Transcript דילמת האסיר

A Non-Probabilistic
Generalization
of the
Agreement Theorem
Knowledge
(ω)
. .
. . .
. .
ω
E
K(E)
Ω – a state space
 – a partition of Ω
(ω) – the element of 
that contains state ω.
At ω the agent knows (ω)
...and also E.
K(E) – The event that the agent knows E.
Jaako Hintikka
Knowledge
(ω)
. .
. . .
. .
ω
Knowledge and Belief –
An Introduction to the Logic
of the Two Notions
K(E) = {ω | (ω)  E }
K : 2Ω

2Ω
1. K(Ω) = Ω
2. K(E)  K(F) = K(E  F)
3. K(E)  E
4. ¬ K(E) = K(¬ K(E))
K(E) – The event that the agent knows E.
Conversely: If K satisfies 1-4 then there exist  such that…
Knowledge
Probability
2/3
1/14
2/3
2/14
. .
. . .
. .
1/14
2/3
2/14
2/3
2/14
1/2
4/14
0
2/14
1/2
P – a prior probability
Fix event E.
The posterior probability of E:
d: Ω  R
d(ω) = P(E | (ω))
E
[d = p] – The event {ω | d(ω) = p}
e.g. [d = 2/3]
Common Knowledge
. .
. . .
. .
1 2 c - coarser than 1 and 2
finest among all such partitions
the common knowledge partition.
K(E) := K1(E)  K2(E)

Kc(E) =  Kn(E)
n=1
The probabilistic
agreement theorem
common
P – a prior probability
… to agree…
d1(ω) = P(E | 1(ω))
d2(ω) = P(E | 2(ω))
p1  p2
… to disagree.
Kc ( [d1 = p1 ]  ([d2 = p2 ] ) = 
It is impossible …
nonA The probabilistic
agreement theorem
P – a prior probability
 | 1(ω))
d1(ω) =P(E
 | 2(ω))
d2(ω) =P(E
δ2
δ1  p
p
+
conditions on
d1 and d2
A set of decisions
Kc ( [d1 = p
δ1 ]  ([d2 = p
δ2 ] ) = 
?
satisfied by the
posterior probability
functions
Virtual decision functions
D : 2Ω  
A decision function
di : Ω  
is derived from the virtual decision function D if
di (ω) = D(i (ω))
Agents are
like minded
if all individual decision functions
Interpretation:
D(E) function.
is the
are derived from the same
virtual decision
decision made if E were the
information
given
Cave, J. (1983), Learning to agree, Economics
Letters,
12. to the agent.
Bacharach, M. (1985), Some extensions of a claim of Aumann in an
axiomatic model of knowledge, J. Econom. Theory, 37(1).
The Sure Thing Principle (STP)
A businessman contemplates buying a certain piece of property. He
considers the outcome of the next presidential election relevant. So, to
clarify the matter to himself, he asks whether he would buy if he knew that
The sure-thing
principle
appropriately
be accepted
the Democratic
candidate
were cannot
going to
win, and decides
that heaswould.
postulate
in the sense
that P1 is,
would
Similarly,
he considers
whether
he because
would itbuy
if introduce
he knew new
that the
undefined
technical
terms
referring
to
knowledge
and
possibility
Republican candidate were going to win, and again finds that he would.
render
useless without
still more
Seeingthat
thatwould
he would
buy it
in mathematically
either event, he decides
that he should
buy, even
governing
these event
terms.obtains,
It will beorpreferable
to regard
thoughpostulates
he does not
know which
will obtain,
as we would
the principle
as too
a seldom
loose one
suggests
formal
ordinarily
say. It is all
thatthat
a decision
cancertain
be arrived
at on the
postulate
well articulated
with
P1. of for the assumption of simple
basis of
this principle,
but except
possibly
ordering, I know of no other extralogical principle governing decisions that
finds such ready acceptance.
Savage, L. J. (1954), The foundations of statistics.
Virtual decision functions
D : 2Ω  
A decision function
di : Ω  
is derived from the virtual decision function D if
di (ω) = D(i (ω))
Agents are
like minded
if all individual decision
functions
are derived from the same
virtual decision function.
The virtual decision
function, D, satisfies the
STP when for any two
disjoint events E, F, if
D(E) = D(F) = δ
then D(EF) = δ.
An agreement theorem
If the agents are
 like minded with virtual decision function D, and
 D satisfies STP,
then it is impossible to agree to disagree.
That is, if the decisions of the agents are common
knowledge then they coincide.
A detective story
A murder has been committed. To increase the chances of
conviction, the chief of police puts two detectives on the case,
with strict instructions to work independently, to exchange
no information.
The two, Alice and Bob, went to the same police school; so
given the same clues, they would reach the same conclusions.
But as they will work independently, they will, presumably;
not get the same clues. At the end of thirty days, each is to
decide whom to arrest (possibly nobody).
Like mindedness
A detective story
On the night before the thirtieth day, they happen to meet … and
get to talking about the case. True to their instructions, they
exchange no substantive information, no clues; but … feel that
there is no harm in telling each other whom they plan to arrest.
Thus, … it is common knowledge between them whom each will
arrest.
Conclusion: They arrest the same people; and this, in spite of
knowing nothing about each other's clues.
Curtain
A detective story
(1999)
Aumann, (1988)
Notes on interactive epistemology,
IJGT.
unpublished.
Is the STP captured?
How do virtual decision functions fit in a
Is the agent more knowledgeable in ω than in ω’ ?
partitional knowledge setup?
Syntactically, they involve knowledge that cannot be
expressed in terms of actual knowledge Ki .
Semantically, at a given state ω the agent’s knowledge
is given by i(ω) and not by any other event.
(Moses & Nachum (1990))
Ki p
¬ Ki ¬ Ki p
ω’
.
.
ω
i(ω)
¬ Ki p
Ki ¬ Ki p
i(ω’)
A remedy
Comparison of knowledge
i knows at ω … more than
… i knows at ω’.
The event that i is more knowledgeable than j:
ω
.
[i  j] :=
 (¬ Kj(E)  Ki(E))
E
intra personal
inter state
interstate
intra personal
interstate
i knows at ω …
more than
.
… j knows at ω.
ω
.
ω’
ω  ¬ Kj(E)  Ki(E)
for each E.
Interpersonal Sure Thing Principle (ISTP)
The decision functions (d1,…, dn) satisfy the ISTP
if for each i and j:
Kj( [i  j] )  [di = δ] )  [dj = δ]
[i = j] := [i  j]  [j  i]
If the decision functions satisfy ISTP, then the agents
are like minded: For each i and j,
[i = j]  [di = di]
Expandability
The decision functions (d1,…, dn)
on the model (Ω, K1 , … , Kn )
are ISTP-expandable
if for each expansion (Ω, K1 , … , Kn , Kn+1 )
there exists a decision dn+1 for agent n+1
such that (d1,…, dn, dn+1) satisfy the ISTP.
An agent is epistemic dummy, if it
is common knowledge that each
other agent is more knowledgeable.
Where n+1 is
epistemic dummy
Officer E. P. Dummy
A non-probabilistic generalization
of the agreement theorem
If the decision functions (d1,…, dn)
on the model (Ω, K1 , … , Kn )
are ISTP-expandable
then the agents cannot agree to disagree.
the list of all known sentences
The decision δ depends
only on the ken.
Binmore’s ken
K
..........
.......
.............
....
........
..........
.......
.............
....
........
......
ken
Alice’s ken
K
......
Why ISTP?
KK
Alice knows that…
Binmore is more knowledgeable: K =
Binmore’s decision is δ:
….. K’
K’ is consistent with
K
…
...
Binmore’s decision is
δ for
.........
each the kens K’ .