INFORMATION AND META INFORMATION Itzhak Gilboa ... - TARK

INFORMATION AND META INFORMATION

Itzhak Gilboa Department of Managerial Economics and Decision Sciences Kellogg Graduate School of Management Northwestern University Evanston, Illinois 60208

ABSTRACT

A model of information is presented, in which statements such as "the information s e t s a r e common k n o w l e d g e " may b e f o r m a l l y stated and proved. The m o d e l c a n a i s o a b e e x t e n d e d to include the statement: " t h i s m o d e l i s common k n o w l e d g e " in a well-defined manner, using the f a c t t h a t w h e n a n e v e n t A i s common k n o w l e d g e , i t i s common k n o w J e d g e t h a t A i s common k n o w l e d g e . Finally, t h e m o d e l may a l s o b e u s e d t o define a "natural" topology on i n f o r m a t i o n .

227

228

Session 3

1.

Introduction I n 1976,

Disagree,"

Aumann p u b l i s h e d

introducing

In h i s

model,

of all

possible

player

t

is

states

for all

assumptions

f r o m more p r i m i t i v e

describing

practical

purposes

regarding

information.

A prevalent

S u c h an a s s u m p t i o n model.

partitions

(i.e.,

Indeed,

Worthy of note

is

this

full

that

the

no l o s s

information

defined

unambiguously as

sets

be s t r e s s e d

Moreover,

defined

manner.

subsets

of ~ (the

information

~,

rendering

this

This last

implicit decade.

it

is

At l e a s t

Kaneko ( 1 9 8 7 ) ,

survey

of these

four last

papers,

informal

information is

itself

that

the

common in the

(p.

1237):

the

information

Included

and P 2 ( ~ )

o f w, and t h a t

it

is

in

the manner in This are

indeed

these

implicit

in Aumann's

may be f o r m a l i z e d will

in the definition

have to

in a wellinclude

o f e a c h member o f

self-referring. has troubled independent

few y e a r s :

and Sam et

that

formalization

sets)

self-reference

i n a game a n d ,

sort

two p e r s o n s .

assumption that

A straightforward

p r o b l e m w e r e made i n t h e (1986),

this

not trivial

definition

of that

we f i n d

Pl(~)

It

to

common k n o w l e d g e .

to the

functions

that

shown i n

(1985)--seems

the

~ the world is

a r e known t o b o t h p l a y e r s .

should

is

for

of generality.

functions

model.

brief

themselves

imparted

implies

is,

o f fl) i s

assumption

of a state

is

This model,

s u p p o s e d t o be e x p r e s s i b l e

implicit

constitutes

description

assumption

i n Aumann ( 1 9 7 6 ) ,

the

which information

the

is

call

that

the partitions

P1 and P2 a r e

Actually,

e.

so f a r .

frequently

the meta-information,

set

when e ~ ~ o b t a i n s ,

of information

encountered

models quite

knowledge.

thus

Ht o f t h e

assumptions--as

the structure

structure

the

partition

o f K n o w l e d g e " ) and B a c h a r a c h

information

partition

o f common k n o w l e d f f e i n game t h e o r y .

of the world ~ such that

H o w e v e r , game t h e o r y

the

"Agreeing to

i n f o r m e d o f t h e member o f H t w h i c h c o n t a i n s

be s u f f i c i e n t

about

paper,

t h a s an i n f o r m a t i o n

Aumann ("An A x i o m a t i z a t i o n

for

seminal

the concept

every player

w h i c h may be d e r i v e d

indeed,

his

(1987).

while

the

many game t h e o r i s t s

trials

to cope with

Tan and W e r l a n g We w i l l content

now g i v e

of Gilboa

in

this

(1985),

Gilboa

a (very) (1986) will

be

Information and Meta Information

229

given in the following s e c t i o n s . Tan and Werlang (1985) begin with a basic u n c e r t a i n t y space ~ and c o n s t r u c t upon i t the spaces of b e l i e f s , b e l i e f s regarding b e l i e f s , and so on in an i n f i n i t e - r e c u r s i o n model.

T h e y r e d e f i n e common knowledge

and show t h a t f o r a given Aumann model t h e r e e x i s t s an i n f i n i t e r e c u r s i o n model such t h a t the two notions of common knowledge coincide. They note t h a t t h e i r r e s u l t s , t o g e t h e r with those of Brandenburger and Deke] (1985), show t h a t the two approaches are e q u i v a l e n t .

One of the

merits of t h e i r model is t h a t they can formally s t a t e t h a t "the information p a r t i t i o n s are common knowledge." However, the p a r t i t i o n s they r e f e r to are those of the "basic" u n c e r t a i n t y space, and not of the " u n i v e r s a l " space obtained from the former by c o n s t r u c t i n g the h i e r a r c h i e s of b e l i e f upon i t .

In o t h e r

words, they do not formalize Aumann's i m p l i c i t assumption but r a t h e r c o n s t r u c t a super-model to deal with the meta-information.

Needless to

say, they do not t r y to formally c o n s t r u c t a model which w i l l i t s e l f be common knowledge. The model presented in Gilboa (1986) i s supposed to be an exact f o r m a l i z a t i o n of Aumann (1976).

I t s f i r s t and most p r i m i t i v e version

shows t h a t the problem we began with can be q u i t e e a s i l y solved:

one

can write down a w e l l - d e f i n e d model in which every s t a t e of the world s p e c i f i e s "the manner in which information i s imparted" to the p l a y e r s - in the sense of subsets of the same u n c e r t a i n t y space ~. However, t h i s i s not a l l we intended to do:

although the basic

s e l f - r e f e r e n c e problem has been solved, the p a r t i t i o n s themselves or, to be precise,

the

knowledge, (1986),

all

that

there

are

such

let alone the model itself.

though not carried out formally,

into the mode[,

that

fact

partitions,

are

not

yet

common

The solution suggested by Gilboa is the introduction of logic

that is to say, explicitly writing down the assumption

t h e axioms o f l o g i c

common knowledge.

a n d the_.mo.del__we.,_havg s_o .far describ_e.d,

are

Using the t r i v i a l r e s u l t t h a t i f an event A is common

knowledge, then i t i s common knowledge t h a t A is common knowledge--one deduces ( r a t h e r than assumes) t h a t t h i s l a s t assumption Is i t s e l f common knowledge, and hence the model i t s e l f (including t h i s assumption) i s

230

Session 3

common k n o w l e d g e . Giiboa results

(1986)

also

which are

included

K a ne ko ( 1 9 8 7 ) to Gilboa

(1976),

notion

"sharing

of

providing

the

knowledge,

suggests

also

extensions

in the sequel.

introduces

in a detailed

to deal

although

logic

into

t h e model

formal way).

the epistemic

tools

o f t h e model and a few o t h e r

He f o r m a i l y

world" defined

as o p p o s e d

discusses

by a p r o p o s i t i o n ,

w i t h a model w h i c h i s

he d o e s n o t e x p l i c i t l y

(but,

itself

the thus

common

assume t h a t

his

model i s

such. The m ai n d i f f e r e n c e least

to the best

between factual knowledge is subsets

and s t r u c t u r a l

the notion

is only defined

model i t s e l f .

uncertainty

that

propositions),

of humann's

to

model?). (1986),

intuition

be j u s t i f i e d

The p o s i t i v e

same way a nd t h e r e

is

Sa m et

of retaining

uncertainty into

the model.

substance

rather

super-

does not

partitions,

t h e game ( " t h e be d e a l t

basic

with

in a super-

t h e p r o b l e m we b e g a n w i t h original

question

equivalence

statements)

also

contains

H i s m odel i s the relationship

space.

(1987)

the

is

and s u c c i n c t

given

in Gilboa

dealt

and e v e n t s

with

in e x a c t l y

which are

subsets

of the world.

(1987)

and m e t a - i n f o r m a t i o n .

about

between statements

of the

Finally,

statements

complete

of the

states

common k n o w l e d g e

are

meta-informational

at

events--

and m e t a - i n f o r m a t i o n

(including set

are

of the world,

in his

is,

framework but rather

solve

answer to this

where information

ganeko

will

T h e s e m o d e l s do n o t e x p l i c i t l y

objects

of which are

intuitive

about

common

statements

(of states

information

factual

Structural

the objects

and t h e m e t a - i n f o r m a t i o n

(can Aumann's

sense

~.

including

t h e Aumann m odel

be r e s t r i c t e d

model.

space,

(1986)

ganeko distinguishes

by humann and i t s

L i k e Tan and Wel ang ( 1 9 8 5 ) ,

uncertainty"),

the

defined

a formalization

suggests

that

and G i l b o a

common k n o w l e d g e :

in a super-model,

(or mathematical

etc.)

judgment of the author,

of the basic

provlde

b e t w e e n Kaneko ( 1 9 8 7 )

He, t o o ,

similar

to Gilboa

of information (1986)

between meta-information

only suggests

However,

t h e m ai n p o i n t

than

formulation

the

a formalization

the

introduction

in Samet's

o f game t h e o r y

paper

in the and t h e of logic

regards

axioms:

the

he shew s

Information and Meta Information

that

one can dispose

every not

player

of

the

knows what

know w h e t h e r still

philosophically

he d o e s

proposition obtain

A is

know ( n a m e l y ,

true,

and

2.

~ e s c E~ P ~ 0.n~. 9..f~..~hP ..Mode ~..~and _~.h~. 8 ~ ~.~_s

characters

(symbols),

sets

of

strings.

sets

of

strings

world,

presented

finite

allow

propositions

as

these

axioms,

we w i l l

and so

"If

be able

an event

A is

A is

common k n o w l e d g e . "

also

that

A is

common k n o w l e d g e ,

be the

that, for

if

fact

A is

then

[it

know

(1976).

(infinite)

these

strings

and

representing

states

of

in

very

the

forth. dealt

with

state

and prove

is

then

Furthermore, so that

the

it

"event"

o n e may a l s o

common k n o w l e d g e

is may prove

that}

nA

ally n ~ I . We w i l l

to

common k n o w l e d g e ,

and

common k n o w l e d g e ,

that

not

t does

meaningless

upon

formally

common k n o w l e d g e

that

player

o f Aumann

with

are

to

if

characters

them as

and meta-information

model,

such

deals

axiom

he does

result

imposed

interpret

game-theoretic

s a m e way i n o u r

of

assumptions

us to

information

(1986)

strings

Only the

events, Since

"common p o s t e r i o r "

in Gilboa

that

t knows that

it),

The model

the

not

controversial

231

also

form partitions

simple

and

is

prove

in

of ~.

identical

our The

to

the

framework

that

intuitive

the

argument

one used

information of

i n Aumann

the

proof

sets is

have very

("An Axiomatization

of

Knowledge"). N e x t we t u r n

to

the

introduction

of

logic

into

the

model.

Intuitively, what that means is to allow the players of the model to ~hJ~,

rather than just ~ n _ ~ facts.

For instance,

in the basic model

the players know what the information sets are, and these form partitions, but the players cannot "understand" this last fact.

In the

extended model it makes sense to state that it is common knowledge that the information sets form partitions.

In fact, in the extended model,

the (extended) model itself is common knowledge;

that is to say, all our

assumptions are common knowledge, and hence everything we may state and prove is also known to every player in the model. Section 5 also suggests extensions of the model by the introduction of topology and of probabiiistic statements.

However, we do not present

232

Session 3

any additional results in this paper, but merely discuss the concepts. This introduction cannot be concluded without an apology. Unfortunately,

all results presented in this paper are trivial.

seems that in this respect too, one may quote Aumann (1976): publish this observation with some diffidence,

appropriate

framework,

On t h e o t h e r triviality, detail,

hand,

the proofs

and t h e

it

reader

is mathematically

in spite

who u n d e r s t a n d s

"We

since once one has the

trivial."

( a n d maybe b e c a u s e )

may be c o n f u s i n g .

It

Therefore t h e m ai n i d e a

of their they are given is

advised

in

to skip

them.

3.

The B a s i c Model Let T denote

there

be g i v e n

a nonempty set

two c h a r a c t e r s ,

"t knows that.

of players.

k t and k t '

For each player

which will

" and "t does not know whether.

t E T let

be i n t e r p r e t e d

as

.," respectively.

In addition, we will need another character, which will be 'c', to denote common knowledge.

Denote K = {ktlt E T) U {kilt E T} U {'c'}.

Let ~ he a nonempty set of characters, state of the world.

Similarly,

each of which denotes a

let E be a set of characters,

each of

which denotes an event. A statement is one of the following: (i)

A sequence w

'E' A where w E ~ and A E ~.

concatenation of three characters, second is the constant character

(That is, the

the first of which belongs to ~, the

'E', and the third is a member of ~.

In the sequel we will drop the apostrophes when they are more likely to

cause confusion that

the (ii)

the event

state

of the world,

~,

A 6 ~.

it.)

The m e a n i n g o f s u c h a s t a t e m e n t

belongs

to the event

A.

The m e a n i n g o f t h e s t a t e m e n t

"A" i s

that

A occurs. A s e q u e n c e kS w h e r e k 6 K and S i s

interpretation

statement

to prevent

A character

(iii)

(lv)

than

of such a statement

d e p e n d s on k,

A s e q u e n c e wS w h e r e w 6 ~ and S i s should

The s e t

be r e a d a s

of all

"S I s

statements

will

a statement.

true

The

as explained

a statement.

earlier. Such a

if w obtains."

be d e n o t e d

by E.

Subsets

of E will

is

Information and Meta Information

be called ~ . ~ K & ~ . We now define a function E: ~ x 2E ~ 2Q as follows:

E(A,L) = (~0 E

That is, according

E(A,L)

C~l"w

is

~ A" E

the set

L)

for A E ~ and L c E.

of states

of the world which are,

to L, members o f the event A.

We define F = K U ~, and denote by F* the set of all finite-length strings of characters in F (including the empty string). For a language L and [ E F* we define a language f(L) by

ZlfS

[(L) = ( S E

w h e r e fS i s

the

For instance, according

kl(L)

t o L.

wh at p l a y e r

statement

generated

will

be t h e

Similarly,

1 w o u l d know,

We now d e f i n e

L)

e

by t h e

language describing

~okl(L) w i l l if

the state

a function

concatenation

denote

the

o f f and S.

w hat p l a y e r

1 know s,

language describing

of the world ~ obtained.

M: 2E ~ 2N by

M(L) = {~ E niL c ~(L)}

for

a n y L c ~.

are

compatible

world are also We w i l l support

That is, w i t h L.

now l i s t

if

it

several

This character 2~

(2)

x

2Z

~

exists

A will

"A" E L i f f

M(L) i s n o t e m p t y .

the

on l a n g u a g e s ,

of the

designed

to

symbols. first-order

consistenc~

a unique A E ~ such that

be d e n o t e d E - I ( B , L ) .

~.)

For any A E ~,

of the states

that:

E ( A , L ) ~ M(L). (3)

requirements

to satisfy

true

For any B c ~ there

of the world which

the definitions

of the various

said

is

of states

on L.

our interpretation

requirements

the set

Note that

dependent

A language L is

(1)

M(L) i s

(Thus,

E(A,L) E-l:

= B.

233

234

Session 3

(4)

For any ~ E O i t (a)

is true

For any B c ~,

that:

either

"~ E - I ( B , L ) '' E L o r "~ E - I ( B C , L ) '' E L o r

"~ E-I(B c,L)'' a L(b)

For any B c ~ and any t E T, exactly one of the following

is

true: "~ k t E - i ( B , L ) '' E L "~ k t E - I ( B C , L ) " E L or

"~

(5)

(6)

kt E-I(S'L)" e L

F o r a n y t E T a n d S E E, (a)

I f k t S E L, t h e n S E L;

(b)

I f k t S E L, t h e n k t k t S

E L;

(c)

I f k t S E L, t h e n k t k t S

E L.

F o r a n y s E E, cS E L i f

and o n l y i f

kS E L f o r a n y f i n i t e - l e n g t h

string

E Un~ 1 ( k t [ t

Requirement any set that

(1)

of states

this

E T) n.

is a grammatical requirement.

of the world will

letter

will

be u n i q u e .

have a letter

It

The s e c o n d r e q u i r e m e n t

those which contain

states assures

of the world which are compatible us t h a t

requirement

there

are such states

is designed

the world resolving

all

to capture

w, f o r e a c h e v e n t B, e i t h e r second part

states

possibilities,

that,

exactly

w i t h L.

B or its

It

Recall

that

requirement

The f o u r t h notion

part

of "a s t a t e

states

that

complement should occur.

a t e a c h w and f o r a n y p l a y e r one o f w h i c h h a s t o r e a l i z e :

t,

there

(a)

t d o e s n o t know w h e t h e r B o r Bc.

Requirement

the following:

(a)

then this

true;

(b)

if

t knows a f a c t ,

knows a f a c t ,

t h e n t knows he knows i t ;

(c)

Its are three (b)

(5) s a y s

fact if

of

at each

t knows B;

t knows BC; ( c )

I f somebody ( t )

M(L)

of the

The t h i r d

(1954)

first

means t h a t

consists

of the world.

Savage's

uncertainty."

it

and

those A's in ~ for

M(L).

i s t h e m i n i m a l e v e n t known t o o c c u r a t L, s i n c e

us t h a t

E to denote it,

those events which are supposed to occur at L (i.e., w h i c h "A" E L) a r e e x a c t l y

assures

is

t does not

Information and Meta Information

know w h e t h e r not

a certain

know i t .

The

fact

last

is

true

requirement

or (6)

not, is

then the

t

knows that

definition

of

235

he d o e s common

knowledge.

We now proceed to define the Sg.condgFdercpns.ister,q~requJre.ments on a language L: (7)

For any string [ E F*, f(L) satisfies the first-order consistency requirements.

(8)

For any string f E F*, any ~ E ~ and any A E ~, ~([(L))

= ~(L)

and E(A,f(L)) Requirement satisfy (1)-(6).

= E(A,L).

(7) means that all languages generated by L will also (Hence,

is again a grammatica]

they will also satisfy

(7).)

Requirement

(8)

one, and ensures that the names of all states of

the world and all events are the same for all languages generated by L. If L satisfies

(7)-(8),

it is called an information set.

Note that

if L is an information set, [(L) is also an information for any string f ~ F*. We w i l l

now t u r n

to

some t r i v i a l

way i n w h i c h m e t a - i n f o r m a t i o n result

says

that,

common k n o w l e d g e

1.

Proposition:

some S ~ E.

Proof:

that

a certain it

is

Suppose

formalized

fact

is

which will in

this

demonstrate

model.

common k n o w ] e d g e ,

the

Our f i r s t

then

it

is

common k n o w l e d g e .

L is

an i n f o r m a t i o n

set,

and

let

cS E L f o r

T h e n c c S a L.

Suppose kl,k2 E Un> 1 (ktlt E T) n.

klk2 s E L. ~i"

if

is

results

Since cS E L, we have

This being true for any ~2' it is true that klcS E L for all

But this is equivalent to ccS E L.//

Before continuing,

we have to simplify our notations.

In view of

requirement fixed,

(8), whenever the information set L under discussion is -1 it may be omitted from the function E and E Furthermore,

these functions may be dispensed with altogether:

since they translate

236

Session 3

elements of ~ i n t o elements of 2O and v i c e v e r s a , no ambiguity may r e s u l t from t h e i r omission.

We w i l l ,

denote elements of ~ as well. -1 read as "E (B,L)"

t h e r e f o r e , use elements of 2O to

For i n s t a n c e , i f B c ~ then "B" should be

Our next r e s u l t i s well-known.

I t s t a t e s t h a t each p l a y e r has a

p a r t i t i o n of ~, such t h a t i f ~ E ~ obtained, the p l a y e r would know the elements of the p a r t i t i o n which c o n t a i n s e.

The i n t u i t i v e e x p l a n a t i o n

of t h i s r e s u l t i s t h a t i f a p l a y e r knows what he knows and knows what he does not know, a t each ~ the s e t of s t a t e s of the world he considers as p o s s i b l e i s e x a c t l y those s t a t e s a t which he would know what he indeed knows.

We remind the reader t h a t t h i s r e s u l t i s obtained in Aumann ("An

Axiomatization of Knowledge"),

though in a d i f f e r e n t framework.

In the sequel, L w i l l be assumed to be an information s e t unless otherwise s t a t e d .

.

Lemma:

P[qo[:

We begin with a few lemmas:

For each ~ E 0, M(~(L))

= (~).

By the d e f i n i t i o n of M and requirement (8), w E M(w(L)).

However, i f f o r some ~ ' E O, ~ ' E M(~(L)), then ~(I,) c ~ ' ( L ) . B = {~}.

I f "Bc'' E ~(L), requirement (2) i s c o n t r a d i c t e d .

requirement (4),

"B" e ~(L), whence "B" E ~ ' ( L ) .

Take

Hence, by

But t h i s implies

w' e M(~'(L)) c B, t h a t i s , w' = ~ . / /

For the next few lemmas we will fix a player "k" instead of "kt",

3.

Lejnma:

Pyoof:

Lemma:

Proof:

For each ~ E 0, w E M(k(L))

(8)

impiies e(k(L))

"kM(k(L))"

By r e q u i r e m e n t

"k",

respectively.

By d e f i n i t i o n of M, w E M(k(L))

requirement

4.

"kt"'

t E T, and write

iff k(L) c ~(L).

i f f k(L) c ~(k(L)).

But

= ~(L).//

E L.

(2),

"M(L)" E L f o r a n y i n f o r m a t i o n

L and,

in

Information and Meta Information

p a r t i c u l a r , "M(k(L)) E k(L).

5.

Lemma:

.Proo_.f:

For each w E ~,

t h i s is equivalent to "kM(k(L))" E L . / /

~ E M(k(L))

F i r s t assume ~ E M(k(L)).

iff

"kM(k(L))"

E w(L).

By Lemma 4, "kM(k(L))" E L, whence,

using requirement (5), "kkm(k(L))" E L, or "kM(k(L))" E k(L). 3 implies, k(L) c ~(L), the f i r s t part is proved.

"kM(k(L))"

E ~(L).

requirement

By requirement

(2), M(k(L)) D M(w(L)).

(5),

As Lemma

Now assume t h a t

"M(k(L))" E W(L), whence, by

However,

by Lemma 2, M(w(L)) = {w),

and w E M(k(L)) has been proved.//

6.

~!~PgD~9~:

For any ~, ~' E ~, ~' E M(k(~(I,))) i f f M(k(e(L))) =

M(k(~'(L))).

Proof:

F i r s t assume t h a t ~' E M(k(~(L))).

k(~(L)) c w'(L).

By the d e f i n i t i o n of M,

However, by Lemma 4, "kM(k(w(L)))" E k(w(L)), whence

"kM(k(~(L)))" E ~(L) or "M(k(~(L)))" E k ( ~ ' ( L ) ) . obtains M(k(e(L))) D M(k(~'(L))).

By requirement (2) one

TO see t h a t the converse inclusion

also has to hold, assume the converse, i . e . , M(k(e'(L))) ~ M(k(w(L))). By requirement (4), exactly one of the following three p o s s i b i l i t i e s is true:

(i) "kM(k(o'(L)))" E o(L); ( i i ) "kM(k(~'(L)))c'' E ~(L); ( i i i )

"kM(k(~'(L)))" E ~(L).

I f (i) were to hold, then "M(k(~'(L)))" E

k(~(L)), which implies M(k(e'(L))) D M(k(e(L))), contrary to our assumption.

On the other hand, ( i i ) would imply

"M(k(w'(L))) c'' E k(w(L)), y i e l d i n g M(k(w'(L))) c D M(k(~(L))), whence M(k(~'(L))) c D M(k(w'(L))), c o n t r a d i c t i n g the nonemptiness of M(k(~'(L))).

Hence we are l e f t with ( i i i ) .

By requirement (5) we

obtain "kkM(k(~'(L)))" E ~(L), or "kM(k(~'(L)))" E k(w(L)).

However,

Lemma 3 implies t h a t ~' E M(k(~(L))) is equivalent to k(~(L)) c ~ ' ( L ) . The former being t r u e , we obtain "kM(k(w'(L)))" E ~ ' ( L ) .

On the other

hand, Lemma 4 y i e l d s "kM(k(~'(L)))" E ~ ' ( L ) , a c o n t r a d i c t i o n to requirement (4).

This completes the f i r s t part of the proof.

assume M(k(~'(L)))= M(k(~(L))). k(~'(L)) c o ' ( L ) .

Now

We note t h a t requirement (5) implies

By the d e f i n i t i o n of M and requirement (8), t h i s i s

237

238

Session 3

equJvaient and that

to ~' concludes

We n o t e relation

that

~'

as possible

by the

it

be denoted

larger

7.

Remark:

equivalent

"~Rt~'

if,

sets.

We f i r s t

Therefore, of player

note

the

Jt

!~

is

~ 2,

for the

set

easy

the

foilowing:

relation."

to

see

player

t

t.

a given

of set

information

will

"The

The e q u i v a l e n c e

player

of states it

the

cardinality

for

of all

In g e n e r a l

instance,

been proved,

when ~ o b t a i n s ,

that

set

T and the

by I(TA'~).

[T! ~ 3 a n d

to

an equivalence

regarding

For

if

6 is

has

proposition.//

t o a few r e m a r k s

set

t h a n C~.

of

partition

unique.

defined

the

e M(k(~(L)))

part

by:

.... is

w'

information

information

wili

second

of Rt form the

essentially

Hence,

proposition

We now t u r n of all

the

R t c ~ x ~} d e f i n e d

considers classes

~ M(k(w'(L))).

sets

of the

be true

~,

the

set

~ is is well

world ~,

that

and

I(T,£I)

is

that

then

II(T.~)! ~ . Proof:

We w i l l show t h a t f o r each s u b s e t M of t i l e n a t u r a l numbers t h e r e

e x i s t s a d i s t i n c t i n f o r m a t i o n s e t LM E I ( T , ~ ) . Assume {1,2,3} c T and A = {~0} c ~. s t a t e m e n t "k l ( k 2 k 3 ) m k l A ''

concatenation for

of

(where

"k2k3" with

Let t h e r e be g i v e n M.

For each m ~ I l e t Sm denote t h e

" (k2k3)m''

itself}.

denotes

Now l e t

the

m-fold

LM b e a n i n f o r m a t i o n

set

which

{m ~ l l S m E LM} = M.

As L M ¢ L M'

for M ~ M',

Note:

If

IT[

I(T,~)

is

denumerable

Moreover,

8.

.Remark.:

= 2,

the

our remark

previous for

in general

II(T,C~}I

is p r o v e d . / /

remark

is

any finite

~.

it

that:

> I~1.

is

true

no l o n g e r

true.

in that case

Information and Meta Information

.~r,~9,,f: Trivial since for any A c ~ there exists a n which,

information set L in

for a given t E T, M(kt(L)) = A.//

This remark may be interpreted as an impossibility result:

tile set

of all information sets I(T,~) is, in a way, the set of a]i conceivable states of the world.

(Technically, not every information set has to

"resolve ail uncertainty."

Hence, not every information set may be a

description of a state of the world.

It is easy to see, however,

that

the previous remar'k is vaiJd even when one restricts one's attention to those information sets which,

indeed,

meaning of this result is, therefore,

"resolve all uncertainty.")

The

that the states of the world in a

model of information (of the type discussed here) cannot exhaust all conceivable states of the world.

The mode] presented in Section 3 is rich enough to formalize phrases such as "it is common knowledge that A is common knowledge" or to prove that the information is structured as in the classical partitions model.

However,

it is not rich enough to formalize phrases

such as "it Js common knowledge that the information is structured as in the classical partitions model."

In other words, the fact that

Proposition 6 proven above is common knowledge cannot be expressed irt the model.

This is quite obvious since our model does not contain any

logical symbols that may be used to describe mathematical

statements.

The point we would like to stress here is that these symbols are (almost) all that is needed to extend the model to include mathematical statements. Since not only

intend

this to

describe

paper define the

already the

changes

has

extended that

a low results/definitions

ratio,

model

therefore,

have

to

formally. b e made i n

We w i l l , the

model's

we do

structure

and assumptions: 1)

The alphabet symbols

2)

(],

of

the

V, A,

model

(,)

has

to

V, B, => . . . .

include

the

standard

logical

).

Any information set has to satisfy additional requirements,

239

240

Session 3

which will guarantee that these symbols have their common meaning.

For instance,

if S e L, then it is false that IS

L, e t c . ) . 3)

Requirements

(1)-(8),

defining

information

translated into formal mathematical

sets,

h a v e t o be

ianguage, and then it

should be assumed that for each information set these requirements are common knowledge. Assuming such a model, we may write:

Informal Propo_s.ltiop: Propositions 1 and 6 proven above are common knowledge.

Infprma] stating

Proof': "it

re.__afi.ing. is

this

under this

knowledge,

repeat

i s common k n o w l e d g e t h a t .

In fact, that

A l l we h a v e t o do i s

informal

model's

and s o a r e a l l

1:

it

may be i n t e r p r e t e d

to assume that

will

h a v e t o be common k n o w l e d g e by P r o p o s i t i o n refer

to

RemaFk~:

In this

"resolving

ali

complete

itself

without

framework,

uncertainty"

and c o n s i s t e n t .

i n c a s e ~ and T a r e

basically,

upon t h a t

requirements. exist, may u s e

(again)

not doubted the

mean t h a t

for

finite),

the existence

of consistent

languages

a nd s u c h a p r o o f

is

consistency

last 1.

defined of the

assumption) Thus i t

may

of state-of-the-world

each ~ E ~,

impossible structure

satisfying

all

other

theorems.

of the model:

to

depends,

such consistent

by G o e d e l ' s

sets.

is

(or even without

of information

o f t h e m odel p r e s e n t e d

of information

~(L)

l a n g u a g e may be e x t e n d e d

we c a n n o t p r o v e t h a t

the recursive

not doubt the existence

the

o n e by t h e a x i o m o f c h o i c e

Of c o u r s e ,

well

d o i n g so f o r m a l l y .

Since a consistent

a com~l_ete and c o n s i s t e n t it,

{i.e.,

the Savagean notion

will

y o u a r e now

the assumptions

common k n o w l e d g e ;

actually

very fact

common

statement

mo d el a r e also

this

2,

as s t a t i n g

is

includinKth~__qne

seemingly meaningless

suffices

of Section

each step.

t h e m odel i t s e l f

the results,

What makes t h i s

Proposition

" before

proposition

assumptions,

the proofs

languages H o w e v e r , we

a reader

in Section

who h a s

2 should

I n f o r m a t i o n and Meta I n f o r m a t i o n

Remark_2:

In the

any fact

we k n o w ,

they

not

are

really

likes

beginning,

model

including

allowed to

let

to the

one should

one can show that

5.

Other

Possible

the

prove player

assume

then

5.1

described

all

above,

the

propositions

these

are

we h a v e

proved.

propositions

"think," formal

proofs

players

as

logic

promised

to know

However,

themselves.

we h a v e to

allowed

If to

one

do

b e common k n o w l e d g e ,

may b e c a r r i e d

out

within

the

in the and

model.

Extensions

Topology:

There is little doubt that information plays a

major role in reality.

Information about the information does not seem

to be as important as the information itself, but it cannot be dismissed as irrelevant.

It is, however, quite rare that the nature of the game

played in reality will depend heavily upon what player I knows about what player 2 knows about what player I knows, etc.

It seems that even

the wittiest spy will get confused after a relatively small number of iterative concatenations of {ktlt ~ T}.

Therefore,

it is preferred that

any solution concept applied to the game w£1i not be too sensitive with respect to the highly complicated statements in the information of the game. This "bounded rationality" type of argument leads us to define a topology on the set of information sets, with respect to which one may require a solution concept to be continuous. defined as follows: of length n.

The topology we suggest is

for each n ~ I, let En denote the statements in

(The exact definition of "length" is immaterial.

For the

sake of concreteness we will suppose that it is the number of characters contained in the statement.)

Consider the topology generated by the

sub-base

{(L e I(T,~)IL

N zn

In this

topology,

L

for

~ ~ ~,

a n d L0 a r e

all

L

~ L0 i f f

= LO N z n } I L 0 e I ( T , ~ ) ,

for

each

identical

n ~ I there as

far

n ~ 1}.

exists

as n-long

~ such statements

that are

241

242

Session 3

concerned.

N o t e t h a t the space of a l l

information s e t s is a metric

space. in case ~ and T are f i n i t e , E is denumerable, enumeration of i t ,

x(E,L)

and f o r any

E: N ~ E, we may map 2Z into [0,1] by the f u n c t i o n :

= Li=l

2-I {E(i)EL} .3

-i

We conclude t h i s s u b - s e c t i o n with the following t r i v i a l remark which may f u r t h e r motivate our choice of topology oil I ( T , ~ ) :

Remark:

If

IT/,

t~21 < ~ ,

L

(n _> 1 ) ,

n

L e 2 Z,

then

the

following

are

equivalent:

(1)

L

L;

~

n

(ii)

O n E 1 U k ~ n Lk

(iii)

F o r some e n u m e r a t i o n

(iv)

F o r ally e n u m e r a t i o n

~:.2.__~.F~babi.~. here

is

the

include extension beliefs, formalize

as

beliefs

b u t we w i l i any notion

Mamuro K a n e k o ,

E,x(E,Ln)

extension

~ x(E,L).

statements.

with probability

1," while

degrees

not expatiate included

grateful

Dov S a m e t ,

~ x(E,L);

o f t h e model t o be d i s c u s s e d

of probability

oil i t

since

it

" K n o w l e d g e " may t h e model may

as well.

of Harsanyi's

Such an

(1967-68)

model o f

d o e s n o t seem t o

i n t h e model o f M e r t e n s - Z a m i r

I wish to thank Professor

I am a l s o

L;

:

E,x(E,Ln)

as a f o r m a l i z a t i o n

not

LR

of probabilistic

with other

Acknowledgmel~!s: guidance.

"belief

may s e r v e

Un~1%~n

The l a s t

introduction

be c o n s i d e r e d

=

to Professors

and Tommy Tan f o r

David Schmeidler

for

(1985}.

his

R o b e r t Aumann, E d d i e D e k e l , comments and r e f e r e n c e s .

References

Aumann, R. J . ,

"An A x i o m a t i z a t i o n

Aumann, R. J . ,

"Agreeing to Disagree,"

(1976),

1236-1239.

of Knowledge," unpublished

paper.

The A n n a l s o f S t a t i s t i c s ,

4

Information and Meta Information

Bacharach,

M.,

"Some E x t e n s i o n s

Model o f K n o w l e d g e , "

o f a C l a i m o f humann i n an A x i o m a t i c

J9,!~).i ~t" EC~L,q.m:i_9__irheg.!:.¥, 37

(1985),

167-

190. Brandenburger,

A. and E. D e k e l ,

Knowledge," research University Gilboa,

i,

Graduate

of Belief

and Common

School of Business,

Stanford

(1985).

"Information

Foerder

paper,

"Hierarchies

Institute

and M e t a - I n f o r m a t i o n , " for

W o r k i n g P a p e r 30--86,

Economic R e s e a r c h ,

Tel Aviv University

(1986). tlarsanyi,

J.,

"Games o f I n c o m p l e t e

Players, " Parts 320-336, K a n e k o , M.,

I - I [ I , ~lajl£~g.e.jn~!_t. . S c i e o c e ,

J.

'Type'

P l a y e d by B a y e s i a n 14 ( 1 9 6 7 - 6 8 ) ,

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Common K n o w l e d g e and F a c t u a l

F.

and S.

and

Zamir,

Hitotsubashi

"Formalization

'Consistency'

Savage,

D.,

L. J . ,

Sons Tan,

"Ignoring

of HarsanyJ's

i n Games w i t h

ignorance

Common K n o w l e d g e , "

University

Incomplete

.irLt~u:nat!.pna..Lg_OUU_Laj_gf_Game.__Whegr~, 14 ( 1 ) , Samet,

159=182,

486--502.

RUEE W o r k i n g P a p e r No. 8 7 - 2 7 , Mertens,

Information

and A g r e e i n g

Notion of

Information,"

1985,

to Disagree,"

!h~_Foup_da.ti0_ns_o_f.._S_tati_stig.s, New Y o r k :

(1987).

1-29. mimeo ( 1 9 8 7 ) . J o h n W i l e y and

(1954).

T. C. C. and S. R. C. W e r l a n g , Knowledge--An Alternative Working Paper Series Chicago,

1985 ( r e v i s e d

"On A u m a n n ' s N o t i o n o f Common

Approach,"

W o r k i n g P a p e r No. 8 5 - 2 6 ,

i n E c o n o m i c s and E c o n o m e t r i c s , 1986).

University

of

243