Set-Up Slide
(Do not read.)
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Monday, August 13, 12
Reasoning About
Games
Melvin Fitting
Opole — August, 2012
2
Monday, August 13, 12
Reasoning About
Reasoning About
Games
Melvin Fitting
Opole — August, 2012
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Monday, August 13, 12
Epistemic game theory tries to analyze
how people behave during interactions
taking into account their knowledge or beliefs
about circumstances of the game,
about other players knowledge or beliefs,
about other players rationality,
about other players knowledge or beliefs
about their knowledge or beliefs.........
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Monday, August 13, 12
I’m a logician.
I don’t want to analyze games.
I want to analyze the analysis of games.
What assumptions lead to what conclusions?
What don’t?
And why?
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Monday, August 13, 12
There is a familiar and
much examined game called
Centipede.
Precisely because it is familiar
and many people know it,
it provides a good subject.
I want to examine the logic behind
the familiar analysis of Centipede.
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Monday, August 13, 12
Formal proofs are tools
logicians use to establish
something follows from something else.
X follows from assumptions S
if there is a formal proof of
X from S.
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Monday, August 13, 12
Models (semantics)
are tools logicians use to show
something does not follow from something else.
X does not follow from assumptions S
if there is a model validating S
but not X.
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Monday, August 13, 12
I will discuss both
proofs and
semantics.
I do not have
deep results to present.
But I do have tools of
versatility and naturalness
to show off.
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Monday, August 13, 12
But first,
what is Centipede?
A
u
?
2, 1
B
- u
?
1, 4
A
- u
B
- u
?
4, 3
?
3, 6
A
- u
- 5, 8
?
6, 5
This picture doesn’t really explain anything.
Let me talk for a bit.
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Monday, August 13, 12
Each player has to know the other player
“understands the situation.”
Each player has to be rational.
Each player has to believe each player
is rational.
Each player has to believe each player
believes each player is rational.
And so on.
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Monday, August 13, 12
What does “know” mean?
What does “believe” mean?
What does “rational” mean?
We won’t answer any of these questions.
Instead we ask, and try to answer:
how do these terms behave in context?
Of course the next question is,
what is a context.
Now let’s begin.
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Monday, August 13, 12
Logics Background
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Monday, August 13, 12
Logics of knowledge and belief
derive from work of Hintikka (1962).
This is now standard machinery.
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Monday, August 13, 12
Epistemic Logics
A finite set of agents, A, B, C, . . . .
For each agent A, a modal operator KA
Read KA X as agent A knows X
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Monday, August 13, 12
Axiomatically KA is a normal modal operator
Tautologies
KA (X
X
Y)
X
Y
(KA X
X
KA X
Y
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Monday, August 13, 12
KA Y )
And some or all of the following axiom schemes
knowledge vs belief
E-1 KA X
X, Factivity.
E-2 KA X
KA KA X, Positive Introspection.
E-3 ¬KA X
KA ¬KA X, Negative Introspection.
commonly assumed
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Monday, August 13, 12
Typical (and easy) theorem.
(KA KB X ^ KC KA Y ) KA (X ^ Y )
(uses Factivity)
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Monday, August 13, 12
Semantics is Kripke/Hintikka
There are states,
and a notion of accessibility
or indistinguishability
for each agent.
Accessibility satisfies various conditions
depending on whether we want
Factivity, or Positive Introspection,
or Negative Introspection.
I’ll skip the details here.
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Monday, August 13, 12
The intuitive idea
is accessible from
for agent A provided
if is how things are,
A can’t tell the situation from
Agent A knows X if
X is so in all states
A can’t tell from
the actual one.
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Monday, August 13, 12
.
hG, RA , . . . , i
states
accessibility
for each agent
KA X i↵
X for every 2 G
such that RA
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Monday, August 13, 12
forces
Next, games involve actions.
There is a logic for this too.
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Monday, August 13, 12
Dynamic Logic (PDL)
Actions:
• atomic actions a, b, . . .
• (↵; ) is an action; ↵ followed by
• (↵ [ ) is an action; non-deterministic choice of ↵ or
• ↵⇤ is an action; ↵ repeated some number of times
• A? is an action; test for A
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Monday, August 13, 12
if ↵ is an action and X is a formula
then [↵]X is a formula
[↵]X is read that
X will be true after action ↵
is (non-deterministically) executed
h↵iX abbreviates ¬[↵]¬X
h↵iX, is read that
at least one way of executing action ↵
leaves X true
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Monday, August 13, 12
Axioms
PDL-1 All tautologies (or enough of them)
PDL-2 [↵](X
Y)
([↵]X
[↵]Y )
PDL-3 [↵; ]X ⌘ [↵][ ]X
PDL-4 [↵ [ ]X ⌘ ([↵]X ^ [ ]X)
PDL-5 [↵⇤ ]X ⌘ (X ^ [↵][↵⇤ ]X)
PDL-6 [A?]X ⌘ (A
X)
and
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Monday, August 13, 12
Induction: either of the following
PDL-7 [↵ ](X
⇤
PDL-8 From X
[↵]X)
[↵]X infer X
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Monday, August 13, 12
(X
[↵ ]X)
⇤
[↵ ]X
⇤
Rules
X
X
Y
X
[↵]X
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Monday, August 13, 12
Y
Kripke Semantics:
hG, R↵ . . . , i
states
for each
action
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Monday, August 13, 12
forces
Special conditions:
• R↵; is the relation product R↵ R .
• R↵[ is the relation R↵ [ R .
•
RA?
just in case
A.
• R↵⇤ is the reflexive and transitive closure of R↵ .
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Monday, August 13, 12
PDL + E
Just combine epistemic (E)
and
propositional dynamic logics (PDL).
This gives us a logic in which
we can investigate
knowledge/action interplay.
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Monday, August 13, 12
Possible Additional
Conditions
There are some connecting
conditions between PDL
and E that we will
sometimes want.
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Monday, August 13, 12
PDLE-1 [↵]Ki X
Ki [↵]X (No Learning)
If an agent knows X after ↵ is performed,
the agent already knew that X would be true after ↵,
so executing the action brought no new knowledge.
1 q ↵- q 2
i
?
?
q - q
3 ↵ 4
i
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Monday, August 13, 12
PDLE-2 Ki [↵]X
[↵]Ki X (Perfect Recall)
If an agent knows X must be so
after action ↵, then after ↵
the agent knows that X.
1 q ↵- q 2
i
?
?
q - q
3 ↵ 4
i
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Monday, August 13, 12
PDLE-3 h↵iKi X
Ki h↵iX (Reasoning Ability)
If an agent could know X after action ↵,
the agent is able to figure that out
and so knows now that X
could be the case after ↵.
1 q ↵- q 2
i
?
?
q - q
3 ↵ 4
i
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Monday, August 13, 12
There is also the very important notion of
Common Knowledge
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Monday, August 13, 12
EX abbreviates KA X ^ KB X ^ KC X ^ . . .
Everybody knows X.
Informally,
CX means
X and EX and EEX and EEEX and . . .
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Monday, August 13, 12
Axiomatically
CK-1 (Axiom Scheme) CX
CK-2 (Rule)
X
E(Y ^ X)
X CY
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Monday, August 13, 12
E(X ^ CX)
Common knowledge has been
much analyzed.
Time presses, and I won’t go into
details here.
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Monday, August 13, 12
Our Cross-Logic conditions
generally extend to common knowledge too.
If each agent satisfies Reasoning Ability
for action ↵ then
h↵iCX Ch↵iX
If each agent satisfies No Learning
for action ↵
[↵]CX C[↵]X
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Monday, August 13, 12
One condition doesn’t quite make it,
though this is a rather technical point.
If each agent satisfies Perfect Recall
for action ↵ then for every n
n
C[↵]X [↵]E X
But
C[↵]X [↵]CX
doesn’t seem to follow
We will sometimes assume it as
Extended Perfect Recall
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Monday, August 13, 12
Next we could discuss
General Game Tree
Knowledge
But time presses. I’ll go directly to Centipede.
A more general is presentation is possible however.
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Monday, August 13, 12
Centipede (finally)
A1
u
R
?
2, 1
B2
- u
?
1, 4
R
A3
- u
R
?
4, 3
B4
- u
?
3, 6
R
A5
- u
- 5, 8
?
6, 5
The diagram doesn’t show anything epistemic.
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Monday, August 13, 12
A1
u
?
2, 1
Centipede
games
could
be
of
any
length.
B
A
B
A5
2
3
4
R
R
R
R
- u
- u
- u
- u
We?want our ?
analysis to?work for ?
all.
1, 4
4, 3
3, 6
6, 5
Keep that in mind as we go on.
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Monday, August 13, 12
- 5, 8
Basic Stuff
Players know the game
A1
u
?
2, 1
R
B
Apropositional
B4 letters.
A5
2
3
R
R
R
Dedicated
- u
- u
- u
- u
?
1, 4
A — player A is to move
B — player B is to move
?
4, 3
?
3, 6
?
6, 5
Axioms
KG-1 A _ B
KG-4 B
KB B
KG-2 ¬(A ^ B)
KG-5 A
KB A
KG-3 A
KG-6 B
KA B
KA A
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Monday, August 13, 12
- 5, 8
Transitions
A1
u
R
B2
- u
A3
B4
A5
R A move
R
R
is a-thing.
- u
- u
u
- 5, 8
A game transition is an action.
?
2, 1
?
1, 4
A
?
choice
4, 3
?
move
3, 6
of
(generally)
triggers an action.
?
6, 5
Moves are represented by propositional letters.
Transitions are represented by PDL actions.
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Monday, August 13, 12
Moves
A1
u
?
2, 1
R
B2
- u
?
1, 4
A3
B4
A5
R
R
- u
- u
- u
There are two moves.
R
We write ri and do.
?
4, 3
KGcent-7
?
3, 6
ri _ do
KGcent-8 ¬(ri ^ do)
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Monday, August 13, 12
?
6, 5
- 5, 8
A1
u
?
2, 1
There is one atomic transition, move right, R.
(Moving down is not a transition
it is a game-ender.)
R
B2
- u
?
1, 4
R
A3
- u
R
KGcent-9 A
KGcent-10
B
?
4, 3
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Monday, August 13, 12
B4
- u
[R]B
[R]A
?
3, 6
R
A5
- u
?
6, 5
- 5, 8
A1
u
?
2, 1
R
At a terminal state,
all plays end the game.
B2 NoRtransitions
A3
4
R areBpossible.
R
- u
?
1, 4
- u
- u
[R]? says terminal
?
4, 3
?
3, 6
hRi> says not terminal
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Monday, August 13, 12
A5
- u
?
6, 5
- 5, 8
A1
u
?
2, 1
R
B2
A3
B4
A5
R
R
R
- u KGcent-11
- u [R]? - KuA [R]? - u
KGcent-12 [R]?
?
?
1, 4KGcent-13
4, 3 hRi>
KGcent-14 hRi>
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Monday, August 13, 12
KB [R]?
?
?
3,K6A hRi> 6, 5
KB hRi>
- 5, 8
A1
u
?
2, 1
B2
A3
R
Aterminal
state
is
u
- u
R
?
1, 4
B4
A5
R
always-possible
to-reach.
- 5, 8
u
u
R
?
4, 3
?
3, 6
⇤
KGcent-15 hR i[R]?
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Monday, August 13, 12
?
6, 5
Rationality
Rationality is operational.
A player who is rational
and who knows what the best move is,
given the limitations imposed by
the knowledge the player possesses,
will play that best move.
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Monday, August 13, 12
The best known move is represented
by a propositional letter.
It is written in a special format,
kbestA (do).
Move do is the best known move for player A.
Similarly for ri, and for player B.
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Monday, August 13, 12
Rationality for a player is also
represented by a propositional letter:
raA or raB
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Monday, August 13, 12
Rationality conditions
RCcent-A (A ^ KA kbestA (do) ^ raA ) do
(A ^ KA kbestA (ri) ^ raA ) ri
RCcent-B (B ^ KB kbestB (do) ^ raB ) do
(B ^ KB kbestB (ri) ^ raB ) ri
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Monday, August 13, 12
Rationality persists:
RPcent-A raA
[R]raA
RPcent-B raB
[R]raB
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Monday, August 13, 12
We can’t represent payoffs directly.
We don’t have arithmetic in our logic.
But payoffs determine local strategy,
and that can be represented.
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Monday, August 13, 12
Endgame strategy:
EScent-A A
([R]?
kbestA (do))
EScent-B B
([R]?
kbestB (do))
If play is at the end node,
down is always best.
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Monday, August 13, 12
Midgame strategy
MScent-A A (KA hRido kbestA (do))
(A ^ hRi>) (KA [R]ri kbestA (ri))
MScent-B B (KB hRido kbestB (do))
(B ^ hRi>) (KB [R]ri kbestB (ri))
If the player to move
knows that it is possible to move right,
but then the next move will be down,
then the best known move is down.
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Monday, August 13, 12
And What Follows?
Let ra abbreviate raA ^ raB .
Cra Cdo
is provable under the assumptions:
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Monday, August 13, 12
The general PDL + E axiom schemes and rules.
The Centipede game tree
knowledge axioms KGcent-1 – KGcent-15.
The Centipede rationality axioms.
RCcent-A and RCcent-B
(a rational player plays
the best move
according to the player’s knowledge)
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Monday, August 13, 12
RPcent-A and RPcent-B
(rationality persists)
MScent-A and MScent-B
(mid game strategy)
EScent-A and EScent-B
(end game strategy)
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Monday, August 13, 12
Also some cross conditions:
PDLE-3, Reasoning Ability
h↵iCX
Ch↵iX
CK-3, Extended Perfect Recall
C[↵]X [↵]CX
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Monday, August 13, 12
What is not needed
Factivity
Positive Introspection
Negative Introspection
NO introspection
is needed
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Monday, August 13, 12
Belief matters,
not knowledge
I omit the proofs,
and instead turn to
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Monday, August 13, 12
Semantics
• Player uncertainty needs representation.
• Game states are not the same thing as
epistemic states.
• Game states are composed of epistemic
states.
• A game transition is a set of transitions from
epistemic states to epistemic states.
Monday, August 13, 12
An augmented game tree
is a game tree
showing the epistemic states
“inside” the game states.
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Monday, August 13, 12
For example
A
- x
↵
B
- x
-
not certain what
epistemic state B
A
B
is
in
after
↵
d
P
↵
- d
d
↵ P
epistemic state for A
epistemic states for B
A has no uncertainty
B uncertain of P
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Monday, August 13, 12
[↵]¬KB P
[↵]¬KB ¬P
at the state
A
B
- Pd - d ↵↵ Pd
KA [↵]¬KB P
KA [↵]¬KB ¬P
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Monday, August 13, 12
¬KB P
¬KB ¬P
at both states
For example
A
- x
↵
B
- x
B
A ↵ : Pd
?
- d
d
P6
↵ z Pd
epistemic state for A
epistemic states for B
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Monday, August 13, 12
-
h↵iKB P , true
h↵i¬KB P , true
¬h↵iKB ¬P , true
B
A ↵ : Pd
?
- d
d
P6
↵ z Pd
So
KA [h↵iKB P ^ h↵i¬KB P ^ ¬h↵iKB ¬P ]
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Monday, August 13, 12
Formal rules for evaluation of
formulas at states can be given.
We skip that here.
A formula is valid in one of these
structures if it evaluates to true at
every epistemic state.
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Monday, August 13, 12
A Three Move Version
A1
u
?
2, 1
B2
- u
A3
- u
?
1, 4
?
4, 3
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Monday, August 13, 12
- 3, 6
Now we consider an
augmented version of this.
The three game nodes are expanded,
and labeled G1, G2, and G3
Each game node has internal structure,
three epistemic states
labeled E1, E2, and E3.
Ellipses represent accessibility.
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Monday, August 13, 12
A1
u
?
2, 1
B2
- u
A3
- u
?
1, 4
?
4, 3
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Monday, August 13, 12
- 3, 6
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Monday, August 13, 12
What is valid here?
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Monday, August 13, 12
For both players
Factivity, positive and negative introspection
at each epistemic state.
But more.
Remember No Learning?
[↵]Ki X
Ki [↵]X
1 q ↵- q 2
i
?
?
q - q
3 ↵ 4
i
Easy to check.
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Monday, August 13, 12
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Monday, August 13, 12
In fact,
No Learning, Perfect Recall, and Reasoning Ability,
all hold for each player, at each epistemic state,
as does Extended Perfect Recall.
Knowledge of the game conditions
holds at each epistenic state.
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Monday, August 13, 12
Midgame and endgame strategy
axioms hold.
As does persistence
of rationality.
Briefly, we have (strong) versions
of all our axioms
true at every epistemic state.
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Monday, August 13, 12
But at (G1, E1)
Cra is not true.
If it were, ra would be true
at every reachable epistemic state
but raA is not true at (G1, E3).
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Monday, August 13, 12
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Monday, August 13, 12
However Era is true at (G1, E1)
that is,
KA raA ^ KA raB ^ KB raA ^ KB raB
Everybody knows that all players are rational.
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Monday, August 13, 12
do is not true at (G1, E1),
so Era do is not derivable
from our axioms for Centipede,
together with strong knowledge assumptions.
Common knowledge,
not mutual knowledge,
is needed.
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Monday, August 13, 12
Of course this is not
a new point.
But we do have sufficient
formal machinery to
establish it rigorously.
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Monday, August 13, 12
Formal proofs show that
for Centipede
Factivity,
Positive introspection
Negative introspection
are not needed.
Semantics shows that
common knowledge of rationality
is needed.
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Monday, August 13, 12
What I’ve presented is
a case study of
appropriate machinery to
examine some of the
generally informal
arguments of game theory.
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Monday, August 13, 12
One can use this machinery
to explore quirky directions.
For instance, what happens if
one or both players
always plays irrationally,
against his or her
best known move?
Case studies of other games
are called for.
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Monday, August 13, 12
There are more games
to be played with games,
but this is enough for now.
Thank you.
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Monday, August 13, 12
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Monday, August 13, 12