virtual move reflect ’s beliefs about the outcome after the move. Just like associating a value to a board position in chess Again, there is guaranteed to be a generalized Nash equilibrium.
Awareness in Games, Awareness in Logic – p. 20/37
Reasoning About Games Game theorists reason about games using knowledge Do you know/believe your opponent(s) are rational?
q vwr xy
q
u
p
form
nCm qop quu tsr
They essentially model knowledge using Kripke structures of the
p
is a set of worlds (states) In game theory:
describes, e.g., the strategies used by
:
if, in world , agent considers
|
tzr
}
is a binary relation on
|
q a|o {zr | }y~
p
the players, the game being played
possible
zr
for all
if
q|o | }y~
q o |m n }y
z
q o |m n y
x
gives meaning to primitive propositions .
Awareness in Games, Awareness in Logic – p. 21/37
Adding Awareness: A Biased History [Hintikka, 1962]: The standard semantics for epistemic logic suffers from the logical omniscience problem: agents know all tautologies and know all the logical consequences of their knowledge. One approach for dealing with logical omniscience: model agent’s lack of awareness [Fagin and Halpern, 1985/88] (FH from now on) This allows us to model, e.g., agents who are not aware of all moves in a game
Awareness in Games, Awareness in Logic – p. 22/37
Capturing Awareness FH model awareness using a syntactic awareness operator. Awareness described explicitly
and
for each agent to the language
Add operators
H
by listing the formulas an agent is aware of at each state
: agent is aware of
a H a
: agent explicitly knows
: the formulas that agent is aware of at state if
is true if implicitly knows
(it’s true at all worlds the agent
H
considers possible) and is aware of it Awareness in Games, Awareness in Logic – p. 23/37
The MR-HMS approach
a
Modica and Rustichini [1994,1999] (MR) took a different approach: A possibly different set
of primitive propositions is associated
is in
only if every primitive proposition in
a
with each world .
{¡
£
holds iff
¢
a¢
is an abbreviation for
is defined as usual (truth in all possible worlds)
Heifetz, Meier, and Schipper [2003, 2008] (HMS) extend the MR approach to multiple agents. The extension is nontrivial: requires lattices of state spaces, with projection functions between them. Awareness in Games, Awareness in Logic – p. 24/37
¦¥¤
Constraints on FH Awareness Can impose constraints on
:
§
:
§
iff he is aware of all of the primitive in
®¬ ®¬ ± ¥°¯ ¤ aª© § «
is aware of
¨ ¥¦aª©¤ «
awareness is generated by primitive propositions (agpp) if an agent
iff
§
primitive propositions in
if
¥°¤ aª© « ± ¥°¤ ³ ©«
a²aª© ³« ¨ ¥´
Agents know what they are aware of (ka): then
ˆ Halpern [2001] and Halpern and Rego [2007] show that MR-HMS models are special cases of the FH awareness model where awareness is generated by primitive propositions and agents know what they are aware of. Awareness in Games, Awareness in Logic – p. 25/37
Capturing Knowledge of Unawareness If awareness is generated by primitive propositions (as in the MR-HMS .
µ
approach), then an agent cannot know that he is unaware of a fact So how do we model knowledge and awareness of unawareness?
ˆ In [Halpern and Rego, 2006/09] (HR), we model knowledge of
¶
·
unawareness by allowing quantification over formulas
µ
,
µ
Formally, allow formulas of the form
¹·
a formula of which he is unaware.
º¹¸
Can say “agent knows
ÁÀ µ¹ Â ÃÄÅ
iff
¾¿
¾¿ Ä ÁÀ ¹º¸ µ ½»¼
½ »¼
The quantification is over quantifier-free formulas for all quantifier-free
formulas Restriction is necessary to make semantics well defined Awareness in Games, Awareness in Logic – p. 26/37
The Good News Ñ ÇHÆ ºÉÈ {ËÊ ÇÉ Ì Í HÎÉÆ Ï Ð
We can capture knowledge of lack of awareness: knows that there is some formula that
Ð
:
knows to be true that he ( ) is not aware of There is an elegant complete axiomatization
Awareness in Games, Awareness in Logic – p. 27/37
. . . and the Bad News (Under standard assumptions) it is impossible for an agent to be
ÖØ Ù Ú Ô ÖHÕ Ø× ÛaÖ Ø Ù
Ü
Consider
Ô ÖHÕ Ô ºØ×
CÓÒ
uncertain about whether he is aware of all formulas.
agent considers it possible that she is aware of all formulas and also considers it possible that she is not aware of all
Ò
formulas. is unsatisfiable! Follows from assumption that in all worlds you consider possible you are aware of the same primitive propositions + KD45 axioms
Awareness in Games, Awareness in Logic – p. 28/37
. . . and the Bad News (Under standard assumptions) it is impossible for an agent to be
áã ä å ß áHà ãâ æaá ã ä
ç
Consider
ß áHà ß ºãâ
CÞÝ
uncertain about whether he is aware of all formulas.
agent considers it possible that she is aware of all formulas and also considers it possible that she is not aware of all
Ý
formulas. is unsatisfiable! Follows from assumption that in all worlds you consider possible you are aware of the same primitive propositions + KD45 axioms + assumption that the language is the same at all worlds! Awareness in Games, Awareness in Logic – p. 28/37
A Better Approach We combine a number of ideas: from FH: the basic framework (syntactic awareness) from HR: quantification to express knowledge of lack of awareness from MR/HMS: idea of allowing a different language at each state. We get an elegant complete axiomatization (a variant of the HR axioms)
è
Extends the HMS axioms to allow knowledge of unawareness a model that satisfies
, and
êì ë
,
êì ë î êï ì
êì í ë ðïî ëñ
êHé ë
don’t need a “syntactic” awareness function in the model: are equivalent. Awareness in Games, Awareness in Logic – p. 29/37
Syntax and Semantics
, and quantification (
øù
ºø÷
óö
start with a set of primitive propositions, close off under ,
óõ
óôò
Syntax: The syntax is identical to that of HR: ,
,
)
Semantics: Extend FH awareness structures to allow a different
ûCú ý þüý þ ÿ þ þ þ þ þ þ
language at each state. An extended awareness structure is a tuple
ÿü
is a set of states
ó ü
is a set of primitive propositions—the language at state
formulas all of whose primitive propositions are in
ÿü
ÿü
ü
At , can be aware only of formulas
the set of formulas is aware of at
.
Awareness in Games, Awareness in Logic – p. 30/37
()*
% '
and
.
for all
#
)
and
for all
, /
!
. E.g.:
3 , 8
654
654
where
where
and another world
, 8
where
9 4
694
:
considers possible a world
7
is satisfiable in a state
,% 1
"
, 0 %$
.
2
,% 1
, +
.
, 0 " &%$
if
and
if
if
quantifier-free sentences
"
and
" %&$
if
to be true at a world , we must have
We give semantics to formulas just as in HR, except that for a formula
The Truth Relation
. Awareness in Games, Awareness in Logic – p. 31/37
is valid in a class
of extended awareness structures if, ?
and worlds
such that
.
In the full paper, we give an axiom system
and complete with respect to structures where
S R Q
GH
P JKI J L J M NM J
O
;
,
D= ? C F E
C
A
? C
;>
B @A
A
for all extended awareness structures
>=
ˆ Leandro Rego
Cornell University
Awareness in Games, Awareness in Logic – p. 1/37
Game Theory Standard game theory models assume that the structure of the game is common knowledge among the players. This includes the possible moves and the set of players Problem: Not always a reasonable assumption; for example: war settings one side may not be aware of weapons the other side has financial markets an investor may not be aware of new innovations auctions in large networks, you may not be aware of who the bidders are ... Awareness in Games, Awareness in Logic – p. 2/37
This talk: Discuss how awareness can be added in games Halpern and Rego: Extensive games with possibly unaware players Lots of other work in the game theory community Discuss how awareness can be added to epistemic logic Halpern and Rego: Reasoning About Knowledge of Unawareness Revisited Work goes back to Fagin and Halpern (1985) Now lots of work in the game theory community!
Awareness in Games, Awareness in Logic – p. 3/37
Nash Equilibrium Arguably, the major goal is to define appropriate solution concepts how a game is/should be played The standard solution concept in game theory is Nash equilibrium (NE) No player can gain by unilaterally changing his strategy But Nash equilibrium does not always make sense if players are not aware of all moves . . .
Awareness in Games, Awareness in Logic – p. 4/37
A Simple Game
One Nash equilibrium of this game (not unique).
plays down
plays across ,
can play down ,
will play down .
is not aware that
But if
Awareness in Games, Awareness in Logic – p. 5/37
We need a solution concept that takes awareness into account! First step: represent games where players may be unaware
Awareness in Games, Awareness in Logic – p. 6/37
Representing lack of awareness
: an underlying standard extensive game.
describes the moves actually available to players is essentially a standard game
An augmented game based on
that also determines for each history
an awareness level,
the set of runs in the underlying game that the player who moves at
is aware of
Intuition: an augmented game describes the game from the point of view of an omniscient modeler or one of the players.
Awareness in Games, Awareness in Logic – p. 7/37
Augmented Games
are aware of all histories of the game;
across , down
is aware of run
is uncertain as to whether player
player
and
players
Consider the earlier game. Suppose that
and believes that
is unaware of it with
that is aware of the run across , down
the type of player
probability ; and
is aware of all histories, and he knows
is
aware that player
is
’s awareness level and knows the probability .
uncertain about
To represent this, we need three augmented games.
Awareness in Games, Awareness in Logic – p. 8/37
are aware of all histories of the underlying game.
considers it possible that
is unaware.
But
and
Both
Modeler’s Game
To represent
’s viewpoint, we need another augmented game. Awareness in Games, Awareness in Logic – p. 9/37
,
is not aware of the run across , down
At node
’s View of the Game
.
We need yet another augmented game to represent this. Awareness in Games, Awareness in Logic – p. 10/37
’s view
neither is
is not aware of across , down
,
!
At node
;
!
( ’s view of)
at
.
Moral: to fully represent a game with awareness we need a set of augmented games. Like a set of possible worlds in Kripke structures Awareness in Games, Awareness in Logic – p. 11/37
"
$%" # ()' & " ( "* +,
Game with Awareness is a tuple
'
A game with awareness based on
, where
;
"
'
is a countable set of augmented games based on is an omniscient modeler’s view of the game
2,
( " ( . / 0, $ / ( "&
1
*-
of game
0 and he is in information set
0
0
2
is
, player believes that the actual game
2
"1
in history
then
".
+& If
"1
( ’s information set) describes the histories where thinks
"#
might be in There are some consistency conditions that
must satisfy
See paper Awareness in Games, Awareness in Logic – p. 12/37
Example Continued
Awareness in Games, Awareness in Logic – p. 13/37
Adding Awareness to Games 3
There are many games of awareness based on an underlying game
, that vary depending on
how players’ awareness levels change over time;
3
players’ beliefs about other players’ awareness level.
67
, we have considered so far, both
4
Example: If in the simple game
5
players were indeed unaware of run across , down
, and this
was common knowledge, then in the modeler’s game of this
36
example, players’ awareness level would not include this run, and we would only need game
to model players’ view of the game.
Awareness in Games, Awareness in Logic – p. 14/37
8
Canonical Representation D D@ for
and ,
M
B@
9A
AE G ;F
G
and
.
8
, where
M
@
K
; < 8 = ; ?@
can be identified with the game
9:
A standard extensive game
This is the canonical representation of
as a game with
awareness.
8
Intuition: as a game with awareness, all
8
In the canonical representation of
players understand the structure of the underlying game
and this
is common knowledge among players. A standard game can be viewed as a special case of a game with awareness, where the underlying game is common knowledge. Awareness in Games, Awareness in Logic – p. 15/37
N
Strategies
N
In a standard game, a behavioral strategy for player is a function
N
N
from -information sets to a distribution over moves.
N
must do the same thing at histories cannot tell apart. A strategy is a universal plan, describing what will do in every possible circumstance. In games with awareness, this does not make sense! A player cannot plan in advance what he will do when he becomes aware of new moves.
Awareness in Games, Awareness in Logic – p. 16/37
Q%O P T)S R T O U VW
Local Strategies In a game
with awareness, we consider a
collection of local strategies, one for each augmented game an
O]
agent may consider to be the true one in some situation. for game
is the strategy that
^
^
^
CZY X [\
Intuitively, a local strategy
O]
would use if were called upon to play and thought that the true game was
. for
O]
ZCY X [\
There may be no relationship between the strategies different games
.
Awareness in Games, Awareness in Logic – p. 17/37
is a generalized Nash equilibrium if for every player , if
believes he is playing game
, then his local strategy
is a
c d Cfe ` gh
b
cd
Intuition:
b
a`_
Generalized Nash Equilibrium
.
`_
best response to the local strategies of other players in .
c a`_
`_
The local strategies of the other players are part of iff
is a
c
is a Nash equilibrium of a standard game
(generalized) Nash equilibrium of the canonical representation of as a game with awareness. Theorem: Every game with awareness has at least one generalized Nash equilibrium.
Awareness in Games, Awareness in Logic – p. 18/37
Awareness of Unawareness Sometimes players may be aware that they are unaware of relevant moves: War settings: you know that an enemy may have new technologies of which you are not aware Delaying a decision you may become aware of new issues tomorrow Chess “inability to compute”
i
“lack of awareness”
Awareness in Games, Awareness in Logic – p. 19/37
can make a move at
that is not aware of,
then
can make a “virtual move” at
j
k
l
If is aware that
j
k
j
l
Modeling Awareness of Unawareness
in ’s subjective
j
representation of the game The payoffs after a virtual move reflect ’s beliefs about the outcome after the move. Just like associating a value to a board position in chess Again, there is guaranteed to be a generalized Nash equilibrium.
Awareness in Games, Awareness in Logic – p. 20/37
Reasoning About Games Game theorists reason about games using knowledge Do you know/believe your opponent(s) are rational?
q vwr xy
q
u
p
form
nCm qop quu tsr
They essentially model knowledge using Kripke structures of the
p
is a set of worlds (states) In game theory:
describes, e.g., the strategies used by
:
if, in world , agent considers
|
tzr
}
is a binary relation on
|
q a|o {zr | }y~
p
the players, the game being played
possible
zr
for all
if
q|o | }y~
q o |m n }y
z
q o |m n y
x
gives meaning to primitive propositions .
Awareness in Games, Awareness in Logic – p. 21/37
Adding Awareness: A Biased History [Hintikka, 1962]: The standard semantics for epistemic logic suffers from the logical omniscience problem: agents know all tautologies and know all the logical consequences of their knowledge. One approach for dealing with logical omniscience: model agent’s lack of awareness [Fagin and Halpern, 1985/88] (FH from now on) This allows us to model, e.g., agents who are not aware of all moves in a game
Awareness in Games, Awareness in Logic – p. 22/37
Capturing Awareness FH model awareness using a syntactic awareness operator. Awareness described explicitly
and
for each agent to the language
Add operators
H
by listing the formulas an agent is aware of at each state
: agent is aware of
a H a
: agent explicitly knows
: the formulas that agent is aware of at state if
is true if implicitly knows
(it’s true at all worlds the agent
H
considers possible) and is aware of it Awareness in Games, Awareness in Logic – p. 23/37
The MR-HMS approach
a
Modica and Rustichini [1994,1999] (MR) took a different approach: A possibly different set
of primitive propositions is associated
is in
only if every primitive proposition in
a
with each world .
{¡
£
holds iff
¢
a¢
is an abbreviation for
is defined as usual (truth in all possible worlds)
Heifetz, Meier, and Schipper [2003, 2008] (HMS) extend the MR approach to multiple agents. The extension is nontrivial: requires lattices of state spaces, with projection functions between them. Awareness in Games, Awareness in Logic – p. 24/37
¦¥¤
Constraints on FH Awareness Can impose constraints on
:
§
:
§
iff he is aware of all of the primitive in
®¬ ®¬ ± ¥°¯ ¤ aª© § «
is aware of
¨ ¥¦aª©¤ «
awareness is generated by primitive propositions (agpp) if an agent
iff
§
primitive propositions in
if
¥°¤ aª© « ± ¥°¤ ³ ©«
a²aª© ³« ¨ ¥´
Agents know what they are aware of (ka): then
ˆ Halpern [2001] and Halpern and Rego [2007] show that MR-HMS models are special cases of the FH awareness model where awareness is generated by primitive propositions and agents know what they are aware of. Awareness in Games, Awareness in Logic – p. 25/37
Capturing Knowledge of Unawareness If awareness is generated by primitive propositions (as in the MR-HMS .
µ
approach), then an agent cannot know that he is unaware of a fact So how do we model knowledge and awareness of unawareness?
ˆ In [Halpern and Rego, 2006/09] (HR), we model knowledge of
¶
·
unawareness by allowing quantification over formulas
µ
,
µ
Formally, allow formulas of the form
¹·
a formula of which he is unaware.
º¹¸
Can say “agent knows
ÁÀ µ¹ Â ÃÄÅ
iff
¾¿
¾¿ Ä ÁÀ ¹º¸ µ ½»¼
½ »¼
The quantification is over quantifier-free formulas for all quantifier-free
formulas Restriction is necessary to make semantics well defined Awareness in Games, Awareness in Logic – p. 26/37
The Good News Ñ ÇHÆ ºÉÈ {ËÊ ÇÉ Ì Í HÎÉÆ Ï Ð
We can capture knowledge of lack of awareness: knows that there is some formula that
Ð
:
knows to be true that he ( ) is not aware of There is an elegant complete axiomatization
Awareness in Games, Awareness in Logic – p. 27/37
. . . and the Bad News (Under standard assumptions) it is impossible for an agent to be
ÖØ Ù Ú Ô ÖHÕ Ø× ÛaÖ Ø Ù
Ü
Consider
Ô ÖHÕ Ô ºØ×
CÓÒ
uncertain about whether he is aware of all formulas.
agent considers it possible that she is aware of all formulas and also considers it possible that she is not aware of all
Ò
formulas. is unsatisfiable! Follows from assumption that in all worlds you consider possible you are aware of the same primitive propositions + KD45 axioms
Awareness in Games, Awareness in Logic – p. 28/37
. . . and the Bad News (Under standard assumptions) it is impossible for an agent to be
áã ä å ß áHà ãâ æaá ã ä
ç
Consider
ß áHà ß ºãâ
CÞÝ
uncertain about whether he is aware of all formulas.
agent considers it possible that she is aware of all formulas and also considers it possible that she is not aware of all
Ý
formulas. is unsatisfiable! Follows from assumption that in all worlds you consider possible you are aware of the same primitive propositions + KD45 axioms + assumption that the language is the same at all worlds! Awareness in Games, Awareness in Logic – p. 28/37
A Better Approach We combine a number of ideas: from FH: the basic framework (syntactic awareness) from HR: quantification to express knowledge of lack of awareness from MR/HMS: idea of allowing a different language at each state. We get an elegant complete axiomatization (a variant of the HR axioms)
è
Extends the HMS axioms to allow knowledge of unawareness a model that satisfies
, and
êì ë
,
êì ë î êï ì
êì í ë ðïî ëñ
êHé ë
don’t need a “syntactic” awareness function in the model: are equivalent. Awareness in Games, Awareness in Logic – p. 29/37
Syntax and Semantics
, and quantification (
øù
ºø÷
óö
start with a set of primitive propositions, close off under ,
óõ
óôò
Syntax: The syntax is identical to that of HR: ,
,
)
Semantics: Extend FH awareness structures to allow a different
ûCú ý þüý þ ÿ þ þ þ þ þ þ
language at each state. An extended awareness structure is a tuple
ÿü
is a set of states
ó ü
is a set of primitive propositions—the language at state
formulas all of whose primitive propositions are in
ÿü
ÿü
ü
At , can be aware only of formulas
the set of formulas is aware of at
.
Awareness in Games, Awareness in Logic – p. 30/37
()*
% '
and
.
for all
#
)
and
for all
, /
!
. E.g.:
3 , 8
654
654
where
where
and another world
, 8
where
9 4
694
:
considers possible a world
7
is satisfiable in a state
,% 1
"
, 0 %$
.
2
,% 1
, +
.
, 0 " &%$
if
and
if
if
quantifier-free sentences
"
and
" %&$
if
to be true at a world , we must have
We give semantics to formulas just as in HR, except that for a formula
The Truth Relation
. Awareness in Games, Awareness in Logic – p. 31/37
is valid in a class
of extended awareness structures if, ?
and worlds
such that
.
In the full paper, we give an axiom system
and complete with respect to structures where
S R Q
GH
P JKI J L J M NM J
O
;
,
D= ? C F E
C
A
? C
;>
B @A
A
for all extended awareness structures
>=
