Efficient random variables

Journal of Mathematical

Economics 2 (1975) 243-252. 0 North-Holland

EFFICIENT

RANDOM

Publishing Company

VARIABLES*

Bezalel PELEG Instituteof Mathematics, The Hebrew University, Jerusalem, Israel

1. Introduction The theory of stochastic dominance was started in Hardy, Littlewood and Polya (1934). For a recent survey of its part which is relevant to economics the reader is referred to Schmeidler (1973). Independently of the existing mathematical theory the problem of comparing random variables with respect to their ‘riskiness’ was explored recently by several economists [see, e.g., Hadar et al. (1969), Hanoch et al. (1969) and Rothschild et al. (1970)]. The outcome of these investigations was, mainly, a precise definition of a partial ordering (‘more risky than . . .‘) on the set of distribution functions together with criteria for comparability in that ordering. Following the above investigations, M.E. Yaari and the present author started [Peleg et al. (1975)] the investigation of risk-aversely efficient random variables (see Definition 2). In that work Yaari and the author obtained a price characterization of efficient discrete random variables. In this paper we extend the investigation of efficient random variables to random variables in L,(S), where S = (T, B,p) is an arbitrary probability space. Our main result (Theorem 1) is as follows. Let R c L, be convex and w(L,, L,)compact [w(L, , L,) is the weak topology of the pairing (L, , I,,), i.e., the weak* topology of L,]. Call a point x E R regularly risk-aversely efficient if a strictly concave utility is maximized at x (see Definition 4). Theorem 1 asserts that the w(L, , &)-closure of the set of all regularly risk-aversely efficient members of R contains the set of all risk-aversely efficient members of R. This is an analogue of the Arrow-Barankin-Blackwell theorem [Arrow et al. (1953, Theorem l)] for our model. Using the above result, and assuming that T is finite, we obtain a price characterization of efficient random vectors (Theorem 2). Theorem 2 is a generalization to vector-valued random variables of Theorem B in Peleg et al. (1975). Furthermore, we show by giving a counter-example (Example 2) that our price characterization theorem (Theorem 2) does not hold when T is infinite. For possible applications of the theory of risk-aversely efficient random variables the reader is referred to Peleg et al. (1975). That paper also discusses *Presented at the Mathematical Social Science Board Colloquium nomics in August 1974 at the University of California, Berkeley.

on Mathematical

Eco-

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B. Peleg, Eficient random variables

the relationship between risk-aversely efficient points and efficient points. For a recent advanced study of efficient points the reader is referred to Majumdar (1972). Strong use is made of Rockafellar’s results on integrals of convex functions in Rockafellar (1971). 2. Preliminaries We denote by Em the m-dimensional Euclidean space. If x, y E Em then we writexh yifx,zy,,i= I,..., m.~>yisx~yandx#y.x~>ifx~>y, i= 1,...) m. The scalar product of two members x and y of E” is denoted by xey, x*y = f

xjyj.

i=l

A real function u on E” is non-decreasing (increasing) if a, b E Em and b 2 a (b > a) imply that u(b) 2 u(a) (u(b) > u(a)). Let u be a concave function defined Em. A vector b E E” is a subgradient of u at a if, u(x)-u(a)

5 b-(x-a)

for all

XE Em.

The set of all subgradients of u at a will be denoted h(a). Let S = (T, B,p) be a probability space and let x and y be two random variables defined on S and taking values in Em. We denote x 2 y if x(t) 2 y(t) a.s. (almost surely). x > y if x 2 y and pit/x(t) > y(t)} > 0. x % y if there exists k % 0 in Em such that x(t) 2 y(t)+k a.s. 3. The model Let S = (7’, B, p) be a probability space. We denote by L”,(S) the set of all essentially bounded measurable functions from T to the Euclidean space E”. Similarly, L’;(S) is the set of all integrable functions from T to Em. For x E L”,(S) and a E L’;(S),

44 = J, 40. xWp(O. From now on we write L”,(S) = L, and L?(S) = L,. Let R c L,. x E R is efficient if there exists no y E R such that y > x. Let x E R be efficient. a EL, , x > 0, is a system of eflciencyprices for x if x(x) 2 z(y)

for all y E R.

Let U be the set of all finite-valued non-decreasing and concave functions on

245

B. Peleg, EjJicient random variables

E”. Intuitively, each u E U may be thought of as a utility function of a risk-

averse decision-maker. For u E U and a random variable x E A,, we denote W4 De$nition 1.

(1)

= 5, u(x(t))d&).

Let x, y E R. We shall say that y dominates x risk-aversely if there exists u* E U such that

Iu(y) 2 Iu(x) for all u E U and, furthermore, Iu*(y) > Iu*(x).

DeJinition 2. x E R is said to be risk-aversely efficient if there exists no y E R that dominates x risk-aversely.

Clearly, a risk-aversely efficient random variable is efficient. Clearly, only risk-aversely efficient points could be chosen by some riskaverse decision-maker if given the set R to choose from. This naturally leads to the following question : Let x E R be risk-aversely efficient. Is there a risk-averter whose expected utility lu is ‘properly” maximized at x? The rest of the paper is devoted to an investigation that gives a complete answer to this question. Essentially, when R is assumed to be convex and w(L, , L,)-compact the answer is ‘yes’ if Tis finite (Theorem 2), and the answer is ‘no’whenTis infinite (Example2). However, the set of risk-aversely efficient points of R which are ‘proper’ maximizers of a functional Iu, u E U, is always a w(L,, L,)-dense subset of the set of all risk-aversely efficient points of R (Theorem 1). 4. Proper maximization

Let R c L, be convex. Definition 3. A functional Iu, u E U, is properly maximized at an eficient point x E R tfx has a system of eficiencyprices ICsuch that 71(t) E &(x(t))

a.s.

Remark 1. If Iu is properly maximized at x E R, then Iu is maximized at x, i.e., lu(x) 2 1u(y) for ally E R. Proof. Let A be a system of efficiency prices for x such that n(t) E &(x(t)) a.s. If y E R then

44

2 n(y),

and u(x(O) 1 u(M) + n(r). (x(t) -Y(O)

a.s.

‘We have to exclude trivial cases where Iu is constant in a (strong) neighborhood

of X.

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B. Peleg, Eficient random variables

Integrating the last inequality we obtain 124(x)2 lu(_v)+ l+c--y) 2 lu(y). We remark that if 1~ is properly maximized at X, then x maximizes A&) subject to the ‘budget constraint’ n(n) 2 n(y). We also remark that if 1~ is properly maximized at x then it is not constant in any strong neighborhood of X. For an increasing u E U maximization implies proper maximization. Lemma I. Let u E U be increasing. If Iu is maximized properly maximized at x.

on R at x then Iu is

Proof: Clearly, lu is a concave functional on L, which is continuous in the strong topology. By the results of Rockafellar (1967) there exists f e Lz, the dual ofL, , such that

for all y E R,

It now follows from (3) and Corollary 2C in Rockafellar (1971) thatfo E &4(x(t)) a.s. Since u is increasingf > 0. The following well-known result is a corollary.

(2)

L1 and

f(t)

Corollary 1. Let u E U be increasing. Iu(x) [see (l)] is upper semicontinuous in the topology w(L, , L,). 5. Density of regular risk-aversely efficient points

Let R c L, be convex. Dejinition 4. x E R is a regular risk-aversely there exists a strictly concave u E U such that

lu(x) 2 lu@) Remark 2.

for all y E R.

efficient random variable if

(4)

If x E R is regular risk-aversely efficient then x is risk-aversely

efficient. Proof. There exists a strictly concave u E U which satisfies (4). Let y E R, y # x. Since R is convex lu(x) > lu(y). Thus, y does not dominate x risk-

aversely.

B. Peleg, Efficient random variables

247

Remark 3. If x E R is regular risk-aversely efficient then there exists a strictly concave u E U such that lu is properly maximized at X. Proof. There exists a strictly concave u E U which satisfies (4). Since u is strictly concave and non-decreasing it is increasing. By Lemma 1, lu is properly maximized at X. Theorem 1. Let R c L, be convex and w(L,, L,)-compact. The w(L,, L,)closure of the set of all regular risk-aversely eficient points of R contains the set of all risk-aversely eficientpoints of R. Proof. There exist a, b E Em, b g a, such that for all x E R, a 5 x(t) 5 b a.s. Let c = a-(1,. . ., 1) and d = b+(l, . . ., 1). Denote Q={rlr~E~,

csrrdd),

V={ulu~U,

u(c)=0

and and

u(d)=l}.

Consider V as a subset of C(Q), the space of all continuous functions on Q. It is well-known that C(Q) is separable. Since C(Q) is a metric (topological) space and Vis a subset of it, Vis also separable. Let ~7, . . ., u:, . . . be a sequence of members of Vwhich is dense in V, and let u be a strictly concave member of V. Letr,, r2,. . . be the rational numbers in the (open) unit interval (0, 1). Form all possible pairs rju,*+ (1 -rJu, i, k = 1,2,3, . . . . They consist of a sequence of strictly concave members of V which is dense in V. Call this sequence ul, . . ., Uk, . . . . For each z E R let

Denote by s+ the non-negative cone of the space s of all real sequences. [The reader is referred to Peleg (1971, especially to Section 2) for our terminology and notation concerning the space s.] We detine y=

{YlY=+,

y 5 p(z)

for some

z E R}.

Now, R is convex and w(L,, L&compact and for k = 1, 2, . . ., Iuk(z) is concave and upper semicontinuous in the w(L, , L,) topology (see Corollary 1). Hence Y is a convex and compact subset of s+. Furthermore, y E Y is an efficient point of Y if and only if there exists a risk-aversely efficient random variable z E R such that y = q(z). Let now x E R be risk-aversely efficient. Then y* = V(X) is an efficient point

B. Peleg, Eficient random variables

248

of Y. By Theorem 6.1 in Peleg (1971) there exist a sequence of efficient points y(l)o Y9 1= 1, 2, * * *, and a sequence ,I(‘) of positive sequences in l,, such that lim y(‘)(k) = y*(k) ,

k = 1,2,

. . .,

(9

Z-rm

(6)

for all y E Y.

Since y(‘) is an efficient point of Y there exists a point z(l) E R such that Y(I) = cp(z”‘), I = 1, 2, . . . . R is w(L,, &)-compact. Hence the sequence z(‘) has a cluster point z. Let {I 1h E D} be a subset of z(‘) which converges to z. Since each component of p is upper semicontinuous it follows from (5) that C&Z)2 y*. Since y* is efficient q(z) = y* = cp(x). We claim that this equality implies that z = X. Indeed, if z # x then &~/2+x/2) 9 V(X), contradicting the fact that &x) is an efficient point of Y. Thus x = lim {zCh)1h E D}. Define now for each 2, 1 = 1, 2, . . .,

v(‘) =

$, ,l(‘)(k)u,.

We shall need now the following lemma. Lemma 2. There exists a strictly concave function u(l) E U such that d”(e) v(‘)(e) for all e E E”, a 5 e s b.

=

We postpone the proof of Lemma 2 and continue with the proof of Theorem 1. By (6) and monotone convergence of integrals [Loeve (1963, p. 124)], Iv(‘)(z(‘)) 2 Iv(‘)(z) for all z E R.

(7)

By Lemma 2 and (7) there exists a strictly concave u(‘) E U such that lu(‘)(z(‘)) 2_ lu(‘)(z)

for all

z E R.

Thus z(‘) is regular risk-aversely efficient. Proof of Lemma 2.

For a $ x I b let

&(‘)(x) = (rr 1x E E”, v(‘)(z) 5 v(‘)(x)+z.(z-x)

6vZ)(x)# 8.

for

a 5 z S b),

B. Peleg, Eficient random variables

249

Define A =

u

(au(‘)(x)( a

6 x =< b} .

Since u”‘(x) is increasing for x E Q there exists f E Em, f % 0, such that n >=f for all 7~E A. By Theorem 24.7 of Rockafellar (1970), A is bounded. For z E E” define u(z) = inf (V(‘)(X)+ x. (z-x)

1a 5 x 5 b, x E &(‘)(x)} .

u E U and is increasing. Furthermore, u(x) = Y(‘)(X)for a 5 x 5 b. For each i, i= 1,...) m, let Wi(~)be a concave and differentiable function which satisfies:

(a) Wiis strictly concave for -co (b) pi = Oforai 5 5 5 bi. (C) jWI(c)l < fi/2 for all 5.

< c 5 a, and for bi 6 r < 00.

Let now U”‘(Z) = U(Z)+ f

Wi(Zi) for

z E

Em.

i=l

By (a) u(‘) is strictly concave. By (b) u(‘)(z) = u(‘)(z) for a =
Let T = (1, 2}, B = 2T and p(1) = p(2) = 3. Let further

R = (x 1~(l)~+(x(2)-1)~

5 I).

Then, as the reader can easily verify, (1, 1) is a risk-aversely efficient point of R which is not regular. 6. A characterization of efficient random vectors

Throughout this section we assume that T is finite, T = (1, . . ., n], and B = 2 ‘, the set of all subsets of T. Furthermore, we may assume that p(i) > 0, i = 1, . . ., n. Under these assumptions we prove a price characterization theorem for risk-aversely efficient vectors (Theorem 2 below). First, we remark, that under the above assumptions both L, and L, are q-dimensional spaces, where q = mn. Now we state:

250

B. Peleg, Eficient random variables

Theorem 2. Let R be a convex and compact subset of L,. If x E R is riskaversely eficient then there exists a u E U such that Iu is properly maximized at x. Remark 4. The assumption that R is compact can be replaced by the weaker assumption that R is closed, with only a slight modification of the proof. Remark 5. It follows from Remark 4 that Theorem 2 implies Theorem B in Peleg et al. (1975). The proof is straightforward. Proof of Theorem 2. Let x E R be risk-aversely efficient. By Theorem 1 there exists a sequence A+‘),I = 1, 2, . . . of regular risk-aversely efficient vectors such that x = lim x(l). l-rm

By Remark 3 for each Z,I = 1,2, . . ., there exists a a(‘) E U such that 1~~‘)is properly maximized at x (‘) . Hence there exists a system of efficiency prices no) of x(I) such that 7&‘)(i) E &(‘)(x(‘)(i)) ,

i = 1,. . ., n.

By p. 238 in Rockafellar (1970) the pair (x(l), Z(I)) is cyclically monotone, i.e., 7c”)(io). (x”‘(&) - x”‘(Q) + 7&l)(&).(x”‘(iJ - x”‘(iJ) + . . . + x”‘(i,). (x(“(i,) -x(‘)(Q)

2 0

for

k = 1, 2, . . . .

Let

Denote A(‘)= n(‘)/a(‘). Then A(‘) is a system of efhciency prices for x(r) and (x(‘) A”‘) is cyclically monotone. We may assume that the limit x = lim A(‘) e&s. Then 1cis a system of efficiency prices for x and the pair (x, ZZ)is cyclically monotone. Define a function u on E” by u(z) = inf (n(i&. (z-x(~~))+?I&-~)~ + . . . +7r(l).(x(i,)-x(1))

(x(ik)-x(ik-l)) 1k = 1,2,.

. .}.

Since (x, n) is cyclically monotone and n(i) 2 0, i = 1, . . ., n, it follows that u is a concave and non-decreasing function defined on E”, i.e., u E U. By Theorem 24.8 in Rockafellar (1970) rc(i) E &(x(i)) for i = 1, . . ., n. Thus, lu is properly maximized at x.

251

B. Peleg, Eficient random variables

7. An example Our main result for discrete random variables is Theorem 2. The following example shows that this result cannot be generalized to the case where T is infinite. Example 2. i=

1,2,3,... R=

Let T = { 1,2,3, . . .}, B the set of all subsets of Tandp(i) = 2/3’, Let SEL, be defined by n(i) = 2’, i = 1,2, 3, . . . Let further

xlO5x(i)61,

i=l,2,3

,...

and

i$l p(Mi)x(i)

6 2) .

Then R is a convex and w(L,, L,)-compact subset of L,. (Here, of course, m = 1.) Let x* E R be defined by x*(i) = 3, i = 1, 2, 3, . . . We claim that x* is a risk-aversely efficient member of R. Indeed, let y E R, y # x*. There exists a natural number h such that y(h) < 3. Let F and G be the distributions of x* and y respectively. Thus,

Hence, y does not dominate x* risk-aversely [see, e.g., Theorem 2 in Hanoch et al. (1969)]. Now, as the reader can easily verify, if x* is a system of efficiency prices for x* then rr* = ss, where s is a positive real number. Thus, if 1~ is properly maximized at x* then there exists a real number s > 0 such that z&(1/2) % s2’ I U’_(1/2),

i=

1,2,3 ,...,

which is impossible. Remark 6. Examples similar to Example 2 can be constructed in L,(O, I) (i.e., when T is the unit interval, B the set of its Bore1 subsets and p is Lebesgue measure).

8. Concluding remarks Remark 7. One can develop a special theory for non-negative random variables. The following modifications are necessary for such a theory. U is to be replaced by U+, the set of all concave, non-decreasing and continuous functions on E,“, the non-negative cone of Em. R is a convex subset of Lz, the nonnegative cone of L,, and one has to add the assumption that there exists k $ 0, k E R. Subject to the above modifications all our results remain true. Remark 8. This work is not only a generalization of Peleg et al. (1975), but also contains a new formulation of the results of that paper. Definitions 3 and 4

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B. Peleg, Ejficient random variables

enable us to deal with vector-valued and continuous random variables and still obtain the results of Peleg et al. (1975) as corollaries. Our formulation also leads to a new interpretation of the results, as presented in section 3. Remark 9. The main reason for restricting our discussion to bounded random variables is that this is the most important case in economic applications [see, e.g., Majumdar (1972) and Radner (1973)]. The most extensive class that could be dealt with, is that of random variables in L, which are bounded from below. The question whether Theorem 1 can be generalized to this class remains open. References K.J., E.W. Barankin and D. Blackwell, 1953, Admissible points of convex sets, in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games, vol. II (Princeton University Press, Princeton) 87-91. Hadar, J. and W.R. Russel, 1969, Rules for ordering uncertain prospects, American Economic Review 59,25-34. Hanoch, G. and H. Levy, 1969, The efficiency analysis of choices involving risk, Review of Economic Studies 36,335-346. Hardy, G.H., J.E. Littlewood and G. Polya, 1934, Inequalities (Cambridge University Press, Cambridge). Lo&e, M., 1963, Probability theory, 3rd ed. (Van Nostrand, New York). Majumdar, M., 1972, Some general theorems on efficiency prices with an infinite-dimensional commodity space, Journal of Economic Theory 5,1-13. Peleg, B., 1971, Efficiency prices for optimal consumption plans II, Israel Journal of Mathematics 9,222-234. Peleg, B. and M.E. Yaari, 1975, A price characterization of efficient random variables, Econometrica 43,283-292. Radner, R., 1973, Optimal stationary consumption with stochastic production and resources, Journal of Economic Theory 6,68-90. Rockafellar, R.T., 1967, Duality and stability in extremum problems involving convex functions, Pacific Journal of Mathematics 21,167-187. Rockafellar, R.T., 1970, Convex analysis (Princeton University Press, Princeton). Rockafellar, R.T., 1971, Integrals which are convex functionals II, Pacific Journal of Mathematics 39,439-469. Rothschild, M. and J.E. Stiglitz, 1970, Increasing risk I: A definition, Journal of Economic Theory 2,225-243. Schmeidler, D., 1973, A bibliographical note on a theorem of Hardy, Littlewood and Polya (Department of Economics, Tel Aviv University, Tel Aviv). how,