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IENSAIO ECONÚt-1ICO N9 137
A REVIE~\' ON THE THEORY OF COIvlMON KNOWLEDGE
Prof. Sérgio Ribeiro da Costa Wer1ang
-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 donewhen the autor visited the Dcpartment of EconQlãcs of the University of
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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 onebel 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 haveseveral 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.
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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.I
I
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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
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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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: ~
! ...
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, thisperson 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
r
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 lheI... occurrence of the cell of his partition which cont<:-\ins
w .
We denoteit 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
"
\
\,..
....
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, thepal'"t it
iDn~;
Df the <:\9cnts cure771
anel77; ,
and the eVE:nt is A.FOI~
conv(-::-n icnce,
~I)C
dr-aw the setn
tllJicc.- Inth~""
+'irst we sec theinfDrmation 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
OJI--+-+----+!:==;:::::==:=:~~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'"mationpartit~on
n; ,
she
.
sees.
the event11;
(w)
As 1(t4J) is cont<.dned inA,
agent i knowsthat 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 theoccunenct:;-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 thisset 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 2In 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 on~·
7T,
ci. n drr;
a I' e c o m OI o n k n o~I}
1 c cf~J
eto 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 iscommon 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~---p6
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 wand is cont:aint-~d in B. Allm;:wn's dcfinition is elegant, c\nd Cé\n be
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~
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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 orderbel 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 set4<..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
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5
!),
,:> r
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'" eJ~.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 emarginaIs of a probabilit:J measure
P
which is defined o~ a product spaceu
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 S· in the set S· _ .. ( ( Si f s~
. . .
) the1 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), l30geand 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
s·
I can be viewed as <::\ joint probabilit!;1distribution (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 thethat the m-th arder bel ief cf agent
(") " -n- 51'1'l. -1,
on the sct ~'-,.. II
d
1'~ lJt:hat is Si .,.,...
is the marginal of
"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
R·
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
R·
JJ,
one has Sj
E
R j ' B~ extending this idea fUl'"ther, we can formal izelonger 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 andtn"I-J.
~ .", - j
j~i , [ / i (s i) J C
if m ~ ~~ Ri
_.
{ si G R· I for alI SI.1PP margS· R· }.J
JI 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 omlllonpartitionS
~or·
theThe 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
..
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 ""asimpl icitl::J
a~::.sl.J.mf2d
b::J1~llm<Hln
.. Given the p<:\rt it ionsTe
J " · .I7T;,
, wesa~ that agent i knol,1ls that c\gent j uses
7Tj
, i f si € P i where thes 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 isstraightfon'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 naturethat 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 n7Tj"
w h i c h c o n t a i n stu .
Indllctivel::J, we candefine 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 ) Tocompare 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) showthe 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· . ..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
..,
..
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
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.
..
•
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.
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 •
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.
t·
I
(1988), ·ThrE:.'E" Views of Commorl. KnoflJledge', in M. Varei i 02d.
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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
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anel Adam Br·anelenbulrger (1988), 'Common KnowlE~dge anel
Theor~', London School of Economics, m~meQ ..
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(1986) , ·On an Axiomat ic Approach to Refinements of Nash
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(1987), ·Common Knowledge With Probc\bilit!;l i", "J Q 1.11:" llil..1-o..f.
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Ecoooms.;..t..t:j_l:.a 55, 1391-1402. :
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S!;lmposium on Foundat ions of Computer Science, West PaIm Beach, Florida.
Geanakoplos, John D. and Heraklis M. Polemarchakis (1982),
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..
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..
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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",
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__________ 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
..
...
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' •
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Know1edge·, EçoQomptrica 49, 219-222.
and Nanc~ Stoke~ (1982), ·Information, Trade and Corumon
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Husbands and Other Stories :
A
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An Introduct ion·, in L. Hurwicz, D. Schmeid1er and H. Sonnenschcin
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,.
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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
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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
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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
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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 - ValdirRamalho 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
... " !',.
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 IvanSi~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 Sirnonscnleal - 1986
78.
FI LOSOnft. E ?OLfTICA ECOt\uMICI\ I: Variações sobre o Fenômeno, a Ciência çseus 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
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
•
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
ESCOLHAPOULICA
- 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
;
-•
..,.
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
-...
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