N9 226
LAWS OF LARGE NUMBERS FOR NON-ADDITlVE PROBABILITIES
James Dow
Sérgio Ribeiro da Costa Wer1ang
PRElIMINARV VERSION COMMENTS WELCOME
Laws of Large Numbers for Non-addltlve Probabllltles
by
James DOW and
Sérgio Ribeiro da Costa
WERLANG--SUMMARV: We apply the concept of exchangeable random variables to the case of non-additive probability distributions exhibiting uncertainty aversion, and in the class generated bya convex core (convex non-additive probabilities, with a convex core). We are able to prove two versions of the law of large numbers (de Finetti's Theorems). By making use of two definitions. of independence we prove two versions of the strong law of large numbers. It turns out that we cannot assure the convergence of the sample averages to a constant. We then modal the case there is a "true" probability distribution behind the successive realizations of the uncertain random variable. In this case convergence occurs. This result is important because it renders true the intuition that it is possible "to learn" the "true" additive distribution behind an uncertain event if one repeatedly observes it (a sufficiently large number of times). We also provide a conjecture regarding the "Iearning" (or updating) process above, and prove a partia I result for the case of Dempster-Shafer updating rule and binomial trials.
- London Business SchooI. Sussex Place. Regents Park. London NW1 4SA .
.. Graduate School of Economics - Getulio Vargas Founda1ion (EPGEJFGV). Praia de Botafogo 19011()Qandar. Botafogo 22253-900. Rio de Janeiro. RJ, Brasil.
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Laws of Large Numbers for Non-addltlve Probabllltles
1. Introductlon
James Dow· Sérgio Ribeiro da Costa Werlang··
In this paper, we explore the applicability of the Law of Large numbers to situations of ambiguity, or Knightian uncertainty (Knight(1921)). The classic paradigm of this type of uncertainty was presented by ElIsberg(1961): consider an urn containing red and blue balls, and a decision maker who must accept or decline lottery tickets that pay off fixed sums conditional on the ball being of a specified colour. A decision maker may behave differently when he has no information about the rei ative frequencies of red and blue, compared to a situation where he knows the frequencies to be 50%.
In the Bayes-Savage model, decision-makers are always represented by a utility function and a subjective probability (Savage(1954». In the model studied in this paper, decision-makers in an ambiguous (Knightian-uncertain) environment, i. e. with little information about the functional form and parameters of the uncertainty they face, behave differently. Their preferences are represented by utility functions and a non-additive probability (Schmeidler(1982, 1989». Intuitively, a non-additive probability is a probability measure where the probability of an event and the probability of its complementary event may sum to less than 1; the deviation from 1 may be thought of as a measure of the amount of uncertainty and induces cautious behaviour by the decision-maker. Thus, a probability zero event (intuitively, an event that almost never happens) need not be an event whosa complement is of probability one (an event that will "definitely" not happen).
A dosely related modal of decision-making under Knightian uncertainty is that of Gilboa and Schmeidler(1989). In this alternative setting, preferences under uncertainty are represented by a sat of additive probabilities, called the core, and the expected utility is computed as the minimum value of the expected utility for lhe additive distributions in the core. In general, when preferences • London Business Scool.
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exhibit uncertainty aversion, under mild assumptions a non-additive probability my be represented by a core. For the precise relationship between the approaches, see Gilboa and Schmeidler(1992a, 1992b, 1993).
These models of decision:-making under Knightian uncertainty give qualitatively different results from the Bayes-Savage model when we consider situations with a single random variable. For example, in earlier research the present authors showed that Arrow's "local risk neutrality" result no longer holds in the same form (Dow and Werlang(1992», that purely speculative trade can occur (Dow, Madrigal and Werlang(1989), that asset prices may be indeterminate (Simonsen and Werlang(1991», and that there may be cooperation in the finitelly repeated version of the prisoners' dilemma (Dow and Werlang(1994».
However, a very interesting class of problems concerns behaviour with many random variable~s. To be specific, we are interested in the question of whether uncertainty "disappear-s" when there are a large number of observations. In other words, we investigate the applicability of the Law of Large Numbers to situations of Knightian uncertainty. Two economic applications come immediately to mind.
The first economic application of the Law of Larga Numbers is to portfolio diversification. If I hold a large portfolio containing many assets, and my beliefs about the distribution of individual asset returns are uncertain, will diversification cause this uncertainty to disappear?
The second application is to leaming. If an agant observes a sequence of random variables, under what conditions will the uncertainty converge to zero over time?
The answer to the question of whether the Law of Larga Numbers holds in this setting tums out to depend on the relationship between the uncertainty in the distributions of the random draws. To return to the ElIsberg urn paradigm, one can distinguish two very different situations:
Experim.ent (1): Balls are drawn (with replacement) from an urn with an uncertain fraction of red and blue balls.
Experiment(2): One ball each is drawn from a sequence of urns. The frequencies of red and blue balls in the urns are uncertain.
Clearly, the second experiment is in some sense more fun~amental although the first is also of interest. It turns out that in the case of experiment (1). we can be sure that the agents' subjective beliefs about the frequencies becomes less uncertain over time and indeed eventually converge to a limit (which may be interpreted as the actual frequency in the urn) (Theorems 5 and 6). Thus. in the experiment (1) the Knightian uncertainty does disappear.
In the case of experiment (2), however, only a much weaker statement is possible (Theorems 3 and 4). This is perhaps unsurprising, since, intuitively, in order to have convergence, one should have a "true" underlying probability distribution on the events. However, the fact that we have a different urn in each stage destroys the notion that there is a "true" distribution behind the phenomenon.
Since the modal in this paper is purely subjective. (being a generalization of the Bayes-Savage model), we start by defining the concept of exchangeability (defined by de Finetti(1937». Successive draws from an unknown distribution cannot properly be termed "independent" from the viewpoint of a decision-maker, since they ali depend on the parameters of the common, unknown distribution. Theyare. rather. "exchangeable" in the sense that rearranging the observations in a different permutation would lead the decision-maker to make the same inferences. De Finetti's theorems (1937) shows that exchangeability ensures that the probabilities of any sequence of observations is the same as if the observations were generated by repeated independent draws from a given distribution (different from the original one. except in the case there is no correlation to start with). Thus, there is a Lawof Large Numbers for exchangeable random variables. We start by exploring the analogue of this result in a Knightian-uncertain environment (Theorems 1 and 2). Then we turn to stronger independence conditions (Theorems 3. 4 5 and 6).
2. Laws of Large Numbers under Knlghtlan Uncertalnty
The concept of Knightian uncertainty is modelled here as in Schmeidler (1982, 1989), Gilboa(1987) and Schmeidler and Gilboa(1989, 1992a, 1992b). We are going to consider the case that we have the space of events as an infinite cartesian product of the same space of states of nature and events (O, A), endowed with the product algebra. We restrict ourselves to non-additive probabilities which are convex (i. e., P(AUB) + p(AnB) à! P(A) + P(B) for events A and B) and for which there exists a closed convex set C(P) of additive probabilities, known as the core of P, such that for ali q E C(P), q(A)à!P(A) for every event A, and such that for ali variables X which are bounded and measurable, one has Ep[X] = minqeC(p) Eq[X). On some conditions for non-additive probabilities to obey these properties, see Schmeidler and Gilboa(1989, 1992a, 1992b).
Deflnltlon If P is a non-additive probability defined in (on, An), we say it is .
syrnmetric, or exchangeable if for ali permutations x of the set {1, ... , n} one has, for ali events K, P(K) = P(KX) , where Kx is obtained from K by changing the coordinates according to
x:
Kx= {(0)1, ... , Wn) E QOI (0)1, ... , Wn) E Kx ~ (Wx-1(1), ... , Wx-1(n) E K}. On the other hand, if P is a non-additive probability defined in (Q'X', ACIO), we say it is symmetric or exchangeable if the restriction of P to any finite dimensional event subspace is symmetric. In the same way, for P in (on, An) we say that random variables (X1, ... ,
Xn)
defined in (O, A) are exchangeable, if for ali events H (in mn), and for ali permutations x of the set {1, ... , n} we have that P«X1, ... ,Xn)
E H) = P«Xx(1), ... , X.(n» E H"), where Hx is defined analogously to Kx. Ukewise,one defines exchangeability for variables on (QD, ACIO). It is easy to see that if (X1, ... , Xn) are exchangeable, then except for a set of P-probability zero (to be precise, of marginal over O of P)we
have, for ali i and j: XFXj. Thus, when (X1, ... , Xn) are exchangeable we can always think of random variables XI and Xj which are "identical", and an underlying probability distribution on (O", An) which is exchangeable. That is what is done from this paint on in the textoDeflnltlon We say that a sat
C
of (additive) probabilities defined on (on, An) is symmetric if for ali q E C, and for ali permutations 3t of the set {1, ... , n} we have that Cf' E C, where Cf' is the probability distribution defined by q"(K)=
q(Klt). Again, it is possible to extend the definition to a sat of probabilities in (CF, ACID).The first auxiliary result shows that the core of an exchangeable non-additive probability P is symmetric.
Lemma 1 If P is an exchangeable non-additive probability P, convex, defined on (QO, An) (n possibly infinite), and C(P) is its core, C(P) is symmetric .
. Proof. Supposa that there is q E C(P) and a permutation 3t of the sat {1, ... , n}, with
qx
~ C(P). This means that there is an event K with qx(K) < P(K). Sut P is exchangeable, so that P(K) = P(Klt) s; q(Klt). Sut this is a contradiction, since we have q"{K)=
q(Klt).QED. The next auxiliary result is central to the proofs. It is shown that in the core of an exchangeable convex non-additive probability, where the core is convex, we have an exchangeable additive probability.
Lemma 2 Let P be an exchangeable convex non-additive probability defined on (QO, An) (n possibly infinite, Q finite), and C(P) its convex core. Then, there exists an additive probability
qP
in C(P) which is exchangeable. Furthermor~, it is the average cf the extremai points of C(P).Proof. Take n finite to begin with. Then the set of extremai points of C(P) is finite (because Q is finite). Also, it is easy to check that if q is in the extremai sat, then for any permutation
x
of the sat {1, ... , n} we have thatqx
is extremai too. Therefore, if we take the average of the extremai points, call itqP,
it must be the case that (qfjx=q',
for any permutation, that is, it is exchangeable. To see that the result is valid for n infinite, consider finite dimensional projections of increasing length. It is immediate to check that restrictions of projections to subspaces coincide with the projections on that subspace. Finally, with the weak topology C(P) is compact, and we take limits of the increasing finite dimensional projections.We are ready to state and prove the first two results.
Theorem 1 (de Flnettls's Weak Law of Large Numbers) Let {XJi=1, 2, 0 0 0 ba an infinite sequence of bounded random variables in the space (Q, A). Let Sk and Sh be, respectively, the average of k and h of the random variables. Assume that the random variables are exchangeable for P in (W, ACIO), and that P is convex, with a closed convex core (C(P)), and such that for any bounded random variable V, Ep[Y]
=
minqeC(p) Eq[V]. Then, given any € >°
and 8> 0, there isko
such that for ali k and h greater than or equal toko:
P(ISk - Shl
>
€) < 8.[In other words, if one takes k and h large enough, the probability that the sample averages Sk and Sh are going to differ by more than € can be made as small as it is wished.]
P roof. For qP of the lemma above is an additive probability which is exchangeable in de Finetti's sense. But P(ISk - Shl > €)$qP(ISk - Shl > E). Hence, it is enough to prove the result for qP. This is done in de Finetti(1937).
QED. Theorem 2 (de Flnettls's Strong Law of Large Numbers) Let {XJi=1, 2, ... ba an infinite sequence of bounded random variables in the space (Q, A). Let S k be the average of the k first random variables. Assume that the random variables are exchangeable for P in (W, A~, and that P is convex, with a closed convex core (C(P», and such that for any bounded random variable V, Ep[V]
=
minqeC(p) Eq[Y). Then, given any E and 6> 0, there is ko such that for ali kgreater than or equal to ko:
P(there is l ~ with ISk - Sk+ll
>
€) < 8.[There is a sharp difference between this law of large numbers and the preceding one. In the first case, it is true that P(ISk - ~+ll > €)
<
6 for any i~. However, the probability thatsome
of the successive Sk+1, Sk+2, ... differs from Sk more than E may be very large, while in the second result it is assured thatthis does oot occur. Of course the second result is stronger than the first.] Proof. Analogous to the proof of the theorem above.
QED. RecaI! that these two laws show that there is zero probability of non convergence. With non-additive probabilities this is different from the fact that
there should be convergence with probability 1. It may even be that convergence also occurs with probability zero. Theorem 5 in the text shows that convergence may occur with probability 1, for a particular class of exchangeable random variables.
The two laws of large numbers above assert that the successive averages get closer and closer the longer the sequence of trials. However, it does not say whether the averages themselves approach something. As it turns out, we may use the same arguments of de Finetti(1937) and characterize the distribution <1>(;)
=
limk_ao qP(Sks;)
(other versions of the results on exchangeable random variables can be seen in Feller(1966) and Loêve(1963». The point is that there is no meaning in this distribution, since the primitive uncertainty is P. In fact, in general the limit of Sk as k increases may be even a random variable with non-additive distribution. We turn now to add independence in the hope of understanding the process a little more.There are two competing definitions of independence for convex non-additive probabilities.
Deflnltlon (Gilboa and Schmeidler(1989» Let P1 and P2 be two convex
non-additive probabilities on (C1. A1) and (C2. A2). respectively (the definition is triviallyextended to more than 2). If the cores of P1 and P2 are C(P1) and C(P2), Schmeidler and Gilboa(1989) define the product non-additive probability
P1®P2 (which corresponds to their definition of the independent combination of P1 and P2). as the non-additive probability defined on (C1XC2. A1®A2) whose core is: C(P1®P2)
=
CO({q1®Q2, where q1 E C(P1) and Q2 E C(P2)}), being co(T) the closed convex hull of the set T.O8flnltlon (Hendon, Jacobsen. Sloth and Tran~s(1993» Let the notation be
as above. Hendon, Jacobsen, Sloth and Tran~s(1993) define the minimal convex product non-additive probability P 1~P 2 (which corresponds to their definition of the independent combination of P1 and P2), as the non-additive
probability whose core is: C(P1~P2)
=
{q additive defined on (C1XC2, A1®~). such that for ali E1 E A1 and E2 E A2, it follows that q(E1XE2) 2: P1(E1)P2(Ev}.Clearty the core of this second definition contains the core of the first, which means that for ali E E A1®~. P1~2(E) S P1®P2(E).
Lemma
3
Let P be an exchangeable convex non-additive probability defined on (go, An) (n possibly infinite, O finite), and C(P) its convex core. Assume that P is a product non-additive probability, of Pj, i=1, .", n, according to any of the two definitions above (that is to say, either P=P1®P2®",®P n orP=P1~P~",~Pn). Then Pj = Pj for ali i and j.
Proof. For any i=1, .", n, and for any event E in the i-th copy of O, define the event Ei = Ox···xQxExQ,,·xO , where the set E is exactly in the i-th position of the n-fold cartesian product. Sy definition Ej is an element of the product algebra
A®A®"'®A (n times repeated) = An. Consider first Schmeidler-Gilboa's definition. To start with, take n finite. P1®p2®··-@Pn(Ej) = minQEC(P1®P2® ... ®Pn) q(Ox···xOxExQ···xO). Sut C(P1®P2®"'®Pn) is the closed convex hull of the
independent combinations of the elements of the core of each Pi (C(Pj». Thus, q(.) is a linear combination of a finite number of independent additive distributions in (Qn, An), being each of the marginais on the i-th copy of O an element of C(Pj). This is true because n and O are finite. Then the marginal of q(.) on the i-th copy of O is a finite linear combination of elements of C(Pj). Sut since C(Pj) is convex by assumption (ali our non-additive probabilities are convex with convex cores), it follows that there is qj E C(Pj) such that q(Qx···xOxExO···xO) = Qi(E), for ali E in A. Conversely, if qj E C(Pj) we can obviously define an element in C(P1®P2®"-@Pn) with the marginal coinciding with qj. Hence, p 1®P2®··-@Pn(Ej) = mirlqiEC(pj) qj(E) = Pj(E). Sy the definition of exchangeability, P1@P2@···@Pn(Ej) = P1@P2@···@Pn(Ei), for ali i and j.
Therefore, for ali i and j, Pj(E) = Pj(E). If n is infinite, it is enough to consider finite marginais of increasing length of the product non-additive probability, and repeat the argument above. Now, take the definition of Hendon-Jacobsen-Sloth-TranéBs. P1~p~··~Pn(Ej) = minqeC(P1~ ... ~n) q(Ox···xQxExO···xQ).
But
by the definition of C(P1~~'~n):q(Qx"'xOxExQ"'xQ) ~ P1(Q)···Pj-1{0)Pj(E)Pj+1{Q)···PnLQ) = Pj(E). Thus, the marginal of q on the i-th copy of Q is an element of C(Pj). Conversely, consider qj E C(Pj). As before, it is immediate to define an element of C(P1~~'~n)
with the marginal coinciding with Qi. Hence, the same argument as above holds, and exchangeability assures that the non-additive distributions Pj and Pj are equal for ali i and j.
The lemma above shows that the interpretation of the exchangeability hypothesis in the non-additive case is similar to the additive case. In a certain sense. exchangeability is the generalization of the assumption of iid distributions to allow for correlation. We may now analyse the behaviour of sample averages of P=Q®Q®··· ad infinitum (or P~~")o
Theorem 3 (Strong Law of Large Numbers for Independent and Identlcally Dlstrlbuted Random Varlables - Independence HJST) Let {Xili=l. 2. 0 0 0 be an infinite sequence of bounded random variables in the space
(Q. A)o Let Sk be the average of the k first random variables. Assume that the random variables are independent (according to the definition of Hendon-Jacobsen-Sloth-TranéBs) and exchangeable for P=~Q~··· in (Qoo. AOO). and that P is convexo with a closed convex core (C(P». and such that for any bounded random variableY. Ep[Y] = mirlqec(p) Eq[Y]. Then. the probability P that the limit points of the sequence {Sk} are outside the closed interval lEQ(Xl). -Ea(-Xl)] is zero.
Proof. It is enough to consider any probability in the core of the form q = p®p®-.. (p defined on (Q.
A»,
and apply the usual strong law of larga numbers.QED. As it was mentioned before. this result is interesting. but it allows a lot of freedom. because it may be that the P-probability that the accumulation points of the sample means {Sk} Iye in [EQ{X1), -EQ(-X1)] is also zero. We are able to prove a stronger result. for the definition of independence of Gilboa-Schmeidler. We do not know whether this stronger theorem is true for the definition of independence of Hendon-Jacobsen-Sloth-Tranas.
Theorem 4 (Strong Law of Large Numbers for Independent and Identlcally Dlstrlbuted Random Varlables - Independence GS) Let {XilI=1, 2, ... be an infinite sequence of bounded random variables in the space (Q, A). Let Sk be the average ofthe k first random variables. Assume that the random variables are independent (according to the definition of Gilboa-Schmeidler) and exchangeable for P::Q®Q®··· in (QaI, AaI), and that P is convex, with a closed convex core (C{P», and such that for any bounded random variable Y, Ep[Y] = mi~p) EqM. Then:
(i) the probability P that the accumulation points of the sequence {Sk} are in the closed interval [EQ(X1), -EQ(-X1)] is one;
(ii) the probability P that the accumulation points of the sequence {Sk} are in any set T strictly contained in [Ea(X1), -EQ(-X1)] is zero, which means that this closed i nte rva I is the smaller set with this property.
Proof. For the definition of independence of Gilboa-Schmeidler, it turns out that it is simpler to prove that a set has P-probability 1. If we can prove that for ali elements of the core C(P) which are independent, i. e., are of the form q1®Q2®··· with q E C(O), it is true that the probability of the accumulation points lying in the set [EQ(X1), -Ea(-X1)] is 1, then the probability under any other of the elements of the core is also 1, since they are convex combinations of the independent distributions. Hence, let us restrict attention to proving this property for the independent elements of the core, and we are done. Consider a generic independent element of the core, q=q1®Q2®-·· with qi E C(O) for ali i. The random variables {XJi:1, 2, ... are fiXed, and uncJer q they are independent. Then, Yi = Xi - EqiXi form a sequence of random variables such that E[Y k I Y 1, Y 2, ... , Y k-1] = O under q, for all~. Define Ik to be the sum of the first k random variables Vi. Then, by Feller(1966, pp. 238, Thm. 2), if it is true that the infinite sum ~,(1rj2)·EqilYi2] is finite, then with q-probability 1: ~Ik tends to
O. But the infinite sum of the variances is finite, because the random variables are bounded, and I~,(1rj2) is finite. Therefore, we have that the set for which ~Ik tends to O has probability 1 according to q. If we recall the definiiton of Vi, Iklk = Sk - (11k)·II=', ... ,k Eqi[YJ it is immediate to check that the set .of accumulation points of the sample averages (Sk) is contained in the closed interval lEa(X,), -Eo(-X,)], because we have that Eo(X1) = minqeC(Q) Eq[X,], and -Eo(-Xd = m8Xqec(Q) Eq[X,]. In olher words, we proved that the set of states of lhe world for which lhe accumulation points of the sequence {Sklb, are in the closed interval above is 1 for any independent distribution on the core of P. Thus, by the argument at the beginning of the proof we proved (i). To
see
that (ii) is true also, we have to notice that if there is a set T C [EQ(X,), -EQ(-X,)] which is different from the interval, then there is some q'EC(Q) with Eq,X,E[EQ(X,), -EQ(-X1)] and Eq,X,~T. Thus, by considering the probability q' = ql®ql® ... and applying the usual strong law of large numbers, T has q'-probability zero, and hence P-probability zero, since q'EC(P).The fact that we cannot prove a stronger result should not come as a surprise, under the weak hypotheses of theorems 3 and 4. Think aOOut the following two experiments. Experiment number 1 is a repeated draw with replacement of a ball from the same urn with unknown number of red and blue balls. Experiment number 2 is a repeated draw of a ball from urns which only have red and blue balls. The difference is that in the second experiment there is a different urn in each stage. If the total number of the balls is the same in ali the urns, say 100, and there is no other information available, both experiments could be modelled, in principie, by means of repeated realizations of a random variable (say it is O if the ball is red and it is 1 if the ball is red) with a non-additive probability distribution.
Let us try to distinguish them in probability terms. In the first case the urn is fixed, 50 that one knows that there is a "true" (possibly interpreted by some as
"objective") probability distribution behind the event of drawing a ball, although this "true" probability distribution is not known. The second experiment does not have such a probability. We argue that the second experiment is suitably formalized by the independence of the non-additive iid probabilities, as in theorems 3 and 4. However, the way to model independence in order to capture the intuition of the first experiment is through a non-additive probability whose core contains only (additive) iid probabilities, and that is what we do in our last result below. Note that distributions in this category are not independent according to the definitions Gilboa-Schmeidler or Hendon-Jacobsen-Sloth-Tranaes, but are exchangeable.
Theorem 5 (Strong Law of Large Numbers - Core of the Dlstrlbutlon Formed Only by 110 Probabllltles) Let {XJt=1, 2, ... be an infinite sequence of
bounded random variables in the space (C, A). Let Sk be the average of the k first random variables. Assume that the distribution Q in
(ao,
A«») of the random variables is convex, with a closed convex core (C(Q» consists only of additive probabilities of the form q=
p®p® ... (p defined on (C, A), and pEC - that is to say that C is the projection of C(Q) in any of the copies of (C, A)), and such that for any bounded random variable Y, Eo[Y] = mirlqec(o) Eq[Y]. Then:(i) the probability Q that the limit points of the sequence {Sk} are outside the cIosed interval [Ea(Xü, -EO(-X1)] is zero;
(iia) O(Upee Cp) = 1 ;
(iib) if wECp, then limk_ao Sk = EpX, (which implies that the O-probability of convergence and, furthermore, of convergence to some value in [Ea(X,), -Ea(-X,)] is one);
(iii) any set strictly contained in the .interval [Eo(X,), -Eo(-X,>1 has O-probability zero of being the set of accumulation points of the sequence {Sk}.
Proof. Sy the strong law of large numbers, the set Cp for which there is convergence of Sk to EpX" when k goes to infinite is of q-probability 1, where q=p®p®.... That is, q(Qp)
=
1. Sut O(Up'EC Cp') = mir1qec(O) q(Up'EC Qp'), and we have that q(Up'EC Qp,) ~ q(Cp)=
1 for ali q in C(O). Thus, O(Up'EC Cp')=
1. Hence, (i), (iia) and (iib) are proven. To prove (iii) we have to notice that if there is a set Tc
[Eo(X,), -Eo(-X,)] which is different from the interval, then there is some p'EC with Ep,X,E[Ea(X,), -Eo(-X,)] and Ep,X,~T. Thus, by considering the probability q' = p'®p'® ... and applying the usual strong law of large numbers, T has q'-probability zero, and hence O-probability zero, since q'EC(O).QED. Here we see that there is room to "Iearn" what is the ''true'' distribution. If the "true" distribution is p*, then the likelihood that the realized averages converge to Ep.X, is 1 (under p*), and that is what the data will show the observers.
Consider the case of the first experiment, and interpret the random variable Xi as 1 if a red ball is drawn in lhe i-th draw, and zero if the ball is blue. Notice that, for ali i, we have that the expected value of Xi is given by Eo[XJ
=
Eo[X,] = mil'lpec Ep[X,], so that when there is a draw from the urn we would evaluate the likelihood of a red ball occurring according to the non-additive probability on ('l, A) whose core is C. As soon as we have a history of draws, we are able to compute Sk for ~1 and we may use this knowledge to update our expectation. For the updating rule of Dempster-Shafer (see Gilboa and Schmeidler(1993) for axiomatizations of updating rules, and Gilboa and Schmeidler(1992a, 1992b) for more discussion about them) , we are able to prove the following theorem. Recall that Ea'(Xi) = 0'(XF1) for any possible i and OI non-additive, due to the definition of the Choquet integration, and the fact that Xi can only attain values O and 1. Also, if O' is a generic non-additive probabilityand B a set such that Q'(BC) <1, then the Dempster-Shafer updating rule is given by: Q'(AIB) = (Q'(AUBC)-Q'(BC» / (1-Q'(BC».
Theorem 6 (Updatlng wlth Oempster-Shafer In the Case of Binomial Trlals, under the Hypotheses of Theorem 5) Consider the case that we have the observation of an event with two outcomes. Take the variable Xi as 1 if one cf the outcomes occurred, and O otherwise. [Think of the event of drawing repeatedly the ball from the um with fixed, but unknown probability distribution -experiment number 1.] Let {X;}i=1, 2, ... be the infinite sequence of trials (Xi means trial in stage i) in the case above, i. e. O with two outcomes, A = P (O) (that is, the power set of O). Let Sk be the average of the first k trials. Assume that the distribution Q in (oao, Aao) of the trials is convex, with a closed convex core (C(Q» consists only of additive probabilities of the form q = p®p®'" with p defined on (O, A) as the distribution which gives probability p to Xi = 1 and 1-p to X; = O, pE[p-, p-] - that is to say that the interval [p-, p-] is the projection of C(Q) in a.ny of the copies of (O, A). Then, if limk __ ao Sk = p*:
(i) p. E ]p-, p-[ ~ limk __ ao Ea[XI<+1ISk] = p.;
(ii) p. E [O, p-] ~ limk __ ao EaPCt<+1ISk] = p_;
(iii)p· E [p-, 1] ~ limk __ ao EaPCt<+1ISk] = p-.
Moreover, by combining this result with theorem 5, we see that: limk ... ao E[Xk+ 1 ISk] = limk ... ao
Stc
in a set of Q-probability 1.[This theorem is exactly about "Iearning" the "true" distribution with the successive observation cf the trials.]
Proof. The conditional expected value above is taken with respect to the Dempster-Shafer updated probability. This means that Ea[Xk+1ISk] = Q[Xtt+1=1ISkl Thus,
we
compute in general Q[Xk+1=1ISk=rlk], and take the limit as k goes to infinity and rJk to p •. Making the calculations, we have:The behaviour of the function f(P) = pr(1-p)s is the following: at O and 1 it attains its minimum, which is O. There is {)nly one maximum, attained at p:r/(r+s), with value f(r/(r+s» = [r/(r+s)]r. [s/(r+s)]s. For values cf p less than this number, it is strictly increasing, and for largar values, strictly decreasing. Thus, in case (i) Um rJk = p. E ]p_, p-[, which means that for large enough k we have that both
maxima in expression (*) above are attained in the corresponding value of r/(r+s). This gives us:
Taking the limit when k goes to infinity and k-r also so that rlk tends to p., the right hand side of (**) tend to:
limk_ao, rlk-p· Q[Xk+1=1ISk=rlk] = 1 - (lIe)·e·(1-p*) = p*.
This shows (i). The other two cases are similar, but easier. The final remark on the statement of the theorem is immediate, given Theorem 5.
QED. In fact, we conjecture that the following result is true: for any sequence of random variables {XVi=1, 2, ... in the hypotheses of Theorem 5 it must be the case that limk_ao Ea[Xk+1ISkl = limk_ao Sk in a set of Q-probability 1. On the other hand, Theorem 6 above is not true if one replaces Dempster-Shafer rule by the analogue of Bayes rule, i. e. by the updating rule defined by Q'(AIB) = Q'(AnB)/Q'(B). An interesting problem is to find the set of (fixed) updating rules (see Gilboa and Schmeidler(1993) for a detailed description of the set of updating rules) for which Theorem 6 would hold.
References
Dempster, Arthur (1967), "Upper and Lower Probabilities Induced by a Multivalued Map" , Anna/s of Mathematica/ Statistics 38: 325-339.
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ENSAIOS ECONÔMICOS DA EPGE
100. JUROS, PREÇOS E DívIDA PúBUCA - VOL I: ASPECi'OS TEÓRICOS - Março Antonio C. M.tiDI, Clovil de Faro -1987 (ngotado)
101. JUROS, PREÇOS E DívIDA PúBUCA - VOL
n:
A ECONOMIA BRASILEIRA - 1971185-Amcmio Sal . . . P. BrmdIo, MaRO Antonio C. M.-timI e Clovil de P.-o -1987 (e..,e.do)
102. MACROECONOMIA KALECKIANA - RubeoI Penha Cyme -1987 (esgotado)
103. O PREÇO DO DÓLAR NO MERCADO PARALELO, O SUBFATURAMENTO DE EXPORTAÇÕES E O SUBPATORAMENTO DE IMPORTAÇÕES - PenJIIIdo de HollOda B.-boIB, RubeoI PeohaCyme e Marcos Co1taHollDIa-1987 (elgotado)
104. BRASIIJAN EXPERIENCE WlTH EXTERNAL DEBT AND PROSPECTS POR GR.OWI'H -FemuIo de HollDda:s.bola aod MBIlUtl Saoçhel de La Cal - 1987 ("lOtado)
105. XEYNES NA SEDIÇÃO DA ESCOLHA PúBuCA - Antonio Mma da Simira - 1987 (
...
)106. O TEOREMA DE FROBENIUS-PERRON - CarlolMn SimOllleD Leal- 1987 (esgotado) 107. POPUlAçÃO BRASILEIRA -Jeaé MeDeio -1987 (tII*do)
108. MACROECONOMIA -
cAPlTOLo
VI: "DEMANDA POR MOFJ)A E A CURVA LM" -Mmo Hlllrique SÚDoDIeIl' Rw-Peaha C,... - 1987 (.II8*do)109. MACROECONOMIA -
cAP1TuLo vn:
"DEMANDA AGREGADA E A CURVAIS" -
Mano Hemique SimoDBeD e RubeDI Peoba Cyme - 1987 (eagotado)110. MACROECONOMIA MODELOS DE EQUlÚBlUO AGREGATlVO A CURTO PRAZO -Mmo Hearique S~. RabnI Peaba
c,... -
1987 ( • ..,..,)111. mE BAYESlAN POUNDA'I1ONS OP SOLU'I1ONS CONCEPTS OP ~ - sqio Ribeiro daCOIta W .. l.,. e Toamy ChiD-aJiu TID -1987 (tII*do)
112. PREÇOS ÚQUIDOS (PREÇOS DE VALOR ADICIONADO) E SEUS DETF..RMlNANTBS; DE PRODtrroS SFl.RCIONADOS, NO PPlÚODO 1980110 SEMESTR.FJ1986 - bal EbaJDai
-1987(",,-,)
113.
EMPRÉSTIMos BANcÁRIos
E SALDO-MÉDIO: O CASO DE PB.PSI'AÇÕES - Clovil de P..., - 1988 (....-.)114. A DlNÂMICA DA lNFLAçÃO - Mllrio Hearique SimoaIeo - 1988 ( • ..,-10)
11'. UNCERTA1NTY A VERSIONS AND THE OPTMAL CHOlSB OP PORTFOUO -JIIII9I Dow e sqão Ribeiro da
coa
Werl.,. - 1998 (tII*do)117. FOREIGN CAPrI'AL ANO ECONOMIC
GR.owm -
THE BR.ASILlAN CASE mIDY - Mmo H...-ique SÍIDOaIen - 1988 ( . . . )118. COMMON KNOWLEDGE - SQio Ribeiro da COIbI Werllq - 1988 (.qutado)
119. OS FUNDAMENTOS DA ANÁLISE MACROECONÔMICA - Mario Henrique SimoDBen e
Rubens Penha Cyme -1988 (esgotado)
I
120.
cAPtroLo xn -
EXPECTA'nVAS RACIONAIS - Mmo Hfllrique SimcmIen -1988 (ngotado) 121. A OFERTA AGREGADA E O MERCADO DE TRABAIJlO - Mario Henrique SimODleD eRubeoB Peoba Cyaoe -1988 (esgotado)
122. lNÉR.CIA INFLACIONÁRIA E INFLAçAO INERCIAL - Mario Heorique SimoDIen - 1988 (el80tad0)
123. MODELOS DO HOMEM: ECONOMIA E ADMINlSTRAçAO - Antonio Maria da Silvein. -1988 ( esgotado)
124. UNDERlNVOlNClNG OF EXPORTS, OVERlNVOlNCING OF IMPORTS. AND mE DOUAR. PREMIUN ON mE BLACK MABXET - Fen.odo de HollDIa ~ RubeaB P ' " Cyme •
MaROB CostaHollDIa-l988 ( • .,..,)
125. O R.ElNO MAGICO DO CHOQUE HETEllODOXO -Femaodo de HolECla BarboA,. ADtoDio Sal .... Peaoa BnmdIo • Clovi. de F .. o - 1988 ( . . . )
126. PLANO CRUZADO: CONCEPÇÃO E O ERR.O DE POÚ'l1CA FISCAL -Rub90s PtDba Cym9 -1988 (ngcàdo)
127. TAXA DE JUROS FLt1'fUANI'E VERSUS CORREÇÃO MONET.ÁRIA DAS PRESTAÇOES: UMA COMPARAÇÃO NO CASO DO SAC E INFLAÇAO CONSTANTE - Clovi. de F.o -1988 ("lotado)
128.
cAPtroLo n -
MONETAR.Y C0RREC'l10N AND REAL INTEREST ACCOUNTINO -R.ubeoI Peoba Cyme - 1988 ( . . . )129.
cAPtroLo m
-lNCOME ANO DBMAND POUCIES IN BR.AZlL -Rubeo8 Peoba Cyme - 1988 (8II80tad0 )130.
cAPtroLo
IV - BRAZILIAN ECONOMY IN 11m EIOH'l'l&1 ANDmE
DEBT CRISIS -R-.. PeabaCyme - 1988 ( . . . )131. THE BRAZILIAN AGRICULTUR.AL POUCY EXPERJENCE: RATlONALE ANO FtmJRE DlR.EC'l10NS - ADtoaio s.l..-P . . . . Bnlldlo - 1988 ( . . . )
132. MORATÓRIA lNTERNA, DÍVIDA PÚBUCA E JUROS REAIS - Mma Silvia B.toI M.-ques e SéQPo Ribeiro da Coa Werlq -1988 ( • ...,..,)
133.
cAPtroLo
IX - TEORIA DO ~ ECONOMICO Mmo Hemique SimoaIen -1988( • .,-10)134. CONGELAMENTO COM ABONO SALARIAL GERANDO EXCFSSO DE DEMANDA -JOiIlp,j ... Vi.iraFwniraLny. Sérsio Ribeiro daCOIta Werl ... -1988 (.",.-10)
13'. AS OlUGP.NS E CONSEQu!NCIAS DA INFLAçÃO NA AMÉRICA LATINA - Feruado de Holaoda Bnoa - 1988 (ngotado)
136. A CONTA-CORRENTE DO GOVERNO - 1970/1988 - Mario Heurique SimonseD - 1989 (
...
)137. A
REVIEW
ON THE THEORY OF COMMOW KNOWLEDGE - SérJio Ribeiro da Costa Werlllll- 1989 (ngotado)U8. MACROECONOMIA - Femaado de HolaodaB.-boIa-1989 (.agotado)
139. TEORIA DO BALANÇO DE PAGAMENTOS: UMA ABORDAGEM SIMPLIFICADA - Joio Luiz Teoreiro Barrolo - 1989 (esgotado)
140. CONTABILIDADE COM JUROS REAIS -Rubens PeobaCyane - 1989 (elgotado)
141. CREDrr RATIONlNG ANO mE PERMANENT lNCOME HYPOTHESIS -Viçtde Madrigal. Toaay T..,
o.u.1
Vi~ Séqio Ribeiro da COIta W ... l ... - 1989 (....-)142. AAMAZÔNIABRASlLEIRA - Ney Coe de Oliveira-l989 (eqotado)
143. DWGIO DAS LFI'B E A PROBABJIIDADE IMPúcrrA DE MORATÓRIA - Maria Silvia BaItoa M .. quea e SérJio Ribeiro daCoIta Werl ... -1989 (eqotado)
144. mE IDC DEBT PROBLEM: A ~nmoRETlCAL ANALISYS - Mmo H...-ique SimoaRa e sqio Ribeiro da Costa Werlllll-1989 (wgcAIIdo)
14'.
ANALIsE
CONVEXA NORn -
Mario Hearique SimoaIen - 1989 (esgotado)146. A CONTROVÉRSIA MONETARISTA NO HEMISFÉRIo NORTE - F ... de Holaoda B.boa-1989( . . . )
147. FISCAL REFORM AND STABn.JZATION: THE BRAZIUAN EXPERlENCE - Fermmdo d. HolllldaBlrboa. Amaaio Sal..-PtIIOaBlutlo • Clovil de F.-o -1989 (wgcAIIdo)
148. RErORNOS EM FJ)UCAÇÃO NO BR.ASlL: 1976/1986 - c..lOl Ivan SimoaIen Leal • Sérgio Ribeiro da Costa Werl .... - 1989 ( • ..,..,)
149. PREPERENCES, COMMON ICNOWLEDGE AND SPECULA'nVE 'fRADE - Jamel Dow, VictID MadripI e sqio Ribeiro da Costa Werlllll-l990 ('lIot1do)
150. FJ)UCAçAO E DlSTRlBtnçAO DE RENDA - CldOlIvla SimoaIeD Leal • S4qio Ribeiro da Coa Werl ... - 1990 (....-)
151. OBSERVAÇOES A MARGEM DO TRABAlBO"A AMAZÔNIA BRASILEIRA" - Ney Coe de Olivein.-I990 (wgcAIIdo)
152. PLANO COlLOR: UM GOUE DE MESTRE CONTRA A INFLAÇÃO? - F.umdo de Hol.ada kbOIa -1990 ( . . . )
153. O EfíHI'O DA TAXA DE JUROS E DA INCERTEZA SOBRE A CURVA DE PHILLIPS DA ECONOMIA BllASlLBIRA -Rinrdo de OIMira Cawlnmi - 1990 ( •• .,....,)
1.54. PLANO COlLOR: CONTRA A PAcnJALIDADE E SUGESTOES SOBRE A CONDUçÃO DA POlÍTICA MONETÁlUA-FISCAL - RubeaI Ptaba Cyme - 1990 (naotado)
155. DEPOmoS DO TESOURO: NO BANCO CENTRAL OU NOS BANCOS COMERCIAIS?
-a-..
Peaba Cyme -1990 ( . . . . )156. SISTEMA F1NANCElRO DE HABrrAçÃO; A QUESTÃO DO DESEQUlÚBRIO DO FCVS -Clovi. de FII"O -1990 (naotado)
157. COMPLEMENTO DO FAScíCULO ND 151 DOS "ENSAIOS ECONÔMICOS" (A AMAZÔNIA . BllASlLBIRA) - Ney Coe de Olmira - 1990 ( . . . )
158. POlÍTICA MONETARIA
óTIMA
NO COMBATE A INFLAçÃO - Permudo de HolllldaBno. -
1990 (esgotado)159. TEORIA DOS JOGOS - CONCFJTOS
BÁSIcos -
Mmo Hearique SÍIDOIIIeD - 1990 (eQObIdo)160. O MERCADO ABERTO BRASILFJRO; ANÁLISE DOS PROCFJ>lMENTOS OPERACIONAIS - Pmmado de HolmdaB.boa- 1990 ( •• geada)
1eSt. A RELAçÃO ARBITRAGEM ENTRE A ORTN CAMBIAL E A ORTN MONETÁlUA -Luiz Guilherme Schyaaade Oliveira-1m ( . . . )
162. SUBADDmvB PROBABlIll'JES ANO POll'l'POUO JNEll11A -Mmo H...-iqae
su..o...a.
SérJio Ribeiro da CoIIa W ... - 1990 (npelldo)163. MACROECONOMIA COM M4 - CIliOlIY1ll SimoDBeD Leal , SQo Ribeiro da Colta WerllDl - 1990 ( . p _ )
164. A RE-EXAMlNA110N OP SOLOW'S GROWl1l MODEL
wrm
APPUCA110NS TO CAPll'ALMOVEMENTS -N...av SavedraRmmo -1990 (.iijObIdu)165.
mE
PUBUC CHOICE smmON: VAlUA110NS ONmE
'I'BEME OP SCIENTIPICWARPARE -
Amaaio Mma daSiJven - 1990 (...,. . . )166. THE PUBUC CHOPlCE PERSPBCl1VE ANO ICNIGRTS lNS'Itl'O'l10NALJST BENT -ADlaaio Mma da Silwira - 1990 (naotado)
167.
mE
00>EI'ER.MlNA110N OP SENtOR - AlDaio Mma da Silveira - 1990 (,...ado)168. JAPANESE DIRECT
JNVESTMENT
IN BRAZIL - NeIdr'o s.v.draRiVlllO -1990 (e . . . ) 169. A CAllT.EIRA DE AçOES DA CORRETORA; UMA ANÁUSE ECONÔMICA - LuizGül'-me Sc:h,....ade 0Iiveira-1991 ( . . . )
170. PLANO COlLOR: OS PIUMPJROS NOVE MESES - Clovil de p." - 1991 (,...ado) 171. . PBR.CALÇOS DA OO>EXAÇÃO EX-ANTE - Clovi. de PII"O - 1991 (naotado)
173. A DINAMICA DA HIPElUNFLACAo - FII'IIIIIdo de Ho'-l& ~ Waldyr Muaiz Oliva e BIvi.M1nb S.1111111 - 1991 ( . . . )
174. LOCAL CONCA VJPIABlUTY OP PREFERENCES ANO DE'TERMINACY OF EQUIIJBlUUM - Mmo Rui Púçoa e Sérgio Ribeiro da Costa Werlaag - maio de 1991 (ngatado)
175. A CONTABILII)ADE DOS AGREGADOS MONETÁRIOS NO BRASIL - C .. lo.lvao Simoosen LMI. Sérwio Ribeiro daCOIta Werl . . -maio de 1991 (."IMa)
176. HOM01lIE'l1C PREFERENCES - J ... Dow e Sqio Ribeiro da COIta Werl.,. - 1991 (
.
.,..,)
177. BARREIRAS A ENTRADA NAS INDÚSTRIAS: O PAPEL DA FIRMA PIONEIRA - Luiz Guilherme SdJymurade 0Iiv.ira-1991 < . . . . )
178. POUPANÇA E CRESCIMENTO ECONÔMICO - CASO BRASILEIllO - Mlrio Heorique SimoaBen - agosto 1991 (eqotldo)
179. EXCESS VOLATIUI'Y OF STOCK PRICES ANO KNIGHI1AN UNCERTAJNTY -J ... Dow .SéqioRibeirodaCoataW ... -1991 < . . . . >
180. BRAZIL -CONDmONS FORRECOVER.Y --Mario H...-ique SÍIDODIm - 1991 (ngatado) 181. mE BRAmJAN EXPElUENCE wrrH ECONOMY POUCY REFORMS AND PROSPECTS
FOR THE FtITURE -Femaado de Hol"" BIrboIa -Dezembro de 1991 ( . . . . )
182. MACR.OD1NÂMICA: OS SlSTF1dAS DINÂMICOS NA MACR.OECONOMIA - Fenumdo de BolllldaBlnoA- DeadHo de 1991 ( • ....,)
183. A EFICI2NClA DA INTF..R.VENçAO DO PSrADO NA ECONOMIA - FerJIIDdo de Hol BIrboD -Dezembro de 1991 (...,)
184. ASPECI'OS ECONÔMICOS DAS EMPRESAS ESTATAIS NO BRASIL: TELECOMUNICAÇÕES, ELETRICIDADE - PtrDECIo de Bo."" BII'bo" M_I Jeremi. Leite CaI_ Mmo Jaqe PiDae H6lio IMIIapArteiro -Dezembao de 1991 ( . . . )
185. THE EX-ANTE NON-OPTJMAUrY OF THE DEMPS'I'ER.-SCHAFEll UPDATJNG RULE FOR AMBIGUOUS
BRIlEPS -
SWsio Ribeiro da CoMaWm . . .
J ... Dow - F..wniro de 1992( . . . . )186. NASH EQUIIJBlUUM UNDEll KNIGHI1AN UNCEllTAJNTY: BRBAJaNG DOWN BACKWARD INDUcnON - J ... Dow e Séqio Ribeiro da COIta WerI ... - Fevweiro de 1992{""")
187. REFORMA DO SISTEMA FINANCEIRO NO BRASIL E "CENTRAL BANlClNO" NA ALFl4ANHA E NA ÁUSTRIA -R-.. PeIà Cyme -Fevereiro de 1992 (...,)
181. A lNDE'TERMINACAO DE SENIOR: ENSAIOS NORMATIVOS - AIâoaio MIIi. da Silveira-M.-çode 1992 ( . . . )
190. HIPERlNFLAçÃO E O
REGIME
DAS POLtnCAS MONETÁRIA-FISCAL - Fernando de Hob .. da B.-bosa e Elvia Mureb Sallum - Março de 1992 (esgotado)191. ACONSTITUIÇÃO. OS JUROS E A ECONOMIA - Clovis de Faro - Abril de 1992 (esgotado) 192. APUCABILIDADE DE TEORIAS: MICROECONOMIA E ESTRATÉGIA EMPRESARIAL
-193. 194.
195. 196.
197.
Antonio Maria da Silveira - Maio de 1992 (esgotado)
INFlAçÃO E CIDADANIA - Fernando de Holanda BarboBa - Julho de 1992 ! A INDEXAÇÃO DOS ATIVOS FINANCEIROS: A EXPERIÊNCIA BRASILEIRA - Fernandol
I
de Holanda Barbosa - Agosto de 1992
i
A INFlAçÃO E CREDmllJDADE - Sérgio Ribeiro da Costa Werhmg - Agosto de 1992 'I: A RESPOSTA JAPONESA AOS CHOQUES DE OFERTA 1973/1981 - Fmumdo Antonio
I
Hadba - Agosto de 1992 I
(JM MODELO GERAL DE NEGOClAÇAO EM UM MERCADO DE CAPITAIS EM
Qt~
NAO,' EXISTEM ThlVESTIDORES "IRRACIONAIS - Luiz Guilhmne Schymura de Oliveira - Setembro"de 1992
I
i
1198. SISTEMA FINANCEIRO DE H."\BITAÇAO: A NECESSIDADE DE REFORMA - Clovis dei
Faro - Setembro de 1992 •
I
BRASIL: BASES PARA A RETOMADÀ DE DESENVOLVJMm.i"TO -RubImS Penha Cysne
-I
I 199. 200. 201. 202. 203. 204. 20S. Outubro de 1992A VISÃO TEÓRICA SOBRE MODELOS PREVIDENCIÁRIOS: O CASO BRASILEIRO - Luiz Guilherme Sçhymura de Oliveira -Outubro de 1992
HIPER1NF'u\ÇAO:
cÂMBIo,
MOEDA E ANCORAS NOMINAIS - Fernando de Holanda,Barbosa - Novembro de 1992 - (esgotado)
I
PREVIDtNCIA SOCIAL: CIDADA.~ E PROVISÃO - Clovis de Faro - Novembro de 1992
!
OS BANCOS ESTADUAIS E O DESCONTROLE FISCAL: ALGUNS ASPECTOS - SérgioI
Ribeiro da Costa Werlaog e Armínio Frap Neto - Novembro de 1992 - (esgotado)i
TEORIAS ECONÔMICAS: A MEIA-VERDADE TEMPoRÁRIA -Antonio Maria da Silveira-I
Dezembro de 1992
t
THE RICARDIAN VICE ANO THE INDETERMINATION OF SEmOR - Antonio Maria da Silveira - Dezembro de 1992
206. HIPERlNFLAçÃO E A FORMA FUNCIONAL DA EQUAÇÃO DE DEMANDA DE MOEDA -Fenumelo de HolandaBarbon.- Janeiro de 1993
207 REFORMA FINANCEIRA - ASPECTOS GERAIS E ANÁLISE DO PROJETO DA LEI COMPLEMENTAR - Rubens Penha Cyme -fevereiro de 1993.
208. ABUSO ECONÔMICO E O CASO DA LEI 8.002 - Luiz Guilbmne Sc:hymura de Oliveira e Sérgio Ribeiro da Costa Werlaog - fnereiro de 1993.
109. Fl.RMRNTOS DE UMA FSTR.ATÊGlA PARA O DPSF.NVOLVJMENTO DA AGRlCULTUllA BRASILEIRA - AIâoaio Sal .... P..,.BnadIo e FJi .... Al'WII - Fneniro de 1993
210. PREVID~CIA SOCIAL PúBUCA: A
EXPERI2NCIA
BRASILEIRA - H'lio Portuca'MO dec.aro,
Luiz Guilherme Sdlyaaa de Oliwira, RtIIIto Praaelli Cnoso e Uriel de M ... ' ... -M .. ~de 1993.211. OS SISTEMAS PREVIDENCIÁRIOS E UMA PROPOSTA PARA A REFORMULACAO DO MODELO BRASILEIRO - Helio POI1uc;aiMO de
c.aro,
Luiz Guilherme Sdlyaaa de Oliv.D, ReDIIto Fraplli COOlO e Uriel de Map'hMs - Mlqo de 1993.111. 'lHE lNDETERMINATION OF SENIOR (OR 'lHE lNDEl'ERMJNATION OF WAGNER) ANO SCHMOII..F.R AS A SOCIAL ECONOMIST - ADtoaio M.-ia da Silveira - Marvo de 1993.
213. NASH EQUlLIBlUUM UNDER. KNIGHTIAN UNCERTAlNTY: BREAKlNG DOWN BACKW ARO lNDUC1'ION (ExteDlively Revi .. d Venion) - JIDe. Dow e Sço Ribeiro da Costa Werl8118 - Abril de 1993.
214. ON THE DIFFEREN'l1ABlLl OF THE CONSUMER. DEMAND PUNCl10N - Paulo Kliupr Monteiro, Mário Rui PéKoa e Sérgio Ribeiro da Coata
WerlIaa -
Maio de 1993.115. DEl'ERMJNAÇÃO DE PREÇOS DE A'I1VOS, AllBlIRAGEM, MERCADO A TERMO E MERCADO FtTI'UR.O - Séqio Ribeiro da Coa
WerlIos •
Flávio Auler -AaoIIto
de 1993. 216. SlS1'EMA MONErÁRIO VERSÃo REVISADA - Mlrio Hearique SimoaIeo e RubeIII PeobaCyaoe -
Aaosto
de 1993.117. CAIXAS DE cONVERSÃo - FenDIIdo Aldaio Hadba - Apto de 1993.
218. A ECONOMIA BRASILEIRA NO PElÚODO MILlTAR -RubeaB Peoba Cyme -AgoBto de 1993 219. IMPOSTO INFLACIONÁRIO E 'I'RANSFER.eNCIAS INFLACIONÁRIAS - Rubeos Peoba
Cyaoe -
Aaosto
de 1993.110. PREVISÕES DE Ml COM DADOS MENSAIS - R.m- Peaba Cyme e Joio Vietor Inler -Setembro de 1993.
221. TOPOLOGIA E cÁLCULO NO
Rn -
RuIMm Peaba Cyme , Humberto Moreira - Setewbi o de 1993.221.
EMPRÉSTIMos
DE MÉDIO E LONGO PRAZOS E JNFLAÇÃO: A QUESTÃO DA INDEXAÇÃO - Clovi. de P.-o -Outubto de 1993.223. ES'l'ODOS SOBRE A INDFATERMlNAçAO DE SENlOR, voL 1 - NellOll R B.rboA, PQ,io N.P. PreitaB, CmOl F.LR. Lapea, M8:C08 B. MOIáiro, Adaaio Mlria
Silveil1l(Coordeoador) e Mati88 V8I1IIIIJID -o.aào de 1993.
224. A SUBSTmJIçÃO DE MOEDA NO BB.ASlL: A MOEDA INDEXADA - Fea"" de Hol B.-boa e Pech Luiz Vali. Pereira - NO't'embro de 1993.
215. FlNANCIAL lNTEGRATION AND PUBUC FINANCIAL INS1TI'tJ'IlONS - Walter NOvae8 e Séqio Ribeiro da COIta Werl-S - Novembro de 1993.
216. LAWS OF LARGE NUMBERS FOR. NON-ADDmvE PR.OBABIUl'lES -Jamn Dow e Sqio Ribeiro da Costa Werlaog - Dezembro de 1993.
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