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ENSAIO ECONÚt-1ICO N9 137

A REVIE~\' ON THE THEORY OF COIvlMON KNOWLEDGE

Prof. Sérgio Ribeiro da Costa Wer1ang

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-A REVIE~v ON THE THEORY OF COt'1MON KNOWLEDGE

by

Sérgio Ribeiro da Cesta Werlang*

*

The author is fran IMPA/Ü'ffi.1 and EPGE/FGV. Part of this research was done

when the autor visited the Dcpartment of EconQlãcs of the University of

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r

..

A REVIEW ON THE THEORY OF COMMON KNOWLEDGE

The conc€pt of l:nOl·Jledg(·~ is centt-al in economics as I,IJE'11 as in I\i<:i,n~

..

othel" sciences lJihoSl;;:' objf:ct of stucJ~'J' is the human bE~ing,

1 i I< e

ps~cholog~, linguistics, and artificial intelligence. Rational

expectat ions models assume not onl~ that economic agents form their

e},~pectatinns accol'-dlng to the medel, bl.lt also th<:\t the~ alI knollJ the

mode1. In the same 1 ines, a game theorist alwa~s assume that the

game i5 known b~ alI the pla~ers to some extent. The famous Lucas'

critique af the application of econometric models to pol ic~, assumes the

.",

publ ic knows the models that arE: being used b~ central planner. The

p.:<amples abovE.' makE',' it clear": I<no"'lledgE~ is fundamental in economics.

In what fol1ows, we wil1 suppose the meaning of the concepts Df

knowledge of a fact, 01" knowledge about the truth of a proposit ion are

well understood.

A

closel~ related nol: ion is that of beljef. If one

bel ieves a fact is true, this does not il\lPl~ lhis fact is trile.

are allowed to hold false beliefs, but nol false knowledge.

SincE: lhe knowledge of a fact b~ one person is itself a faet, we

ma~ think of the I<nowledge b~ an individual, sa~ i, about lhe knowledge

of another individual, sa~ J, about a facto In gener a I on e ma~ rE~-peat

\

..

this iterative process as man~ times as wished. Hence, ir we have

several agents, we mc\~ sa~ that a faet is I<nown up to leveI m b~ these

,

,

..

a 9 e1l t s i r : • e ver ~i o n e k n 01.>,,1 s t h a t ( e v <': r ~ o 11 e k n e til s t h a t ) Ir! - i t h e f a c t • i s

true.

r

times, common I<nOt,\Jledg€,' of a faet i r the faet before the words ·the fact·. i 5 In the same known '..Ip to fashion lf~vel m for alI we define Ill.

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I

I

I

..

ever~one knows that ever~one knows it, eve~~one knows that ever~one

knm'Js th,,"lt -~VE.'r"~.Ionc knml,fs it, ••• , ad infinitum.

lhe concept of knowledge has bccn discusscd for a 10n9 time b~

philosophers, <;I"t least sincc Plato. However, the ideas of common

I .. kno"Jlcdge anel of thc impot-t<Hlce clf hi9h('::I'" levcls "of knowledgc :::'~I'"C

rE.'cent. lhe~ were introduced b~ David Lewis in 1966, so that he could

stud~ social convcnt ions.

papers on games with incomrlete information (1967-1968) notices the

importance of itcrated gucsses about the unknown parameter of the game.

A gamc with incomplete informat ion is such that the pa~off functions of

thE" upon an Ilnk nOl-'Jn pal'"ameter (possibl~

~ muI t i d í me n s i o n a I ) • R ou 9 h 1 ~, h e <":\ r 9 u e cI t h a t i f a p I a ~ e r i s f a c E dw i t h a

game of incomple~e informat íon, the act ion chosen b~ this pla~cr wil1

depend on the beliefs she/he has about the values of the unknown

paramcter of the game. Therefore, this pla~er wOllld know that the same

should occur with the other pla~ers. As the act"ion Df an~ pIa~er

influences the pa~off of an~ other pla~er, a pla~er should tr~to gucss

~\what

othel"

pla~E"~rs

at-e guessing abollt the unknown pcwameter, guess

~Jh<:\t

other pla~ers are gUessin9 abollt the guesses of the other pla~ers, and

I - so on.

I

I-When we see these definit ions of higher leveIs of knowleclge,

we promptl~ wondET what is the impot-t:::tnce of levi.ds greater than the

r

I

first. In order to illustrate the point that higher leveIs of knowledge

of a fact a~e linked with ver~ different sitllations, we will Fefer to a

well known puzzle. Long time ago a king decided to give amnist~ to a

large gFOllp of PFisoners. lhe ro~al prison was a ver~ special one: the

prisoners were not allowed to speak to each other. In ol"der to

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I

I

: ~

! ...

put on a hat. which th~ wcre told the~ cculd not see the top, under an~

c i rculllst,·Hlce. If the~ did 50, the~ ~'Ioulcl bc k i11cd. AlI but two hats

had a white topo Those two hats Were red topped. AlI the prisoners

could sce the hats of the othcrs, but not his own.

The kin9 delivered the following speech: as ~ou noticed, most of

!;!OU ha'.'E~ a white hat. Sut some have a red hat. From now on, eYer~ da~

~ou IIJill be brought to this roem. The da~ ~ou finei out the color of

~ou are free to 90.

If

an~ of ~ou guess the wrong color, this

person is 90in9 to be behaded·.

How llIan~ da~s did it take 'to the prisoners with the red hat to

figure out the calor of lheir hats?

Let us reason da!;! b~ da~. On the first da~. after the speech, lhe

prisoners with white hat see two red hats, anel the prisoners with red

hat see onl~ one red hat. Hence, there is nothing the~ can concllJde

abolJt the colar Df lheir own hats.

As a reslJ1t, on lhe second da~ eYet-~ pr isoner 'is bacl< in the main

room. Again. the ones with a red hat see one red hat onl~, and the

othsrs two r-t:-d hats. But t hel'"e is <.\n aciditional information: one da~

has passed. A prisoner ~ith a red hat knows that the other prisoner

with the t-ed hat did not leave. As this other recí hai:i..t-: • ..i I-'f ison2r knows

ther€' is a re-d hat, th is olher PI'" isoner must have seen another rE:'d _ hat.

It fol1ows that this other red hat has to be his own, since he sees no

other. Therefore, the prisoners with a red hat leave the prison on the

second da::!.

What would happen if we had three hats instead of two? A similar

arglJment wOlJld would appl~, and the PFisoners with Fed hat would leave

on the third da~. It should be clear that this reasóning could 90 on

for an~ number of Fcd hats, PFovided the number of white hats is gFcatcF

3

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than or equal to th2 numbcr of red hats.

This puzzle serves the purpose to show that different leveIs Df

knowledge of a faet are connected with ver~ distinct situations. In

f,,\ct, in t he case of tJ,'JO hat s, a person w i t h a red hat sees at I east onE

red hat, but does not know whether the other person sees at least one

I!J'" red hat. 50, in the ca~)e of t~'w hats, ever~one sees a r~'d hat, but it

i5 not the case that eVEr~one knows that ever~onE sees at lea5t one red

hat. In lhe case of three red hats, the persan with a red hat sees two

red hats, and knows thal ever~one sees at leat one red hat. Thus, now

ever~one sees a red hat, and eve~~one know5 ever~one sees a red hat. We

could go on inductivel~ to show that each higher leveI of knowledge of

~ the fact ever~one sees at least one red hat· is attained for versions

of the puzzle with different numhers of red hals. As lhese puzzles with

different numbers of red hats are ver~ distinct problems (in particular,

their solutions are different), we can see lhe importance of the higher

leve1s of knowledge of facts.

The formalization Df common knowledge is not an immediate task. To

~

\ begin with, i t i s nece:;;sar::J to define what are the obJects of knowledge.

,

The first formal ization waS due to Robert Aumann in 1976. In his model,

knowledge refers to events which are subseis ui ;;~ates of the

wor 1 d. Ca 11 it

n.

AI.1mann repre~-ent s the agents b~ informalion

~

..

-,-,-;.

ThE-se partitions work in the

/ partitions,

fo 11 ow i n 9 "'Ja~ : i f a stale

WEQ

occurs, then agent observes lhe

I... occurrence of the cell of his partition which cont<:-\ins

w .

We denote

it b~ 7f(w). The descr ipt ion of the set of stales of lhe wOI~ld anel of

the information structure above, has to be laken to be common knowledge

in a primitive sense. Then, one agent know~) an evenl A at: thE;.' statew

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"

\

\

,..

....

are (;\ l i t t l c mDn"" subtlc. The~ depend cntil--el~-j on the pl'"imitive common

knowledge of the information structure. In the picture below, we can

see ,?-r\ e;-:ample wit:h tv-1C) <'-I.9cnts. The set

fi

is an intetval, the

pal'"t it

iDn~;

Df the <:\9cnts cure

771

anel

77; ,

and the eVE:nt is A.

FOI~

conv(-::-n icnce,

~I)C

dr-aw the set

n

tllJicc.- In

th~""

+'irst we sec the

infDrmation partition of the first agcnt, and in the second we see the

information partition Df the sccond agcnt.

FIGURE 1

(

r

1T,(w)

~1

)

L V" .-J

A

.o.

(

r

1'f

~ z(W)

"I

)

- I

OJ

I--+-+----+!:==;:::::==:=:~~r___:._!_-~I Agent 2

'-= .J

v

A

We will vcrif~ the knowledge of the event A in its various la~ers,

b~ both of the agents, when the state Df the world that has -occl.ll'"red

isW, as

displa~ed.

Because agent i has the infol'"mation

partit~on

n; ,

she

.

sees

.

the event

11;

(w)

As 1(t4J) is cont<.dned in

A,

agent i knows

that t he: event A o c c: IH-I'" e d • Sim i 1 ar"l ~ , foI'" agent '1 c.. , as 7{(w) is contained

in A, ~_gent 2 I<nows A. As age:-nt i sees 1((vJ) I and she knows agent 2 has

the part it ion

11;. ,

she I<now:; that agent 2 has obset-ved the

occunenct:;-of -n;{fIJ~ for some W' rr;rw). -So, she I<no\,lJs that agent 2 has seen the

occurrence of the union of the cel1s 1Ç(w') f 01--

w '

in 77f(UJ). As this

set is cGntaincd in A (see Figure 2),agent 1 knows that agent 2 I<nows A.

FIGURE 2

(

.

/"".

71i IW) ~ -

'1

Agent 1

(

w _

)

Agent 2

(8)

In fact,

(:l.s a9cnt 2 sees, 77;(w) , i t tl.ll·-n':; Ciut that the st.:.-\te W sho\lJn i n F i ~,I.H"e

2 could be the real state of the world. As he knows agent 1 has

P<:i.t-t it ion

7!i '

he knD'lJ-::;· <:1.9<::nt t c01l1d b,::: observin::;,

o/w),

,:\nd hence

~:;he

~ w01l1d be unccrtain abollt the occurrence of the event A. It follows that

Bgent 2 does not know whether ~gent 1 knows A •

....

Suppose now that we are considering the knowledge of the event 8,

drB~'Jn i n F i 9 1l1~ C 3 • Th e· s~~·t CI( (w ) is observed b~ both agents.

F u r t h e

~"

I\l o t" C ,

~". ~:.

t h

,,~

i n 1-o I'" IH a t i C) n p :::r ,." t i t i o

n~·

7T,

ci. n d

rr;

a I' e c o m OI o n k n o

~I}

1 c cf

~J

e

to b0:'Ç"~ in with, bath C1.~H?nts knm,) (n~ch oth~:::'I~ see CI« W) , know the~ knal.l}

C K (W ), a n d ~> () o n • T h i s se t h <'1.'::;' a s Clt"" t o f f i ~.~ f~ d p o i n t p r o p e r t ~ : C K ( W )

~ belongs at thc same time to the sct of evcnts that agent 1 observes in

heF informat íDn strllcturc and the set of events that agent 2 observes in

his inform,,1.t ion ~.truct1.ln::·. That is to S;3.~, CK(W) belongs to the meet

of the two partitions,i.f:~., the finest comman coarsening of lhe

patitions, d,~noted b~

7!;A7í';.

As the set B conb."-\ins CK(w), and it is

common knolJ.lleclge that the set CK(W) is observed b~ both, B

knmJledge.

FIGURE 3

Cl«Wj

~:-========TF'

===uJ~·

~\~========~e-:":l-\

~)Lt

Agent 1 ~ -v~---p

6

C

\

cf.,

.. , } \

Agent 2

--

... --~_ ...

--

...

---Ck(w}

b

is common

Based Oli the intuition above, Aumann defines an event B to be

common knC!w1 ;::'d!Je at W ir then? is ,ln elemenl of

1!.;A-rr;,

which contains w

and is cont:aint-~d in B. Allm;:wn's dcfinition is elegant, c\nd Cé\n be

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allmt)s for t- ichcl'" infcJl'"IIl,~t ion s;tIr 1.1 C tl.llres, 9cn(::t-ated b~ 1T--<~1gebr<1s. On

t h i s r 12 t' 12 r t o N i E-~ 1 5 12 n ( l 9 B 4) a n d B r' a n d 12 n !:)Ij 1" 9 e I" a n d D 12 k e 1 ( 1 9 B 7 ) •

The pl"'oblcm with his definition is the difficult~ with i ts

ap p 1 i c;::ü i on • Imagine if we triEd to formal ize the fal10wing statement:

.,. ·the modEl is common knovJled::-JE ,:\I'llong lhe (:\~J.;:?nt<.:;". The fil"'st thing I:Je

would have to do, wauld be to embed ·the modElo as an event on a large

SP~CE af states of the world. Also it would be necessal"'~ to figure out

what were the informalion structul"'es which represented the agents in

th i -:::' \JJor 1 d • Not on 1 ~J t h i s, but this representation should be

intuitivel~ common knowledge a priori.

Aiming at avoiding these problems, and at appl~ing knowledge and

p commun knowledge to games, an alternat ive formalization was developed b~

l-r

Tan and Werlang(i985) (see also Brandenburger and Dekel(1985), Tan and

Werlang(1984,1988,i988b), and Werlang(1986». The idea behind i 5 the

direct modelling of la~Ers of knowledge <to be precise, this is a model

of belief with probabil it~ one). Suppose there are n agents, and eaeh,

of thE n agents are uncertain about some basic set of states of the

world. To be consistent with the discussion up to now, sa~ this set

is

n

The Ba~esian view holds that each agent has a first order

bel i ef QVt':t- t he poss i b 1 e oceurn?nees of..f2..

orde:" bel ie-F' of agent i. That , 1 S, 5 1. i i 5 an E'lement of

L1

(.!2). The set

4<..n.) is the set of probabilit~ distribljtions on..Q. We ~'Jill avoid <",11

c:ompl ications, b~ assuming that alI mathemat ieal objeets here are wel1

dE'fined. The ver~ curious reader should reTer to the papers eited

abovl~ • 8eeause beliefs about beliefs matter, agent i wil1 have 5Ecand

order bel i efs S{i , a second

order' bel iefs: if we cal1 the set of first

belief Df agent is ,'" point (;" 2 in

6.

<.D.

x

7r

5

!),

,:> r

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sEcond arder bel ief is a prababil it~ distribution over the statEs of

nature and the first order be1iefs of alI other agents. Call lhe set of

sE'cond arder bel i e f·::; of agent i b:J t) ~2 i • Indl.lctivel:J an m-th cwdf2r

belief, i S <:lo b E·l i c f on.D. X

JI

S.~-\.Jh

(;:1'" e

J~.I. d is tho::~ set of

(m-j. )·-th

ordel~ belief af agent J. Formal1:J ane has: b,:dongs

The i S sunHnal~ i zed b:J

h i el"arch~'J of ••• ) € .m~.L

-rr

5.~

~

.

This

:1T" ""-1

t o

li

(!lx I' S. ).

)*4 J

his infinite

SE:·t is t 00

largE.'. It al10ws the existence of extremeI:J inconsistent beIiefs. We

impose a minimum consistenc:J requirement: whenever lhere is an event

whose probabilit:J can be evaluated b:J the m-th order beIief, as weIl as

b~ the p-th order belief, then the probabilit~ assessment given b:J beth

~ orders of bel iefs must coincide. For exampIe, this implies that the

'f'"',

f i r s t 01'-d i2 t- bel i (.:-f fi) 1.1 S t b e t h e m:;1 r 9 i n :::\ I o f t h e s e c a n d o r d e I~ bel i e f o n

n :

f 2

si:: m a r 9

n

(

s i ) ' R e c alI t h (,). t t h e m a r Ç.I i n a 1 d i s t r i b I j t i o n s , o r sim p 1 ~ t h e

marginaIs of a probabilit:J measure

P

which is defined o~ a product space

u

X Vare defined b:J margu(A)=P(A X V), and similarl:J margV(B)= P(U X B)

for an!;1 event A in U and an!;1 eVt:-nt B in V. Hence, agent is

c:haracter' i zed b:J a point in the set S· _ .. ( ( Si f s~

. . .

) the

1 1 1

beIiefs sat ist'!;1 the minimum consistenc:J requirement ). B;J a t heOl~em

first Pt-oved b!;1

BO~H.d1974)

(and later b!;1 Armbruster and 80ge(i979), l30ge

and Eisele(1979), Mertens and Zamir(1985) and Brandenburger and

Dekel(1985», a point in this set of pS!;1chologies has a ver~ important

propert;J: C· •

=>1 in

I can be viewed as <::\ joint probabilit!;1

distribution (belief) on the occurrence of nature and on the other

agents' ps~chologies • In formal terms, lhere is an homeomorphism

1l(Jlx7L~')' Moreovet", this homeomol~phism has the

that the m-th arder bel ief cf agent

(") " -n- 51'1'l. -1,

on the sct ~'-,.. II

d

1'~ lJ

t:hat is Si .,.,...

is the marginal of

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"f'

The í n t e 1·-P I'" et <:"t t i on given to the support of the nlar·9ínal

S 1.1 P P m

<.~

nJ s. [ 1·1 (s '1) ] i s

~~ in

the heart of the defínit íon of knowledge. RecaI1 that the support of a

probabilit~ measurc is the srual1cst closed sct with probabil it~ one. We

S·I.1PP 11i:::l.l'~lS. [ f i (si) J ':l.:> the ':)~t Ot p~>::lcholo9 i~.?s 01- the j·-th

J .

agent that agent i considers possible. In othel" wOI"ds, it is the set of

Let

J bE ;:;\ subset of c) .

~ J ' j=1, ••• ,n. Suppose one wants to

Tonlla 1 i ZE:.': knmJs that agent j is in Rj •• Making use of the

i , could possibl~ be tr~e

this means: for an::l j;6i,

must satisf::l

J

J,

one has Sj

E

R j ' B~ extending this idea fUl'"ther, we can formal ize

longer sentences where the verb ·know· appears nlan~ times. Given s' I

and R1,···,Rn t-i.S above, IIJ€' define that the sets Ri,···,Rn aI'" e I<nown IlP

Á. ".,

R~

to 1 (;:'ve 1 I< in the e::les of <:I.gent i, i t s·e-{\ R· I '711": .1 I

,

"Jh,~re I - Ri and

tn"I-J.

~ .", - j

j~i , [ / i (s i) J C

if m ~ ~~ Ri

_.

{ si G I for alI SI.1PP margS· R· }.

J

J

I t is eas~ to see that b::l taking k=OO Wf2 obtain the definition of

common knowledge (in the e~es of agent i).

This fl'"amework is more complex than Aumann's from the mathematical

point of view. However, it al10ws the dil'"E.'ct fOl'"mal izat ion of higher

leveIs of knowlEdge. FoI'" example, an event

knowledge without an~ I'"efel'"ence to the infol'"mation

tnl€ state of the world

w€Sl...

In fact, lc:d:

A

C.n..

ma~ b e c omlllon

partitionS

~or·

the

The set Ai I'"epl'"esents the ps~chologies of agent rOI'" which the

first ordel'" beliefs al'"e concentrated in a subset of A. This means that

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..

def i n(:: th,ü A is common knowledge (in the e~es of agent i ) i f

wh 121'" E-~ and for m ~ 2

We

ma~ also define common knowledge of the part itions, ~\Jh i ch ""as

impl icitl::J

a~::.sl.J.mf2d

b::J

1~llm<Hln

.. Given the p<:\rt it ions

Te

J " · .I

7T;,

, we

sa~ that agent i knol,1ls that c\gent j uses

7Tj

, i f si € P i where the

s d Pi:::{ 5iéiSi for

Jt-

i , (o.), Sj)EsIlPP m:;:l"I'"~1[s6i(Si)] ~

--=) SIJ.PP marg..n.[pj(Sj)]

::"11[(1.() ).

Again, the int.::!'rpretation is

straightfon'J<:\t-d. I-f ;:"1" st<:d~f~ of the ~'Jorld CU is thought possible to OCCIl''"

jointl::J with a ps~cholog~:J S j ' it must be the case that has a

bel i ef over

n

~'Jhose SlJ.ppot-t co i nc ides l'" i th thE" set of sb""-\tes of nature

that c",g.=."nt j observes. As we have SE"en before, this set is7f(W) the

e"1 e m e n t o f t h e p

~

r t i t i o n

7Tj"

w h i c h c o n t a i n s

tu .

Indllctivel::J, we can

define that the information part itions are common knowledge (in the e~es

of agent i) i f , where P

~

= P i above, and for m

~

2/

~ ,/, "111-1

for alI j r i , supp rnar gS

j

[ r i ( s i ) ] C P j ) To

compare this framework with Aumann's, we stil1 need to formalize the

fact that the state W has occllrred in the e~es of agent i • Clf~arl~

this the same as SIlPP mar'},.[fi(hl)] =

7[

(W). Tan and Werlang(1985) show

the following theorem:

Theorem. Assume that in the e~es o~ agent the infornl,üion

part i t i ons

7!j, .. '/

?T::;

are common knowledge and that state tu occlll'"red.

Then an event A is common knowledge at W in the sense of Allmann if,

and on 1::J i f, in t he e::Jes of agent i A i s common k nowl edge".

Observe that we alwa::Js mentian that the knowledge or common

knowledge occurs in the e::Jes of agent i . The reason for that was seen

beTore. We farmalized a belief with probabil

it~

one. As a matter of

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· . ..1

.,

..

invalidate an~ of the results above.

The infinite hierarchies Df bel iefs allowed us to state precisel~

the notions of knuwledge and common knowledge of rationalit~ in a game

(see Tan and Werlang(i984,i988b) and Werlang(1986».

There is a third wa~ to medel common knowledge: through the

109ic-theoretic framewerk. This consists basical~ Df an

formal izat ion as above, but with a less rich mathematieal structure.

The main source for that is Fagin, Halpern and Vardi(i984). WR will not

appl icabil it~. Since the mathematical structure behind is not rich

enough, it tut-ns out to bE' VE't"~ h<3,t-d to fOI·-m<..;,) ize even simple st<3,tenH::-nts

1 ike ·rat ional it~·.

There are several appl ieat ions Df the concepts Df knowledge and

common knowledge in economics. Among them are the Ba~esian foundations

of solution concepts of games, no-trade results for speculative markets,

rational expectations models, games with incomplete illformation, the

mechanism Df acquiring knowledge through information transmission, and

the stud~ Df games whose pa~offs depend on the kllowledge about the

outcomes. Outside the realm of economics, we call name some other

applications: the theor~ of so~ial conventions, the theor~i of

communication and language, and the theor~ Df information sharing in

parallel processing. AlI the topics melltioned here are covered in the

l'

list of references. We will review olll~ two of the most i mpol" t an t

consequences in economic theor~.

notions of equilibrium in games. The theor~ of knowledge and common

knowledge applied to games helps the understanding of the imp) icit

assumptions on the pla~ers which underlie some Df these solut ion

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..,

..

wil1 concentrate on the stud~ of norm,11 form

noncooperat ive games. The interested reader should refer to Tan aod

WerlangC1984,1988,i988b). lhe program of this literature is simpl~

stated: given a solution concept, find the impl icit assumptions about

the knowledgc and common knowledge of the pla~ers which wil1 lead to

this s01ution concept. Doe should be aware of another trend to

investigate the foundat ions of solut ion concepts, the evolutionar~ view

(see SamuelsonCi988) and Ma~nard Smith(1982». AIso, there are some

dcveloprncnts on the 8a~esian foundat ions in the extcnsive formo On this

see Ren~(1988), Binmore(1988) and Bicchieri(1988).

lhe following solution concepts have been studied in the normal

..

forOl: itE:rative E~liOlinati()n of strietl~ dominated strategies,

rat i ona 1 i z,:,.b 1 €. strategic behavior (defined b~ Bcrnheim(i984> and

Pearce (j. 984) ) , correlated equilibrium (defined b~ Aumann(1974», Nash

equilibrium, perfeet equilibrium (defined b~ Selten(i975» and proper

equilibrium (defined b~ M~erson(i978». We will anal~ze two of them,

namel~ iterative elimination of strietl~ dominant strategies and Nash

equi1ibrium. We sa~ that a pla~er is Ba~esian rational (or simpl~

rational) when this pla~er chooses an action which maximizes her

expected utilit~ given the bel iefs she has about the aetions Df lhe

other pla~ers. lhl.ls, in the game Df Figure 4,

I

Jl

/C.

) i ({OOO, \000) {-toDO qoo} ; '

r

ti..

(qUO,! fOCO) (1oo o/900)

~ic;uA.E 4

pla~er 11 has a strictl~ domioant strateg~, which is 1. Hence,

whatever pla~er II thinks pla~er I will do (i.e. given an~ belief about

(15)

pla~iCt- I'~-; actions), pla::H::-r 11 ~Jil1 (',I,ll,\.I::l:Js pla~ 1, if sh~Z' is

ratiana1-Ir pla~er I knows that pla~er 11 is rational, then p1a~er r, being

himself rational, wil1 pla~ u. This example shows us the I inks bet~een

la~ers of know1edgc Df rat iona1 it~ and iterations in the eliminat ion Df

strict1~ dominated strategics.

Wer1CingO<;'8Gb) that i f Ba~Jesian r'c\t iona1 it~ i5 known up to leveI OI, then

the ~la~ers will pla~ strategies which survive m+1

elimination of strictl~ dominated strategies. Carr~ing this argument to

the limit, if it i'::, COlnmC)11 kncI'Jl";'dg'::: th,1t pl<ÕI.~iClr.;:> alre B',\;Jf::~,i'~.n r::'Õl.tion:::I.l,

then the outcome of the game has to survive iterative elimination of

strictl~ dominated strategies. Converses of lhe results above are also

tn.H:.' : in a game with n pla~ers, an~ n-tuple which survives m+l round5

of elimination of strictl;J dominated strategies is pla~ed b:J some

n-tuple of pla~ers (represented b~ their ·ps~chologies· or ·t~pes·) for

whom Ba~esian rationalit:J is known up to leveI m. The same goes for m=~

( i . e., c ommon k n ow 1 Ec! 9 e) •

When we come to Nash equilibrium, matters are not so nice. There

are several wa~s to derive Nash behavior from the 8a~esian point of

view, none of them entirel~ sat isfactor~. In the case of two pla~ers,

as in Annbrl.lster and BOgE·(j.979), •• W2 cou d require common 1 knowíedge UT

rational it~ as wel1 as knowlec!ge of each other's beliefs. Or as in

Wer 1 an H ( j. 986) , Tan and Werlang(1988b) and Bacharach(1987), i t i s

possible to shol,L, that if one requires a solution concept to be 5ingl€

valljed, then the on1~ solution concept which i5 consistent with common

know1edge of rat iona1 it~ anel of itself is a Nash equi1 ibrium. Thi5 last

explanation for Nash behavior shows the strong coordination requiremcnts

behind this solution concEPt. Not onl~ rational it~ has to be corumon

knowledge, but also the actions taken, before the~ are taken.

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..

time that in an econom;1 ,."ith no aS;1mmetr;1 of' information, thE're is no

rcason for the existence of speculat ive markets. In fact, consider the

case of the stock market. (.1 stock is a claim on futut-e random profit~;

of a firmo Since ever~one knows the same probabil it~ distribution Df

the future profits (because there is no aS;1mmetr~ of information), the

onl~ reason to trade stocks would be differences in risk aversion. Thcn

lhe role of the stoek exehanges should be taken b~ insurance eompanies.

show a mueh stronger result. Thc::-.1

consider the tracle of' a I~isk~ <:r.sset in an econom~ with three pE'riods:

toda~, tomorrow and the da~ aftcr tomorrow. Toda~, the state of the

world is not realizecl ~et, and we can trade the risk~ asset •

the state of the world occurs, and each trader has access to his own

private information. The value of the ass~t is not ~et diclosed, so the

traders are al10wed one more round of exchange. The da~ after tomorraw,

the value of the risk~ asset becomes common knowledge, and no more trade

makes sense. This is an econom~ with as~mmetric information, where

trade can occur before and after part of the inform~t ion is revealed to

the agents.

Their main result is: suppose tomol"row (after the ~;!~a:':e of the

world is realized, but onl~ partiall~ revealed to the agents through

their informat ion structure) it is camman knowlE~dfJe that the agents wc\nt

to trade. Then the onl~ trade possible has to be indifferent to

no-trade for ever~ agent. The idea of the proof is simple:

common knowledge ex-post that the agents benefit from trade,

i t I· C' -,

t hen

c;:-ante there would be a feasible contract which could be written and would

I e a v e e v c r ~ o n e b e t t e r o f f, s o t h ,;( t t h e I" e w o 1.11 d h a v e b f~ e n t r a d e e:-: -,HI te.

(17)

r--~----.- .. _-.---.----.-_.

I ..

~,

EVEr~ene wants to trade ex-post, but lhe fact the~ want to is not commen knowledge.

Their result is a problem for the theer~ of speculative markets:

as~mmet~ic informalion alone cannot be rcsponsible for the existence of

1 Cl. r 9 E.' S t o c k ~'~.~ c h a n 9 e s • A v E~ Ir ~ i 01 P O r t a n t r e s e a r c I··, p r o j e c t i n t h e f i n a n c e

literature is to find where Milgrom-Stoke~'s model derarts frem real it~.

It is a point which IS crucial for lhe undcrstanding of the ver~ complcx

speculative markets we see newada~s.

This essa~ did not pretend to be exhaust ive. For surve~s of the

which conb:a.in more diverse material, see Binmore and

Brandenburger(1988), Tan and Werlang(1988) and Brandenburger(1987) •

Armbruster, W. and l.J. B0ge (1979), 'Ba~esian Game Theor~' , in

o.

North-~ Holland, Amsterdam.

Aumann, Robert (1974), 'Sub~iect ivit~ and Corr·elat ion in Random i ;·~ed

S t r a t e 9 i e s ., ..J Q ur n.$1.L.,;;tLl:1.a.t.h.f:.ru.:;i.l.i~E.c..o.n.Qmls~~ 1, 67 - S' 6 •

( 1976) , • I~g r e e i n 9 t o D i s a 9 r e e', Ih.e...é.n~S..t.i";t~.LL.!::..s. 4,

1236-1239.

__________ (1987), 'Correlated Equilibrium as an Expression of Ba~esian

R a t i o n a 1 i t ~ ., EJ;;..Qll.Qlü.e.tL.i.c..a. 55, i - 18 •

Bacharach, Michael (1985), • SomE.' E~·~tensions to a Claim of Aumann in an

A ~.: i o m a t i c Mo cI e I o f 1< n o w 1 e d 9 e', J .. Q!.J.r:.ll.:;l.L.r.:LL.E.c..o.n..Qlll.L!:"'-.Il).~..Q.ci 3 7, 167 - 1 <"/0 •

(18)

27, 17-55.

(1981', • Se en es an cI Dt h E~r S i tua t i on s', J..o..'../Lll..a.l ___ O.1.

P.hiJ.~tSSJ.pJd LXXV I I I, 369-397.

I

(1988), ·ThrE:.'E" Views of Commorl. KnoflJledge', in M. Varei i 02d.

Conference, held in Asi1omar, CA, Morgan Kaufman.

Basu, Kaushik ( 19B5', • St r a t eg i c I F1~ a t i on ,1.1 i t ~ in Extensive Games·,

Institute fclt"" f~dvancE~d Stuclies, Pr·ince:·ton, DLL.UU;.'J1 ••

Bet'"nheim,

B.

D. ·Rationalizab1e Bt~hav i or· ,

..

B:..mUHJl.cl..t.:..lJ:;j;j. 52, j ~~ 0 ;7 - 1 0 2 8 •

(1987), ·A:domat ic Char·actE~lr izat ions of Rat iona1 Choic~ in

488.

Bicchieri, Cristina (1988), 'Common Know1edge anel Baekwarel Ineluetion: A

Solut ion to the Paraclo>~', in M. Varei i

~..o.n.l..o..9..-..ê..b..o.lJ..L . .K.ll..Q.J:JJ_E.~JigS;_J...I. , C o n f e r e n c e h e 1 d i nAs i 10m a r, C A , M o r 9 a n

Kaufman.

Binmore, Ken (1988), ·Model1ing Rationa1 Pla~er·s I anel 11', J.oJ..!.!:.lli;\l of

anel Adam Br·anelenbulrger (1988), 'Common KnowlE~dge anel

Theor~', London School of Economics, m~meQ ..

81ume, L. (1986), 'LE~:dcClgraphic RefinemE~nts of Nash Equilibrium',

Depd.rtml~'nt of EC:Cll1omic:s, Universit~ of Michigê\n, DWu.c_O .••

(19)

and Th. Eisele (1979), ·On Soll.1t ion~;; of Games" ,

Brandenburger, Adam (1987', ·The Role of Common Knowledge Assumrtions

in Game Theor!;l· i n F" H a h n e d "' J..h..fL..E.l:;J;)11.Q..IlUJ:;..S..-O . .J. Iof o.r...lll.a.tJ..o.n~j;t!ll.c..s ...

and Eddie DekE~l (1985>, ·Hierarchies of BeliE~t's and Common

Knowledge·, Research Paper No. 841, Graduate School of Business,

Stanford Universit!;l.

(1986) , ·On an Axiomat ic Approach to Refinements of Nash

Equilibrium", Department of Economics, Universit!;l of Cal ifornia at

Berkele!;l, ruJ..Il~~ ••

(1987), ·Common Knowledge With Probc\bilit!;l i", "J Q 1.11:" llil..1-o..f.

Mat h 1:·'lll.a.t~1..L..E.c..o.n .... "llil.i..c;..s 16, 237-245.

(1987) , ·Rationalizabilit!;l and Correl~ted E qu i 1 i b r' i a • ,

Ecoooms.;..t..t:j_l:.a 55, 1391-1402. :

'. Cave, Jonat han (1983),· Learn i ng to Agree·, tJ:..o.D..O.nl i c Let t ~ 12,147-152.

Fagin, Ron<:\ld, Joseph Y. Halpern and Moshe Vardi (1984), ·A

Model-Theoretic Anal!;lsis of Knowledge: Preliminar!;l Report·, Proc. 25th IEEE

S!;lmposium on Foundat ions of Computer Science, West PaIm Beach, Florida.

Geanakoplos, John D. and Heraklis M. Polemarchakis (1982),

O i s a 9 r e e F o r e v f~ r ", J..Q.I.JI_n.a.Lj:LL.E.c: . .o.n..o..llLiJ:.J.b..f;.Q~ 28, 1 92- 200 •

·We

Canil

Gilbel.-t, M'1.lr garet (1981>, "Game The(Jlr~.i and Convcntions", S.!wthr.>!:iJ:. 46,

(20)

..

I -..

..

• Ag I~ ecmcn t ~>, Con vcn t i on s an d Lan glla9 e', B.::I.n..th.ese 54,

375-4~;4 •

Gilboa, Itzahk (1.9fi8), "Information and Met~\--Information", in M. Var'di

e d. Jj-.l.1;.QL_c.Li.!':;'.~.L.A;;;Jlf.;:.í~ .. L;~_!.:LÍ: ... _.R..r--";il.;}_t1n.j .. n~L

.

.â.b.fJ.!.l.t.:....KD_Q.!d.~d..9?._ll , A n n <~. 1 s o f t h e

1988 Conference, held in Asilomar, CA, Morgan Kallfman"

__________ and David Schmeidler (1988), 'Information-Dependent Games

Can Common Sense be Common l<nm·J1edgE~?· in ~i. Vardi

~..e..r.-L~~. ___ Q_f..Jte..a:;:\.cw.JJJ..9 .. _.oJ.l.Q!.t.t_.KD.!:l!.!1J.f,,;_cLg.f;;_I~ , A n n aIs o f t h e C o n f e r e n c e h e 1 d

in Asilomar, CA, Morgan Kaufman •

HalpE~rn , Joseph Y.

IbJ;..QL..P...ti..r..SJ 1 o f _.EJ:;j-~';;~.Q.nl.r.isL.ú.b . .r.L!.J.L.liJl!~j] .. ÜJ:·..d..9?. , A 5 i 1 o m a t'", C A , Mo t'" 9 a n 1< <111 f m a n •

and Ronald Fag in (1985), "A Formal Model of Knowledge,

Action and Commllnicat ion in Distt'"ibllted S~stems : Preliminar~ Report",

fL.OJ: ___ .ftC.tL.S:J Dl

e

D..!;·U.!.JJlLOjL.Er..i.lLí,;j.J.U~_Q.:L.Dj ~i t r i b_I..lt..f:..d-.-C.o..olE..1J t a t i

o.n..

__________ and Yoram Moses (1986), "Knowledge and Common Knowledge in a

Distributed Envit'"onment" , IBM Research Report RJ 4421, Januar~"

Hat'"san~i, John (1967-68>, 'Games With Incomplete Information Pla~ed b~

\Ba~es i an IPI a~H:~rs', tiaoj;1.!1f;;.lUf'..nL-..~_{;;.L(;.'.nt:..e 14, 159-182, 320-334, 486-502"

Kaneko, Mamoru (1987), 'Structural Common Knowledge and Factual Common

Knowledge', RUEE Working Paper No. 87-27, Hitotsubashi Universit~"

and Takashi Nagashima (1988), 'Pl<:\~er's Deductions and

Deductive Knowledge and Common Knowledge on Theorems" , Working Paper

No. E88-02-01, Derartment of Economics, Virginia Pol~technic Instit'Jte

(21)

..

...

1<1~1:·l\{e, Saul USt>3) , ·Semantical Anal~sis of Modal Logic·, ~~Li.f..t

i~.u:

__

.t.1.a.tl)..c.lll.aLL~'.1;b.f.'-.L.C).~Ü1L...!.l11.f.L..Gr_!J.n..dli;li.U'.J.L_d_~r..Jia.tbJ:·:J!h-UjJt 9, 67- 9 6 •

L e lIJ i "!'.; , D a v i d ( j. 966 ), .C..Q.n~~.c.~f.i.tjJ:r.n~--P-b.il.o.!2.QE.ljs.:_a.L-.s..t..!.J..d..':l. , C a m b r i d 9 e ,

Massachusetts, Harvard Universit~ Pressa

Ma~ n ar d S m i t h , J oh n ( 1982 ) , EY.o..l.!J...t...L.o.n-l~h..P Tb~..t::::L-.O.i--..G..dJÜ.~,

Cambridge Universit~ Pressa

McKelve~, R. and T. Page <1.986), ·Common I<no·wledge, Consenslls, and

Aggreg·ate Infonnat i on·, WJ.llilill.c_lrjJ;jl 54, 109-127.

Mertens, Jean François and Sbmllel Zamir (1985) , ·FOt-mulat ion of

Ba~esi<:r.n tlnal~sis for G<:r.tnE·f. with Incomplete InfOt-mat ion·, In..tsx.n..atls,HLal

J.!l1J.r.J1.al . .J.i..LJiiõl..Ul.f,;_Ib..c-,:u::..:J. 1 4 , j. - 2 7' •

M i 19rom, Paul (1981.), ·An Axiomatic Cbaracterization of Corumon

Know1edge·, EçoQomptrica 49, 219-222.

and Nanc~ Stoke~ (1982), ·Information, Trade and Corumon

K n o w 1 e d 9 e·, J..o.!.lt:lli.l.L.-O..:L..E:J:;..o.LU'..ill..i.L-~ 26, 1. 7 - 29 •

Moses, Yoram, Dann!:,: Dol·~v and Josepb Y. Ha1pern (1986). ·Cbeat ing

Husbands and Other Stories :

A

Case Stud~ of Know1edge, Action, ~.nd

C o m m uni c a t i o n ., llill.r:l.b..,.J..t.;:..d_c..o.lllnJ.l.LLn..9. 1, 167- 176 •

M!:Jel~son, Roger (1978), • Ref i nement S of t be Nash Equ i 1 i br i um Concept s· ,

.l.D-U~r..n..a.t...LOJ1.a.L..> . .!.o.!JL.O..aL.o..:L.G;;Ullf'--.ItLc..o.r...':l. 7, 7 3 - 8 0 •

__________ (1985), ·8<.'\~esian Equi1ibrillm and Incentive-Colllpatibilit!:J

An Introduct ion·, in L. Hurwicz, D. Schmeid1er and H. Sonnenschcin

e d 5 . , B . .o.tiaL_(io..oJ..;i_jw..d-.SJJ.f.:-Lal.-i!r.,jl?nl.z.:.ll.l.,Q.n : E ~:i i;..a..':I~_.l.u.-tl~.J.llW:'L..O.1.-L1··L:.,..llil

(22)

,.

Ne"'lI11an, P et 121'" (1988), ·Common f(n()J."Jl·~elge and the G<:\me of Red H<.üs·,

WOl'"king Papers in Economics "210, The Johns Hopkins Univer-sit~.

Nielsen, L. (1984), ·Common l(noI,<Jleclge, Communicat ion, and Convergemce

Parikh, Prashant (1988), ·Language and Strateg ic Inference·, Ph.D.

Dissertation, Stanford UniVErsit~.

Paril<h, Rohit anel Paul Krasucki (j.987), ·Communication, Consenslls c\nd

Knowledgc", Dcpartment of Mathematics, CUNY Graduate Center, New York,

N y, nÚlll.t:.o." "

Pec\"'ce, David (1984), "Rationalizable strategic Behavior and lhe

Problem of Perfection", EL;.~:~nJ:llll...;:d.:..r:.l.wi 52, 1029-10~30.

Ren~, Ph i 1 i p (j. 988), • Rat i ona 1 i t ~, Common Knowl edge and t he Theor~ of

Games·, Ph.D. Thesis, Princeton Universit~i Chapter 1.

Rubinstein, Ar-iel (1988), ·A Game with Almost Common Knm ... ledge An

E}~ample, London School of Economic,:" milu§'.o. ••

Samet, Dov (1987), "Ignoring Ignol~<:\nce and Agreeing to Disagree·, MEDS

Discussion Paper, Northwestern Univcrsit~.

Samuelson Larr~ (1988), ·Evolutionar~ Foundat ions of Solution Concepts

for Finite, Two Pla~ers, Normal-Form Games·, in M. Vardi ed. ,

tonference held in Asilomar, CA, MOl'"gan Kaufman.

S a v a 9 e, L e o n a r d (1 954 ),

n~_...E..o.!.ul.d

.. a.Li .

..Q~..f....J3..t.ili...!ltJ...c.5.

• N e w Y o r 1<, W i 1 (?

~

(23)

Selten, Reinhardt (1975), "Reexaminat ion of the Perfectness Concept ftir

E q 1.1 i 1 i b r i I j lU fi o i n t s i n E s t e n s i v c G :':l. me'.:; ", I.n..tJ.·;;-.r:.n.:;i.L.i.r..:o.n.?;l.l ... J.n-'.LL.n.i;~L ... .ct:L_Gx.uII';';.

I.h.f;..o.r:.~ 4, 25-5::'i "

Shin, H~l.ln 50ng (1987), "L.ogical Strl.lctur·e of Common Knowleelge, I anel

11', Nl.lffield Col1ege, mLl1lf;;;JJ .• "

Tan, Tomm~ Chin-Chiu anel Sirgio Ribeiro da Costa Werlang (1984), "The

Ba~esian FDundations of Rational isable Strategic BehaviDur anel Nash

Eqljilibl~ium BehaviDl.lr", Princeton LJniversit~, m..i...oll~""

<1.985), "On Aumann's Notion Df Common Knowledge An

Alterni:\t ive (.lPPt"o~c:h", Princeton LJniversit~J, nÜlll.f;.~.C~".

(1988) , "On Aumann's NotiDn Df Common Knowledge An

Alternative Approach", Extensivel~ Reviseel Version, CARESS Worl<ing

Paper, UniYersit~ of Penns~lYania.

(1988a), "A GI.liel2 to Knowledge anel Games", in M. Varei i ed. ,

Ih..e.or~..t i ç a 1 A ~iJ2.f.;~---O..f_,E..c.S1...s.º.J) ... Ln.9.-Alw..!.L.L.KD-'J,lti~d.9JLlik , A n n aIs o f t h e

Conference held on Asilomar, CA, Morgan Kal.lfman.

(1988b), • The Ba~es i an FOl.lndB.t i ons of 50 11.lt i on Concept s of

G a m e s " ~..t..o..w::.nj;)..L .. Q f E c ClitQ.nLi ... c:_1M..Q~. 45, 370 -391 •

Va r di, Mos h e ( e d .) (i 988 ), Er .. !'::oJ:..e.!.:.d.llJ..9 .. !L .. Q.1_ .. tb..c S ~.t:..Q.r.I..d...-..CS.Ul .. :r.r:.J.:.E.I1..C..P---..Qn.

Ihf,; .. o~t..lJ:;..al...A5Jl.~ .. ~ll..-Jj .. L...E .. ~'..a.~iJ:L') ... Ln .. sl...é.b .. Q_'.!.1.. ... .K.r.l.CH~/ .. lJ;::.d..9~ , A s i 1 o 111 a r, C I~ , Mor' SI ê\ n

Kal.lfman.

Werlang, Sér'gio Ribeiro ela Costa 0.986), 'Cammon KnowlE.'dge and Game

(24)

EI/:;I\IOS lCO!!0:':ICOS D/X U'GE

(a parti,. de li? 50)

50. JOGOS DE IN~O~~~~~O INCOMPLETA: UMA I~TRODUÇ~O - S~rgio Ribei ro da Costa

\·Ir;.rl.:>n2 - 198'~ (';~'QOL;do)

51. A TEOl\i/\ 1',Oi':~TJ\;{ll\ h0[)~f{N.l\ E O [QUILfsRIO G[~.l\L ,\·I.'\LPJ\SIMlO C011 Ui'1 I~OI-\ES.O

INFINITO DE BENS - A. Araujo - 1984, (esgolado)

...,. 52. r~ INDETErUilt,:,lI.ÇJ\o DE hOf{G~NSTERt! - Antonio f'briô da Silveira':' 1981, «(,St~ot~l!O)

'+

53.

O PROBL[M~ DE CREDICILiDADE EM POLfTICA Eco~b~iICA - Rubens Penha Cysne

-1984' (csgotac1o)

S/L 1Jt-~i\ ANÁL I $[ ESTAr rs TI CA Df-\S CAUSf.S DA ou 55/\0 DO CHEQUE SEH FU~1DOS:

FOf\:·:U-l{{çii.O DE L'H PROJEI-O PILOTO - Fernando de Holc:r.da Garbosas Clovis de Farc e

AJor~io Pessoa de Araujo - 1984

55. POLfTIC;\ i'!I\CROECOr·lÔHIC{-\ NO BR!\SIL: 1961~-66 - Rubens Penha Cysne 1935

-(esgotéldo)

56. E\lOlUÇ.7í.() DOS PU'.NOS BÁSICOS OE FII!f\l,CIN',Et-iTú P,l'.p..~\ l-'\QlIISIÇÃO Dr= C/\S/\ Px0P!:I.L,

DO BM';CO ilí\CiOf'UI.L DE HABITAÇÃO: IS6l'-lS31j - Clovis de Faro - 1985 (cssoté)~,:ú)

57. liOEOA I tJDEXADA - Rubens P. Cysne - 1985 (esgotado)

58. INFL'l.CÃO [ SJI.LÁRIO REAL: A [XPERltNC:A BR/\SILEI/1A - Raul José Ekerman

-1985 (csgotudo)

59.

O ENFCQUE MONETARIO DO BALANÇO DE PAGAMENTOS: UM RETROSPECTO - Valdir

Ramalho de Melo - 1985 (esgciadc)

60. '·\OED/\ E p;;r:ÇOS RELATIVOS: EVI DtNCIA EI~prRI CA Antonio SaLazar P. I3r"nd.)c

-1935 (esgotado)

61. INTFnPkETAç~O ECONDMICA, INFLAÇ~O E INDEXAÇ~O - Antonio Maria da Si l~cil~

-1985 (esgotado)

62. W\CROE'CONOl-lIA - CAPrTULO I - O SISTE/·IA HONETI\RIO - Mario Henrique Si!110ilSCIl

e Hulwl)s Penha C)'5nc - 1985 hsgotado)

63. HACROECOtIO/-lll\ - CAPrTULO 11 - O ÜflLl\liÇO D~ PAG,tU·íENTOS - gario Henriqu(~

Simonsen c Rubens PCnhL) C~/Sr.c - 198) (esgotodo)

611. HACROr-:CO/10lll/\ - CAPrTULO 11I - I\S COlH!~S W\CION/\IS - t1ario Hcnric;uc Sirl'()I!'.(ól

(! P.ubcns Penha Cysn(: - 1985 (esgotado)

65.1\ DH1NJO/\ rOR DIVIDF.NDOS: UH'" JUSTIFICI\TIVA TÊÓrUCA - TOH/W CHIN-CliIU TN!~;

$6r9io Ribcir~ da Costa Wcrlang - 19U5 (esg()tado)

6G. BI:l VE RnnOS PECTO Dl\ ECONOI1! J\ 15;,/\$ I U: I nA EI\T(~E 1979 c 198~ - l\ulJcns P(:!I:';1

Cy!> ne - 1985

(25)

... " !',.

T

r '

. !

... " ' , ", , \ .,

68.

INFLAÇ~O E POLrTICAS DE RENDAS - Fernando de Holand~ Barbosa e Clovis de Faro - 1985; (esgotado)

. I

69. SP.AZIL ItnETU~/\TlOtl/\.l TRt~DE ANO ECONOHIC GRO\.JTH - Hario Henrique

Si monsen - 1986

70. CAPITl\LlZ,lIÇfiO corarNUA: APLICAÇÕES - Clovis de Faro - 1986 (esgotado)

71. A RATIONAL EXPECTATIOt~S PARADOX - Mario Henrique S'irnonsen - 1986 (esgotado)

72.

A

BUSINESS CYCLE SlUDY FOR THE U.S. FORM 1889 TO 1982 - CarlQs Ivan

Si~onsen Leal - 1986

73. DINÂMICA HI\Cr-OECONÕ~HC;'\ - EXCRCrCIOS RESOLVIDOS E PROPOST0S.- Rubens Penha

Cysnc - 1986 (esgotado)

74. COMMON KNOWLEDGE AND GAME THEORV - S~rgio Ribeiro da Costa Werlang - 19C6

75: HYPERSTABILlTY OF N/\SH EQUllIORIA - Carlos Ivan Simonsen Leal - 1986

76. lHE BHOHN-VOtJ NEUi~.l\NN DI FFERENTI AL EQUATI.0il FOR B 111ATRI X GAl1ES

-Çarlo5 Ivan Sirr.onsen Lea)' - 1986 (esgottido)

77.

EXISTEl-lCE OF A SOLUTlO;'i TO THE PRINCIP/'.L'S PROBLa~ - Carlos Ivan Sirnonscn

leal - 1986

78.

FI LOSOnft. E ?OLfTICA ECOt\uMICI\ I: Variações sobre o Fenômeno, a Ciência ç

seus Cientistas - Antonio Maria da Si lyci r~ - 1986

79. O PP.EÇO DA TERRA NO BRI\SIL: VERIFICAÇÃO DE ALGUHAS HIPÓTESES - Antonio

Salazar Pessoa B~and3o - 1936

80. HÍ:TOOOS HATEi'1ÃTlCOS DE r.STATfsTICA E ECONOI-1ETRIA: Capitulos 1 e 2

Carlos Ivan Simonsen Leal - 1986 - (esgotado)

81. BRAZlllAU INDEXING ANO !NERTIAL INFLATlON: EVIDENCE FROH TlHE-VARYING

ESTIMATES DF AN INFLAT!CH TRANSFER FUNCTIOH .

Fernando de Holanda Barbosa e Paul D. McNelis - 1986

82. CONSÓRCIO VERSUS CR!:DITO DIRETO EM UM REGII'lE DE HOEDA ESTÃVEL-- Clovis de Faro

:. '1986

83. NOTAS DE AULAS DE TEORI.I\ ECO:~ÔHICA AVANÇADA - Carlos Ivan Simonscnlcill-1986

a~. FILOSOFIA E pOlfTICA ECONÔMICA I I - Inflaç~o c Indcxaçio - Antonio Maria da

Si lvcira - 1936 - (esgotado)

85. SIGN/\LLlNG II.ND fd~I3ITRlI,G!: - Viccllt~ l1adrigal e Tornmy C. Tan - 19116

86, N;SEssor.1 A ECONNíl CA P/\R/\ A ESTRATrGl1\. DE GOVi:RNOS ESTADL'I\.I $: ELI\UOI\l\c('ES

(26)

88. INDEXAÇÃO E ATIVIDADE AGRICOLAs: CONSTRUÇÃO E JUSTIFICATIVA PARA A ADoçÃO DE

UM INDICE ESPECIFICO - Antonio Salazar P. Brandão e Clóvis de Faro - 1986

89. MACROECONOMIA COM RACIONAMENTO UM MODELO SIMPLIFICADO PARA ECON<J1IA ABERTA - Rubens Penha Cysne, Carlos Ivan Simonsen Leal e Sergio Ribeiro da Costa

Wer1ang - 1986

90. RATIONAL EXPECTATIONS, INCOME POLICIES AND GAME THEORY - Mario Henrique

Simonsen - 1986 ~ ESGOTADO

91. NOTAS SOBRE MODELOS DE GERAÇÕES SUPERPOSTAS 1: OS FuNDAMENTOS ECONÔMICOS - Antonio Sa1azar P. Brandão - 1986 - ESGOTADO

92. TOPICOS DE CONVEXIDADE E APLICAÇÕES Ã TEORIA ECONÔMICA - Renato Frage11i

Cardoso - 1986

93. A TEORIA 00 PREÇO DA TERRA: UMA RESENHA

- Sergio Ribeiro da C03ta W~r1ang - 1987

94. INFLAÇÃO, INDEXAÇÃO E ORÇAMENTO DO GOVERNO - Fernand"o de Holanda Barbosa - 1987

+ 95. UMA RESENHA DAS TEORIAS DE INFLAÇÃO

- Maria Silvia Bastos Marques - 1987

96. SOLUÇÕES ANALtTICAS PARA I. 1AXA lNl:~RNA DE RETORNO

- Clovis de Faro - 1987

97. NEGOTIATION STRATEGIES IN INTERNATIONAL ORGANISATIONS:

A ~~ - THEORETIC VIEWPOINT - Sergio Ribeiro da Costa Wer1ang - 1987

(27)

99. llil '1'E~lA llliVISITADO: A lmSPOSTA DA PRODUÇÃO AGR!COLA AOS PRELOS NO BRASIL

- }'ernando de Holanda Barbosa e Fernando da Silva Santiago - 1987

100. JUROS, PREÇOS E D!VrDA PUBLICA VOLU~lli"I: ASPECTOS TEORICOS

-- Marco Antonio C. }fartins c Clovis de Faro -- 1987

101. JUROS, PREÇOS E D!VIDA PliBLlCA VOLmm 11:" A ECONOt-lIA BRASILEIRA (1971/1985)

- Antonio Salazar P. Brandio, Clóvis de Faro e Harco Antonio C. Hartill.s - 1987

102. MACROECONOHIA KALECKIANA - Rubens Penh;l Cysne - 1987

103. O pRR~lIO DO D6LAR NO NERCADO PARALELO, O SUBFA'fUHANENTO DE EXPORTAÇÕES E O

SUPERFATURAHEN'l'O DE INPORTAÇÕES - Fernando de Holanda. Barbosa - Rubens Penha

Cysne c ~larcos Costa Holanda - 1987

194. BRAZILIAN EXPElUENCE WITIl EXTERNAI. DEB! AND PROSPECTS FOR GROHTU

-FernanJo de Holanda Barbosa and Manuel Sa"nchcz de La Cal - 1987

105~ KEYNES NA

SEDIÇÃO DA

ESCOLHA

POULICA

- Antonio Maria da Silveira - 1987

lObo O 'l'EORENA DE FROBENIUS-PERRON - Carlos Ivan Simonsen Leal - 1987

107. POPUUç.ÃO HHASILElRA - "Jessê 1'1onte11o 't". 1987

108. MACROECONOMIA - CApITULO VI: "DEMANDA POR MOEDA E A CURVA !.Mil - Mario Henrique

Simonsen e Rubens Penha Cysne - 1987

109. MACROECONOMIA - CAPITULO VII: "DEMANDA AGREGADA E A CURVA IS"-Mario Henrique

(28)

;

-•

..,.

111. THE BAYESIAN FOUNDATIONS OF SOLUTION CONCEPTS OF GAMES - Sergio Ribeiro

da Costa Wer1ang e Tanmy Chin - Chiu Tan - 1987

112. PREÇOS LíqUIDOS (PREÇOS DE VALOR ADICIONADO) E SEUS DETERMINANTES; DE PRODUTOS

SELECIONADOS, NO PERíODO 1980/19 SEMESTRE/1986 - Raul Ekerman - 1987

113. EMPRÉSTIMOS BANCÁRIOS E SALDO-MtDIO: O CASO DE PRESTAÇÕES - Clovis de Faro - 1988

114. A DINÂMICA DA INFLAÇÃO - Mario Henrique Sie~ns~n - 1988

115. UNCERTAINTY AVERSION AND THE OPTIMAL CHOISE OF PORTFOLIO - Jaroes Dow e Sergio Ribeiro da Costa Werg1ang - 1988

116. O CICLO ECONÔMICO - Mario Henrique Siroonsen - 1988

117. FOREIGN CAPITAL AND ECONOMIC GROWTH - THE BRAZILIAN CASE STUDY

Mario Henrique Siroonsen - 1988

118. COMMON KNOWLEDGE - Sergio Ribeiro da Costa Wer1ang - 1988

119 •. OS FUNDAMENTOS DA ANÁLISE MACROECONÔMICA - ProL. Mario Henrique Simonsen e Prof. Rubens Penha Cysne ~ 1988

120. CAPITULO XII - EXPECTATIVAS RACIONAIS - Mario Henrique Siroonsen - 1988

121. A OFERTA AGREGADA E O MERCADO DE TRABALHO - Prof. Mario Henrique Simonsen e Prof. Rubens Penha Cysne - 1988

122. INERCIA INFLACIONÁRIA E INFLAÇÃO INERCIAL - Mario Henrique Siroonsen - 1988

123. MODELOS DO HOMEM: ECONOMIA E ADMINISTRAÇÃO - Antonio Maria da Silveira - 1988

124. UNDERINVOICI~G OF EXPORTS, OVERINVOICING OF IMPORTS, AND THE DOLLAR PRE~UM

ON THE BLACK MARKET - Prof. Fernando de Holanda Barbosa, Prof. Rubens Penha

Cysne e Marcos Costa Holanda - 1988

125. O REINO MÁGICO DO CHOQUE HETERODOXO - Fernando de Holanda Barbosa, Antonio

Sa1azar Pessoa Brandão e Clovis de Faro - 1988

(29)

-...

J

127. TAXA DE JUROS FLUTUANTE VERSUS CORREÇÃO MONETÁRIA DAS PRESTAÇÕES: UMA COMPARA

çÃO NO CASO DO SAC E INFLAÇÃO CONSTANTE - Clovis de Faro - 1988

126. CAPITULO 11 - MONETARY CORRECTION A1~ REAL INTEREST ACCOUNTING

Rubens Penha Cysne - 1988

129. CAPITULO 111 - INCOME ANDDEMAND POLICIES IN BRAZIL - Rubens Penha Cysne - 1988

130. CAPITULO IV - BRAZILIAN ECONOMY IN THE EIGHTIES AND TRE DEBT CRISIS

Rubens Penha Cysne - 1988

131- THE BRAZILIAN AGRICULTURAL POLICY EXPERIENCE: RATIONALE AND FUTURE DlRECTIONS

Antonio Sa1azar Pessoa Brandão - 1988

132. MORAT6RIA INTERNA, DtVIDA PUBLICA E JUROS REAIS - Maria Silvia Bastos Marques e

Sergio Ribe.iro da Costa Wer1ang - 1988

.

133. CAP1'IUID IX - TEORIA IX) CRESCIMEN'ID ECDNOOro - Mario Henric::ue Sironsen - 1988

134. mNGErAMEN'IO roM A.OONO SAIARIAL GERANJX) EXCESSO DE DEMANDA - 1988

Joa0Ui.m Vieira Ferreira Levv

~io Ribeiro da Costa Werlang

135. AS ORIGENS E roNSE((Jt:NCIAS DA INFIJ\.ÇÃO NA AMERICA LATINA

Fernancb de lb1anda Barrosa - 1988

136. A CONTA-CORRENTE DO GOVERNO - 1970-1988 - Mario Henrique Simonsen - 1989

137. A REVIEW ON TRE THEORY OF COMMON KNOWLEDGE

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